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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.

For a normalized level-one cusp form Hecke eigenform of weight $k$, fix a prime number $p$ whose local quadratic has distinct roots. Hecke multiplication relations give $b_e=x{b}_{e-1}-b_{e-2}$ for $b_e=a_{p^e}/p^{e(k-1)/2}$ and real $x=a_p/p^{(k-1)/2}$. If $|x|<2$, write $x=2\cos \theta$; then $b_e=\sin ((e+1)\theta )/\sin \theta$. Periodicity when $\theta /(2\pi )$ is rational, and density of irrational rotations otherwise, give infinitely many $b_e\ge 1/2$. If $|x|>2$, the distinct real reciprocal roots give positive unbounded even-index terms. Reality follows from Hecke operators are self-adjoint for the Petersson inner product.

The Fourier expansion of a normalized Eisenstein series and the one-dimensional weight-twelve cusp form space give $691E_{12}=691E_4^3-432000\Delta$. The coefficient of $q^n$, for $n\ge 1$, reduces modulo $691$ to $566{\sigma }_{11}(n)=566\tau (n)$. Since $566$ is invertible modulo $691$, this proves the congruence. Integrality follows from $E_4=1+240\sum {\sigma }_3(n)q^n$ and the product $\Delta =q{\prod }_{n\ge 1}(1-q^n{)}^{24}$.

For two weight-$k$ level-one cusp forms, use the primitive real-analytic Eisenstein series and $d\mu =dxdy/y^2$. The Rankin–Selberg unfolding identity isInitially this follows by absolute convergence, unfolding to $0\le x<1$, and integrating the Fourier series in $x$. Exponential cusp decay and polynomial growth of the primitive real-analytic Eisenstein series make the integral holomorphic for $\mathrm{Re}w>1$, continuing the Dirichlet series to $\mathrm{Re}s>k$.

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.

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 $G_4,G_6$.

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.

A nonzero weight-fourteen cusp form would have order at infinity at least one and order at $i$ at least one, since $i^{14}=-1$. The valence formula for the modular group would then give $1+1/2\le 14/12$, which is impossible.

There are no weight-four cusp forms at level two. A nonzero form would have a weight-twelve coset norm of a modular form with modular cusp order at least two, contradicting the simple modular cusp order and interior nonvanishing of the modular discriminant. The two independent displayed Eisenstein series span the weight-four space, since its cusp-constant map embeds it in a two-dimensional space.

The weight-sixteen cusp form space is spanned by $E_4\Delta$. Its leading coefficient and the Fourier expansion of a normalized Eisenstein series give $3617E_{16}=3617E_4^4-3456000E_4\Delta$. The coefficient multiplier $-3456000$ is congruent to $16320$ modulo $3617$, and is invertible there. Coefficient comparison gives the displayed modular congruence. Integrality of $E_4\Delta$ follows from the modular discriminant product.

The effective group ${\Gamma }_0(11)$ is torsion-free, has index twelve and has two regular cusps. If its weight-two cusp form space is one-dimensional, any nonzero member has order one at each cusp and no interior zeros by the regular-cusp valence formula on a torsion-free modular curve. Its twelfth power and $\Delta (\tau )\Delta (11\tau )$ have the same weight and divisor. Their ratio is a holomorphic function without poles on the compactified modular curve, hence a nonzero constant. Normalizing the leading Fourier coefficient to one proves the displayed formula, with the holomorphic root chosen to start with $q$. It equals $q{\prod }_{n\ge 1}(1-q^n{)}^2(1-q^{11n}{)}^2$.

For two level-one weight-$k$ cusp forms, $F(s)=\sum a_n{\overline{b}}_n{n}^{-s}$. Unfolding against the nonholomorphic lattice sum normalized by ${\pi }^{-w}\Gamma (w)$ gives the displayed completion as the integral of $f\overline{g}{y}^k$ against that Eisenstein series. Its functional equation is $\mathcal{R}(s)=\mathcal{R}(2k-1-s)$, and its only possible simple poles are at $k-1,k$. The raw series is obtained by dividing out the gamma and zeta factors, so it does not inherit the same unqualified pole statement; zeros of the denominator zeta factor can yield additional possible poles. The raw series is regular at $k-1$ and can have a simple pole at $k$.