= Solution
<Hecke operators> give a commuting family of arithmetic symmetries of <modular forms>. They preserve the weight and <cusp forms>, and diagonalizing them turns the <Fourier coefficients> of a modular form into multiplicative arithmetic data. Odd weight spaces are zero by the action of $-I$; in weight zero the operators act on constants by $T_n1=\sigma_{-1}(n)$. For the substantive theory, fix an even positive weight $k$ and $\Gamma=SL_2(\mathbb Z)$, and use the following normalization consistently.
Extend the <slash operator for modular forms> to $GL_2^+(\mathbb Q)$ by
$$
(f|_k\alpha)(\tau)=(\det\alpha)^{k/2}(c\tau+d)^{-k}f(\alpha\tau),
\qquad\alpha=\begin{pmatrix}a&b\\c&d\end{pmatrix}.
$$
It is a right action and positive scalar matrices act trivially. Let $\mathcal D_n$ be the integral matrices of determinant $n>0$. The <determinant-n matrix representatives for Hecke operators> for $\Gamma\backslash\mathcal D_n$ are
$$
\begin{pmatrix}a&b\\0&d\end{pmatrix},\qquad ad=n,\quad a,d>0,\quad0\leq b<d.
$$
Integer row operations reduce the first column to $(a,0)$, after which a row shear reduces $b$ modulo $d$; uniqueness follows from these same conditions. The <Hecke operator> is
$$
\boxed{T_nf=n^{k/2-1}\sum_{\alpha\in\Gamma\backslash\mathcal D_n}f|_k\alpha
=n^{k-1}\sum_{ad=n}d^{-k}\sum_{b=0}^{d-1}f\left(\frac{a\tau+b}{d}\right).}
$$
Right multiplication by $\Gamma$ permutes the left cosets, so $T_nf$ is again invariant under the weight-$k$ <slash operator for modular forms>. Each summand is a <holomorphic function> on the <complex upper half-plane>, and the coefficient formula below proves <holomorphy at a cusp>. Thus $T_n$ acts on $M_k(\Gamma)$ and preserves $S_k(\Gamma)$. Geometrically, these operators sum over index-$n$ sublattices, or equivalently the associated finite-degree correspondences between complex tori; the factors above compensate for the weight's response to lattice scaling.
Write $f=\sum_{r\geq0}a_rq^r$. The sum over $b$ is zero unless $d\mid r$, in which case it is $d$. Substituting into the displayed definition proves the <Fourier coefficients of a composite-index Hecke operator> formula
$$
a_r(T_nf)=\sum_{d\mid\gcd(n,r)}d^{k-1}a_{nr/d^2}(f),\qquad r\geq0,
$$
with $\gcd(n,0)=n$. In particular, $a_0(T_nf)=\sigma_{k-1}(n)a_0(f)$ and $a_1(T_nf)=a_n(f)$. At a prime this becomes
$$
T_pf=p^{k-1}f(p\tau)+\frac1p\sum_{b=0}^{p-1}f\left(\frac{\tau+b}{p}\right),\qquad
a_r(T_pf)=a_{pr}+p^{k-1}a_{r/p},
$$
where $a_{r/p}=0$ if $p\nmid r$.
The <Hecke multiplication relations> are
$$
\boxed{T_mT_n=\sum_{d\mid\gcd(m,n)}d^{k-1}T_{mn/d^2}.}
$$
They can be checked by applying the coefficient formula twice and regrouping the divisors. More explicitly, coprime indices factor independently, so $T_mT_n=T_{mn}$ if $(m,n)=1$. At a prime, separating the terms in the divisor sum according to whether one more factor $p$ is available gives
$$
T_pT_{p^r}=T_{p^{r+1}}+p^{k-1}T_{p^{r-1}},\qquad r\geq1.
$$
Iteration gives $T_{p^a}T_{p^b}=\sum_{j=0}^{\min(a,b)}p^{j(k-1)}T_{p^{a+b-2j}}$; combining the prime factors gives the full relation. Hence all <Hecke operators> commute, $T_1=1$, and the <Hecke algebra of modular forms> is generated by the $T_p$ together with scalars.
The <Petersson inner product> on the finite-dimensional <cusp form> space is
$$
\langle f,g\rangle=\int_{\Gamma\backslash\mathbb H}f(\tau)\overline{g(\tau)},y^k\,\frac{dx\,dy}{y^2}.
$$
The integrand is invariant and the integral converges because <cusp forms> decay exponentially at the cusp. The <Hecke operators are self-adjoint for the Petersson inner product> at level one. To see the underlying adjoint calculation, unfold the finite correspondence for a determinant-$n$ matrix and change variable $\tau\mapsto\alpha\tau$. The invariant hyperbolic measure and the determinant-normalized slash factors transfer $\alpha$ from one side of the inner product to $\alpha^{-1}$. Scalar matrices act trivially, so one can replace $\alpha^{-1}$ by the integral adjugate $n\alpha^{-1}$, which again has determinant $n$. This reverses the correspondence and permutes the same collection of terms, with the same real prefactor $n^{k/2-1}$. Summing gives
$$
\langle T_nf,g\rangle=\langle f,T_ng\rangle.
$$
Commuting <self-adjoint operators> on a finite-dimensional <inner product space> admit a common <orthonormal basis> of <eigenvectors>. Thus $S_k(\Gamma)$ has a basis of simultaneous <Hecke eigenforms>, with real <eigenvalues>.
A nonzero simultaneous cuspidal <Hecke eigenform> has $a_1\ne0$: if $T_nf=\lambda_nf$ and $a_1=0$, comparison of the first coefficients gives $a_n=\lambda_na_1=0$ for every $n\geq1$, so $f=0$. Normalize $a_1=1$. Then
$$
\boxed{T_nf=a_n(f)f.}
$$
Consequently the full <Fourier series> is determined by the eigenvalues, and a simultaneous eigenvalue system has a one-dimensional eigenspace. The <Hecke multiplication relations> translate into
$$
a_ma_n=\sum_{d\mid\gcd(m,n)}d^{k-1}a_{mn/d^2},\qquad
a_{p^{r+1}}=a_pa_{p^r}-p^{k-1}a_{p^{r-1}}.
$$
In particular the coefficients are multiplicative at coprime indices and are determined by the prime coefficients. This is the elementary multiplicity-one statement for normalized level-one <Hecke eigenforms>.
The <Hecke eigenvalues are algebraic integers>. One way to explain the arithmetic input is to use the rational structure supplied by the graded ring $M_*(\Gamma,\mathbb Q)=\mathbb Q[E_4,E_6]$. The rational <cusp form> space has a basis with integral <Fourier coefficients>, for example from appropriate monomials in $E_4,E_6,\Delta$. Its subgroup of forms whose entire expansions are integral is a full lattice: a finite initial coefficient map is injective by the <valence formula for the modular group>, so it embeds this subgroup into a finite-rank integer module. The coefficient formula shows that every $T_n$ preserves this lattice, and therefore has a monic integral characteristic polynomial. Its <eigenvalues> are <algebraic integers>. Simultaneous diagonalization also makes their number field finite over $\mathbb Q$; conjugating the rational coefficient data gives another eigenform, whose eigenvalues are again real by self-adjointness. Thus the coefficient field is totally real.
For a normalized cuspidal <Hecke eigenform>, define its <L-function of a cusp form> by $L(f,s)=\sum_{n\geq1}a_nn^{-s}$. The coefficient relations yield the <Euler product of a Hecke eigenform>
$$
\boxed{L(f,s)=\prod_p\left(1-a_pp^{-s}+p^{k-1-2s}\right)^{-1}.}
$$
Indeed the prime-power recurrence gives $\sum_{r\geq0}a_{p^r}X^r=(1-a_pX+p^{k-1}X^2)^{-1}$, and multiplicativity assembles these factors. These identities hold analytically in an initial right half-plane, as well as formally. For instance boundedness of $y^{k/2}|f(\tau)|$ on the full <complex upper half-plane> gives $a_n=O(n^{k/2})$, so $\operatorname{Re}s>k/2+1$ suffices for absolute convergence.
The <Mellin transform of a cusp-form L-function> relates this arithmetic to the modular transformation:
$$
\Lambda(f,s)=(2\pi)^{-s}\Gamma(s)L(f,s)=\int_0^\infty f(iy)y^{s-1}\,dy.
$$
Splitting at one and using $f(i/y)=i^ky^kf(iy)$ gives
$$
\Lambda(f,s)=\int_1^\infty f(iy)\left(y^{s-1}+i^ky^{k-s-1}\right)\,dy,
\qquad\boxed{\Lambda(f,s)=i^k\Lambda(f,k-s).}
$$
Exponential cusp decay makes the last integral entire in $s$. This proves the analytic continuation and functional equation of the <L-function of a cusp form>, complementing its <Euler product>. The root number is $i^k=(-1)^{k/2}$.
Two examples show the scope of the theory. For even $k\geq4$, the <Eisenstein series> $E_k$ is an eigenform with $T_nE_k=\sigma_{k-1}(n)E_k$. This follows from its divisor-sum expansion and the same multiplicativity relations; its associated positive-index Dirichlet series is proportional to $\zeta(s)\zeta(s-k+1)$. The constant coefficient gives $M_k(\Gamma)=\mathbb CE_k\oplus S_k(\Gamma)$, so an eigenbasis of the cusp space together with $E_k$ gives an eigenbasis of the whole modular-form space. The ordinary <Petersson inner product> above is only used on the cusp space, where it converges. In weight twelve the <valence formula for the modular group> gives $\dim S_{12}=1$: cancellation of the first coefficients of two independent cusp forms would give a nonzero form with order at infinity at least two, exceeding $12/12$. Thus the normalized <modular discriminant> is a <Hecke eigenform>; writing $\Delta=\sum_{n\geq1}\tau(n)q^n$ gives the <Ramanujan tau function> relations $\tau(mn)=\tau(m)\tau(n)$ for coprime $m,n$ and $\tau(p^{r+1})=\tau(p)\tau(p^r)-p^{11}\tau(p^{r-1})$.
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