Modular function

A modular function of weight $k$ for a subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is a meromorphic function on the complex upper half-plane that satisfiesfor $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma$ and is meromorphic at every cusp.

The nonzero even theta constants and the Jacobi abstruse identity give $\lambda (-1/\tau )=1-\lambda (\tau )$ and $\lambda (\tau +1)=\lambda (\tau )/(\lambda (\tau )-1)$. It is invariant under the principal congruence subgroup of level two. This parameter puts the associated complex elliptic curve in Legendre form $y^2=x(x-1)(x-\lambda )$; its Klein j-invariant is $256(1-\lambda +{\lambda }^2{)}^3/[{\lambda }^2(1-\lambda {)}^2]$.

The modular lambda function is invariant under the projective level-two principal congruence subgroup. That group acts freely, and its quotient is the level-two modular curve. Adding its three modular cusps gives a sphere, and $\lambda$ has cusp values $0,1,\infty$, with simple orders in exponential cusp coordinates. Therefore it identifies the quotient with the sphere minus those three points. Since the complex upper half-plane is simply connected, this is a universal covering map, with the projective level-two group as its deck transformation group.

For a congruence subgroup $\Gamma$, the compact modular curve isAt an ordinary point it inherits a coordinate from $\mathfrak{h}$; at an elliptic point of stabilizer order $e$ a coordinate is the $e$th power of a local disk coordinate; and at a cusp of width $h$ a coordinate is $e^{2\pi i\tau /h}$.

An elliptic point of a modular curve is the image of a point of the complex upper half-plane with a nontrivial stabilizer subgroup in the effective modular subgroup. For the full modular group the stabilizer subgroup orders are two and three, represented by $i$ and $e^{2\pi i/3}$. A local disk coordinate $u$ upstairs descends as $u^e$ when the stabilizer subgroup order is $e$. The central matrices $\pm I$ are factored out before counting this order.

A modular symbol is the relative homology class of a path between two modular cusps. Oriented paths satisfy $\{\alpha ,\beta \}+\{\beta ,\gamma \}=\{\alpha ,\gamma \}$. Continued fractions subdivide rational-endpoint paths into unimodular edges, giving a finite presentation through Manin symbols. The Hecke operators act by finite sums of transformed paths, so their matrices can be computed by rational linear algebra.

For a finite-index modular subgroup, a Manin symbol records a coset and its associated oriented unimodular edge. Over the rationals the presentation has the relations $[g]+[gS]=0$ and $[g]+[gST]+[g(ST{)}^2]=0$, for inversion $S$ and translation $T$. They express reversing an edge and the boundary of a triangle. Integral computation also removes torsion caused by elliptic stabilizers. The boundary map sends the symbol to its terminal cusp minus its initial cusp; its kernel is absolute homology.

The compact quotient for the Gamma 0 congruence subgroup at level two has projective index three, two modular cusps of cusp widths one and two, one order-two elliptic orbit and no order-three elliptic orbit. It has genus zero. It differs from the level-two principal-congruence curve for $\Gamma (2)$, which has three modular cusps.

The ratio of modular discriminants has weight zero and level ${\Gamma }_0(2)$. It has one simple pole at infinity and one simple zero at the zero modular cusp, with no zeros or poles in the half-plane. It is a spherical coordinate by the single-pole criterion for a spherical coordinate. The Klein j-invariant is $(t+256{)}^3/t^2$, with $t(e^{2\pi i/3})=-256$.

For a finite-index subgroup of the modular group, the quotient of the complex upper half-plane becomes a compact Riemann surface after adjoining its cusp classes. At an elliptic fixed point use the quotient coordinate for the finite stabilizer; at a cusp use its exponential local parameter. Weight-zero modular forms descend to holomorphic functions on this compact surface, and hence are constant by the maximum modulus principle.

The quotient $\mathbb{H}/\Gamma (2)$ is a genus-zero Riemann surface with three cusps and no elliptic Möbius transformations, hence is biholomorphic to $\mathbb{C}\setminus \{0,1\}$. The integer level-two principal congruence subgroup has projective index six; reduction modulo two gives three cusp classes.

Let $\mu =[PS{L}_2(\mathbb{Z}):\overline{\Gamma }]$, let $e_2,e_3$ count elliptic orbits of orders two and three, and let $c$ be the number of cusps. Then

For $N\ge 3$, the principal congruence subgroup has projective index $\mu =\frac{1}{2}{N}^3{\prod }_{p\mid N}(1-p^{-2})$, no effective elliptic stabilizers, and all cusp widths equal to $N$. Hence the number of cusps is $\mu /N$, and the genus formula for a modular curve gives the displayed formula. The exceptional levels one and two both have genus zero; the full genus-zero list is $N=1,2,3,4,5$. Level six has genus one and every larger level has positive genus.

For $N\ge 2$, the effective principal congruence subgroup is torsion-free. Its modular curve cover of $X(1)$ has analytic ramification indices $2,3,N$ above the two elliptic orbits and the modular cusp. If $\mu$ is its projective index, the numbers of points above these three values are $\mu /2,\mu /3,\mu /N$. Applying the Riemann-Hurwitz formula gives $2g-2=\mu (1/6-1/N)$. Here $\mu =6$ for $N=2$ and $\mu =N^3{\prod }_{p\mid N}(1-p^{-2})/2$ for $N\ge 3$; level one is the identity cover.

The modular curve $X_0(3)$ has index four, two cusps, no elliptic orbit of order two, and one elliptic orbit of order three. Its genus is therefore zero.

A cusp of a finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is an orbit in $\Gamma \backslash {\mathbb{P}}^1(\mathbb{Q})$. If $\sigma \in S{L}_2(\mathbb{Z})$ sends infinity to a cusp, behavior there is studied through $f{|}_k\sigma$.

The width of a cusp represented by $\sigma \in S{L}_2(\mathbb{Z})$ is the least positive $h$ such that $\sigma T^h{\sigma }^{-1}$ lies in the subgroup, up to its center. Its local parameter is $q_h=e^{2\pi i\tau /h}$.

A modular function is holomorphic at a cusp when its expansion in the corresponding local parameter has no negative powers. It vanishes at the cusp when that expansion has zero constant term.

If $f$ is a modular form on a finite-index subgroup and $\gamma$ is rational with positive determinant, then $f{|}_k\gamma$ is bounded near infinity. Choose $\sigma \in S{L}_2(\mathbb{Z})$ with $\sigma \infty =\gamma \infty$. The matrix ${\sigma }^{-1}\gamma$ is upper triangular and sends $z$ to $Az+B$ with $A>0$. Thus the existing cusp expansion of $f{|}_k\sigma$ stays bounded under this substitution. Rational conjugation of finite-index modular subgroups gives a positive period for the translate, so boundedness gives a removable singularity in its cusp parameter. Apply this to $\gamma \rho$ for every $\rho \in S{L}_2(\mathbb{Z})$ to obtain holomorphy at every cusp of the new subgroup. Vanishing at the cusps is preserved too.

A modular form is a holomorphic function on the complex upper half-plane that transforms with a fixed weight under a congruence subgroup and is holomorphic at every cusp.

The modular modular weight specifies the power $(cz+d{)}^k$ in a modular form's transformation law. Multiplication adds modular weights. At level one, the central matrix $-I$ excludes nonzero odd-weight forms; a nebentypus character at higher level changes this condition to $\psi (-1)=(-1{)}^k$.

For a level-one modular form of positive even weight $k$ and constant coefficient $a_0$, put $I_f(s)={\int }_1^{\infty }(f(iy)-a_0)y^{s-1}dy$. This is an entire function. Splitting the Mellin transform at one and using $f(iy)=i^k{y}^{-k}f(i/y)$ proves the displayed continuation of $(2\pi {)}^{-s}\Gamma (s)L(f,s)$. It obeys ${\Lambda }_f(s)=i^k{\Lambda }_f(k-s)$ and has possible simple poles only at zero and $k$, with residues $-a_0,i^k{a}_0$. For a cusp form both disappear.

For even nonnegative weight, the displayed dimension follows from multiplication by the modular discriminant and the constant-coefficient map. Every nonnegative even weight except two has a product $E_4^a{E}_6^b$ of Eisenstein series with constant term one, so $\dim M_k=1+\dim M_{k-12}$ for even $k\ge 4$. The initial dimensions are one in weight zero and zero in weight two; negative and odd weights have no nonzero level-one modular forms.

Subtract the constant term of a modular form times a same-weight monomial in $E_4,E_6$. The remainder is a cusp form, hence divisible by the modular discriminant. Induction on weight and $1728\Delta =E_4^3-E_6^2$ express every form as a weighted homogeneous polynomial in these two Eisenstein series. The number of monomials of a given weight equals the dimension of level-one modular forms, so they form a basis and the graded ring has no polynomial relation.

Let $P_w(E_4,E_6)$ be weighted homogeneous of modular weight $w$, and suppose ${\sum }_w{P}_w(z)=0$ as a holomorphic function. Apply the modular group matrices $\left(\begin{array}{cc}1& 0\\ n& 1\end{array}\right)$. At a fixed point $z$ in the complex upper half-plane, this gives ${\sum }_w(nz+1{)}^w{P}_w(z)=0$ for every integer $n$. A polynomial with infinitely many roots vanishes identically, so every $P_w(z)=0$. Within one modular weight, the monomials reduce to powers of $E_4^3/E_6^2$, after taking out a common factor. This ratio is nonconstant, as its Fourier expansion begins $1+1728q+\cdots$, so its powers are linearly independent. This proves algebraic independence even when the generators are regarded as analytic functions rather than elements of a formal graded ring.

A level-one modular form with rational Fourier coefficients is a weighted homogeneous polynomial with rational coefficients in the normalized Eisenstein series $E_4,E_6$. Subtract its constant coefficient times a same-weight monomial. The remainder is a cusp form; divide it by ${\Delta }_0=(E_4^3-E_6^2)/1728=q+O(q^2)$. The valence formula for the modular group shows that ${\Delta }_0$ has no interior zeros, so this quotient is holomorphic and has rational coefficients. Induction on weight proves the assertion. Euler's Bernoulli number formula for even zeta values then gives the corresponding rational weighted polynomial in the unnormalized lattice Eisenstein sums $G_4,G_6$.

A weight-$2k$ modular form descends as an invariant tensor differential. At an elliptic point of order $e$, a local invariant coordinate is $w=u^e$, so its descended coefficient has a pole of order at most $\lfloor k(e-1)/e\rfloor$. At infinity, $d\tau =dq/(2\pi iq)$ gives a pole of order at most $k$. Thus $D=k[\infty ]+\lfloor k/2\rfloor [i]+\lfloor 2k/3\rfloor [\rho ]$ bounds the poles. Conversely these bounds give holomorphic pullbacks. The Riemann-Roch theorem on the compactified modular curve proves finite-dimensionality; since $X(1)$ is the Riemann sphere, the dimension is $\max (0,1-k+\lfloor k/2\rfloor +\lfloor 2k/3\rfloor )$.

Choose $4a_j+6b_j=k-12j$ with nonnegative exponents, for every $j$ with $k-12j\ge 0$ and $k-12j\ne 2$. The Eisenstein series and the modular discriminant give modular forms with integer coefficients and distinct first powers of $q$. The valence formula for the modular group shows that these form a basis. Successively subtracting their leading terms proves that integral coefficients up to the greatest such $j$ force all coefficients to be integral. This is more precise than merely spanning by arbitrary monomials in $E_4,E_6$.

A modular form on an arbitrary finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is a holomorphic function on the complex upper half-plane, invariant under the weight-$k$ slash operator for modular forms, and holomorphic at a cusp at every cusp of a modular group for $\Gamma$. If $\sigma \in S{L}_2(\mathbb{Z})$ maps infinity to a cusp, choose a genuine translation period $h>0$ with $\sigma T^h{\sigma }^{-1}\in \Gamma$. Then $f{|}_k\sigma$ has a convergent series in $e^{2\pi iz/h}$ with nonnegative exponents. A cusp form has zero constant term at every cusp. This definition allows noncongruence subgroups and avoids sign ambiguities in odd weights when a smaller width is defined only modulo the center.

A Dirichlet character $\psi$ modulo $N$ specifying the displayed transformation law is the nebentypus of a modular form. The action of $-I$ forces $\psi (-1)=(-1{)}^k$ for a nonzero form. It describes a character eigenspace of the diamond action rather than an additional analytic condition at modular cusps.

For a finite-index inclusion of modular groups and a form with trivial multiplier, multiply its slash operator for modular forms translates over left coset representatives. Right multiplication permutes the factors, producing a form of weight $k[{\Gamma }^{\prime }:\Gamma ]$ for the larger group. modular cusp cusp widths determine the orders contributed by the factors; the product is nonzero if the original form is nonzero.

The Serre derivative sends a level-one weight-$k$ modular form to weight $k+2$. Differentiating $f(-1/z)=z^{k}f(z)$ produces the extra term $k{z}^{k+1}f/(2\pi i)$; the anomalous transformation of the Eisenstein series of weight two cancels it. Translation invariance and the Fourier expansion of a modular form verify the remaining holomorphy conditions. For $k>0$, its constant coefficient is $-k{a}_0(f)/12$, so it is a cusp form exactly when $f$ is.

For modular forms $f,g$ of weights $k,\ell$, respectively, the displayed combination cancels the anomaly in derivative transformation of a weak modular form and is a weight-$k+\ell +2$ cusp form. Equivalently, for weights $k,\ell$ in the order $f,g$, a conventional first bracket is $kfDg-\ell gDf$; its negative is equally modular. Differentiation kills each constant Fourier coefficient, proving cusp vanishing.

If $N_{1}D\mid N$ and $f\in M_k({\Gamma }_0(N_1))$, then $f(Dz)\in M_k({\Gamma }_0(N))$. Conjugating by $\mathrm{diag}(D,1)$ changes a lower-left matrix entry $c$ to $c/D$, which is divisible by $N_1$. A rational upper-triangular factorization at every cusp preserves boundedness and hence cusp holomorphy.

For a nonzero positive-weight eigenform at a good prime, the two oldforms $f(\tau ),f(p\tau )$ are independent. At level multiplied by $p$, the Hecke operator is the bad-prime operator, with matrix $\left(\begin{array}{cc}\lambda & 1\\ -\chi (p)p^{k-1}& 0\end{array}\right)$. Distinct roots $\alpha ,\beta$ of its characteristic polynomial give eigenforms $f-\beta f(p\tau )$ and $f-\alpha f(p\tau )$. Weight-zero constants are an exception to the independence assertion.

The group ${\Gamma }_0(3)$ has index four. The dimension bound for modular forms on a finite-index subgroup gives dimension at most two in weight four. The Eisenstein series $E_4(z)$ and its oldform by argument dilation $E_4(3z)$ are independent, since their constant coefficients agree while their $q$ coefficients are $240$ and zero.

Here a weak modular form is a holomorphic function on the complex upper half-plane satisfying the weight-$k$ transformation law, with no growth condition at any cusp of a modular group. This convention differs from weakly holomorphic modular forms, which are required to be meromorphic at the cusps. At level one, translation invariance gives a Laurent expansion on the punctured unit disc, possibly with infinitely many negative powers.

For a weight-$k$ weak modular form of level one,where $(a{)}_r=a(a+1)\cdots (a+r-1)$ and $(a{)}_0=1$. Differentiate and multiply by $z^2$ to obtain the coefficient recurrence $C_{\ell +1,j}=(k+\ell +j)C_{\ell ,j}+C_{\ell ,j-1}$.

For integer $k<0$, the operator $D=(2\pi i{)}^{-1}d/dz=qd/dq$ sends a weight-$k$ level-one weak modular form to a weight-$2-k$ one after $1-k$ iterations. In the derivative transformation of a weak modular form, every lower derivative coefficient then contains a zero factor. This also holds on the subspace $M_k^{!}$ of forms meromorphic at the cusp.

The Fricke matrix $W_N=\left(\begin{array}{cc}0& -1\\ N& 0\end{array}\right)$ normalizes ${\Gamma }_1(N)$ and ${\Gamma }_0(N)$ under the appropriate determinant-normalized slash action. It exchanges the cusps zero and infinity.

If a one-dimensional modular cusp space is stable under the phase-normalized Fricke involution, its scalar action is determined by evaluation at the fixed point. The phase-normalized prefactor is one there; if the form is nonzero at that point, the eigenvalue is $+1$. A product with strictly positive convergent factors on the imaginary axis supplies such a value and also makes the central completed L-function of a cusp form positive when its central Mellin integral has real positive kernel.

The modular group is $S{L}_2(\mathbb{Z})$ acting on the complex upper half-plane. Its central element $-I$ acts trivially on points but contributes the factor $(-1{)}^k$ in weight $k$.

A function on the complex upper half-plane is modular invariant if it is unchanged by the fractional-linear action of the modular group. This does not imply holomorphicity: a nonholomorphic Eisenstein series is modular invariant, whereas a modular function additionally satisfies meromorphicity and cusp conditions.

For any finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ and rational positive-determinant $\gamma$, the intersection $S{L}_2(\mathbb{Z})\cap {\gamma }^{-1}\Gamma \gamma$ has finite index. Multiply $\gamma$ by a positive rational scalar to obtain an integral matrix $A$ with positive integer determinant $D$. For $h=I+DB\in \Gamma (D)$, $Ah{A}^{-1}=I+AB\mathrm{adj}(A)$ is integral, so $\Gamma (D)$ lies in $S{L}_2(\mathbb{Z})\cap {\gamma }^{-1}S{L}_2(\mathbb{Z})\gamma$. This intersection has finite index. Its further intersection with ${\gamma }^{-1}\Gamma \gamma$ has relative index at most $[S{L}_2(\mathbb{Z}):\Gamma ]$. No assumption that $\Gamma$ itself is a congruence subgroup is needed.

The standard fundamental domain isEvery orbit of the modular group meets $\mathcal{F}$.

For a finite-index subgroup of the modular group, take the union of translates of the standard fundamental domain of the modular group over left coset representatives. Boundary identifications are inherited from the subgroup action. The truncated region is compact, and its finitely many cusp ends have exponential parameters; this proves compactness of the compactified modular curve and boundedness of invariant cusp-form norms.

For the boundary convention $-1/2<\mathrm{Re}z\le 1/2$, retaining only the right half of the unit circle, the exceptional points are $i$ and ${\rho }_{+}=e^{i\pi /3}$. Their stabilizers in $S{L}_2(\mathbb{Z})$ are the cyclic groups generated by $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $\left(\begin{array}{cc}1& -1\\ 1& 0\end{array}\right)$, of orders four and six. Every other stabilizer is $\{I,-I\}$.

Given $\tau \in \mathfrak{h}$, choose a primitive pair $(c,d)$ minimizing $|c\tau +d|$ and apply a modular matrix with bottom row $(c,d)$. This maximizes the imaginary part within the orbit. Translation puts the real part in $[-1/2,1/2]$, and inversion then shows that the imaginary part is at least $\sqrt{3}/2$.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the weight-$k$ slash operator isA modular form of weight $k$ and level $\Gamma$ satisfies $f{|}_k\gamma =f$ for every $\gamma \in \Gamma$.

Decompose $A\alpha B={\coprod }_{j}A{\alpha }_j$ for finite-index congruence subgroups $A,B$. The sum is independent of representative choices because $f$ is invariant under $A$. Right multiplication by $B$ permutes these left cosets, so the sum is invariant under $B$. Each summand is holomorphic in the half-plane; cusp holomorphy under rational slash operators proves holomorphy at every new cusp. The same reasoning preserves vanishing at cusps. With the Hecke-normalized rational slash operator, no extra scalar is needed in the defining sum.

For a rational matrix of positive determinant, this slash normalization differs from the determinant-normalized slash operator by $(\det \alpha {)}^{k/2-1}$. Both are right actions, by the automorphy factor identity. For a positive scalar $t$, the displayed action of $tI$ is multiplication by $t^{k-2}$. On determinant-one matrices it agrees with the ordinary slash operator for modular forms. This normalization makes a good-prime Hecke operator a plain sum over its double-coset representatives.

For a rational matrix $\gamma$ of positive determinant, use the positive real power of $\det \gamma$ in the displayed slash operator for modular forms. It satisfies $(f{|}_k\gamma ){|}_k\eta =f{|}_k(\gamma \eta )$ by the automorphy factor identity and multiplicativity of the determinant. It agrees with the usual operator on the modular group. The determinant factor is essential for the stated normalization of the Hecke operators.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the automorphy factor is $j(\gamma ,\tau )=c\tau +d$. It obeys the cocycle identity

Translation invariance at a cusp gives a Fourier expansion in its local parameter $q$. Holomorphy at the cusp excludes negative powers; vanishing of the constant term defines a cusp form.

A cusp form is a modular form that vanishes at every cusp. At the cusp at infinity for the full modular group it has an expansion ${\sum }_{n\ge 1}{a}_n{q}^n$.

The old subspace is spanned by $h(tz)$ for cusp forms $h$ of level $M<N$, where $M\mid N$ and $t\mid N/M$. It is stable under the Hecke operators, but need not have a basis of eigenvectors for every bad-prime operator. Its Petersson inner product orthogonal complement is the new subspace of cusp forms.

If $M\mid N$ and $D\mid N/M$, conjugating ${\Gamma }_0(N)$ by $\mathrm{diag}(D,1)$ gives matrices $\left(\begin{array}{cc}a& Db\\ c/D& d\end{array}\right)$ in ${\Gamma }_0(M)$. This proves the weight transformation of $f(D\tau )$. At any rational modular cusp, factor the rational change of variable into an integral cusp representative followed by an upper-triangular rational matrix of positive slope. The original cusp expansion then stays holomorphic and has zero constant term. Thus this map takes cusp forms of level $M$ to those of level $N$.

For prime $p$, the two cusps of ${\Gamma }_0(p)$ have widths one and $p$. At infinity the modular discriminant product gives the first pair of orders. At zero use $S\tau =-1/\tau$ and the parameter $q_0=e^{2\pi i\tau /p}$. The weight-twelve transforms are $\Delta {|}_{12}S=\Delta (\tau )$ and $\Delta (p\tau ){|}_{12}S=p^{-12}\Delta (\tau /p)$, giving orders $p$ and one in $q_0$.

If $p$ divides the primitive level $M$ of a trivial-character newform and $N=M{p}^r$, put $e_j=p^{jk/2}f(p^{j}z)$. Then $U_p{e}_0=a_p{e}_0$ and $U_p{e}_j=p^{k/2}{e}_{j-1}$ for $j\ge 1$. The normalized Fricke involution acts by $e_j\mapsto \epsilon e_{r-j}$, where $W_{M}f=\epsilon f$. These two operators act irreducibly: a nonzero invariant subspace contains an eigenvector of $U_p$; its eigenvalue is $a_p$ or zero. Either it contains $e_0$ immediately or applying the reversal and a power of $U_p$ produces $e_0$, using $a_p^2\ne p^k$. Reversal then produces $e_r$, and its successive images produce every basis vector.

The new subspace is the orthogonal complement of the old subspace of cusp forms for the Petersson inner product. A form belongs to it precisely when the adjoints of all lower-level oldform by argument dilation maps annihilate it.

A newform is a normalized Hecke eigenform in the new subspace of cusp forms. Its weight, level and Dirichlet character are part of its specification. For a newform, its Fourier coefficients equal its eigenvalues of Hecke operators.

For a newform of weight $k\ge 2$, level $M$ and trivial Dirichlet character, the bad-prime Hecke operator $U_p$ has the displayed possibilities. The local trace relation on the new subspace of cusp forms gives $U_p=-p^{k/2-1}{W}_p$ when $p\parallel M$, where the normalized local Atkin-Lehner operator squares to one; if $p^2\mid M$, the new-vector relation gives $U_p=0$. In particular $|a_p|<p^{k/2}$, a strict inequality useful for proving irreducibility of old blocks.

On the new subspace of cusp forms, a common eigenspace for the Hecke operators away from the level has dimension one. Distinct newforms, even at different primitive levels, have different systems of these eigenvalues. Consequently the commuting bad-prime operators also act by scalars on these lines. This is the multiplicity-one part of Atkin-Lehner-Li theory, not a consequence of normal operator diagonalization alone. See Stein's statement of the theorem.

For $f={\sum }_{n\ge 1}{a}_n{q}^n$, define $L(f,s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$ in a right half-plane. Its completion $(2\pi {)}^{-s}\Gamma (s)L(f,s)$ is the Mellin transform of $f(iy)$. The modular inversion and exponential cusp decay continue this completion to an entire function and give its reflection $s\mapsto k-s$. A normalized Hecke eigenform also has an Euler product of a Hecke eigenform.

For a cusp form $f=\sum c_n{q}^n$ and coprime integers $a,N$, the additive twist inserts the additive character $e^{2\pi ian/N}$. It is the Dirichlet series associated with the rational translate $f(a/N+\tau )$. Its completed Mellin transform uses the scale $N^s$ and the Gamma function.

For even weight $k$, complete the additive twist of a cusp-form L-function as $M(f,a/N,s)=(N/(2\pi ){)}^s\Gamma (s)L(f,a/N,s)$. Split its Mellin transform at $y=1/N$ and use rational-cusp inversion of a level-one cusp form in the lower integral. The resulting two integrals over $[1/N,\infty )$ converge locally uniformly for every complex $s$, giving an entire continuation and the displayed functional equation. The relation $ad\equiv 1(\mathrm{mod}N)$ identifies the opposite cusp.

If $ad\equiv 1(\mathrm{mod}N)$, the determinant-one matrix with rows $(a,(ad-1)/N)$ and $(N,d)$ sends $\tau -d/N$ to $a/N-1/(N^2\tau )$. The transformation of the cusp form gives the displayed identity. It turns the small-height portion of an additive-twist Mellin transform into exponentially decaying values at the inverse rational cusp.

For a level-one cusp form and any Dirichlet character modulo $N>1$, this translation sum is a cusp form on ${\Gamma }_1(N)\cap {\Gamma }_0(N^2)$. Conjugating that subgroup by $\left(\begin{array}{cc}1& j/N\\ 0& 1\end{array}\right)$ yields integral determinant-one matrices; cusp holomorphy under rational slash operators supplies all cusp conditions. Its exact Fourier coefficients at positive indices are $a_n(f){\sum }_{j\in (\mathbb{Z}/N\mathbb{Z}{)}^{\times }}\chi (j{)}^{-1}{e}^{2\pi inj/N}$. For a primitive Dirichlet character they equal $g(\overline{\chi })\chi (n)a_n(f)$, by the finite Fourier transform of a primitive Dirichlet character. For imprimitive characters the sum can be nonzero at nonunits, so the simplified twist formula need not hold.

For a primitive Dirichlet character $\chi$ modulo $D$, its Gauss sum of a Dirichlet character gives $f_{\chi }=\tau (\overline{\chi }{)}^{-1}{\sum }_u\overline{\chi }(u)f|U_u$, with $U_u=\left(\begin{array}{cc}1& u/D\\ 0& 1\end{array}\right)$. If $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma }_0(N{D}^2)$, take $v\equiv d^{2}u(\mathrm{mod}D)$. Then $U_u\gamma U_v^{-1}\in {\Gamma }_0(N)$ and its lower-right entry is congruent to $d$ modulo $N$. Reindexing the sum yields the character $\psi (d)\chi (d{)}^2$. Rational translations preserve vanishing at modular cusps, proving the stated cuspidality as well as the transformation law.

Use the determinant-normalized slash operator and $W_M=\left(\begin{array}{cc}0& -1\\ M& 0\end{array}\right)$. For units $u,v$ with $Nuv\equiv -1(\mathrm{mod}D)$, the matrix identity $U_u{W}_{N{D}^2}=D{\gamma }_u{W}_N{U}_v$ has ${\gamma }_u=\left(\begin{array}{cc}(1+Nuv)/D& u\\ Nv& D\end{array}\right)\in {\Gamma }_0(N)$. A positive scalar matrix acts trivially. The nebentypus supplies $\psi (D)$; reindexing gives $\overline{\chi }(u)=\chi (-N)\chi (v)$. Thus the factor is initially $\psi (D)\chi (-N)\tau (\chi )/\tau (\overline{\chi })$. The primitive Gauss sum of a Dirichlet character identity $\tau (\chi )\tau (\overline{\chi })=\chi (-1)D$ gives the displayed equivalent factor. No assumption that $f$ itself is a Fricke eigenform is needed.