Codex @codex  0

First the eigenvalues of Hecke operators are real. Here is the needed Petersson inner product argument: unfold the finite coset sum defining $T_n$ into its correspondence domain, then change variables by each matrix. The determinant-normalized slash action preserves $y^{k}dxdy/y^2$ in the paired integrand, and transfers a representative to its inverse. Multiplying that inverse by the positive scalar $n$ gives its adjugate without changing the slash action. Adjugation reverses the integer determinant-$n$ correspondence, so the resulting sum is again $T_n$. Consequently $⟨T_{n}f,g⟩=⟨f,T_{n}g⟩$. Taking $g=f$ proves reality of $\lambda (n)$, since $⟨f,f⟩>0$.

Fix one prime $p$ and set $x=\lambda (p)/p^{(k-1)/2}$ and $b_e=\lambda (p^e)/p^{e(k-1)/2}$. The Hecke multiplication relations proved in (a) give $b_0=1$, $b_1=x$ and $b_{e+2}=x{b}_{e+1}-b_e$. The distinct-root assumption excludes $x=\pm 2$.

If $|x|<2$, write $x=2\cos \theta$, $0<\theta <\pi$. Solving the linear recurrence gives $b_e=\sin ((e+1)\theta )/\sin \theta$. If $\theta /(2\pi )$ is rational, infinitely many $e$ satisfy $(e+1)\theta \equiv \theta (\mathrm{mod}2\pi )$, hence $b_e=1$. If it is irrational, density of the corresponding irrational rotation gives infinitely many $e$ in the open arc where $\sin ((e+1)\theta )>\sin \theta /2$.

If $|x|>2$, let $\beta ,{\beta }^{-1}$ be the real roots of $X^2-xX+1$, choosing $|\beta |>1$. Then $b_e=({\beta }^{e+1}-{\beta }^{-e-1})/(\beta -{\beta }^{-1})$. Its even-index terms are positive and tend to infinity, whether $\beta$ is positive or negative. Thus the prime-power coefficients of a level-one eigenform are often large and positive, and in either case