Hecke–Fricke characterization conjecture for cusp forms

Let $N$ be a positive integer, let $k$ be a weight, and let $\chi_0$ be the trivial character modulo $N$. Let $\mathcal H=\{x+iy\in\mathbb C:y>0\}$, and let $S_k(\Gamma_0(N),\chi_0)$ denote the space of cusp forms of weight $k$ and character $\chi_0$ for the Hecke congruence group $\Gamma_0(N)$. For an analytic function $f:\mathcal H\to\mathbb C$, interpret the Fricke and Hecke relations as $f|H_N=\pm f$ and $f|T_p=a_p f$ for every prime $p$. Hecke–Fricke characterization conjecture. If $f$ is periodic with period $1$ and satisfies these Fricke and Hecke relations, then

$$f\in S_k(\Gamma_0(N),\chi_0).$$

The conjecture asserts that the invariance property defining cusp forms follows from the period-one condition together with the Fricke and Hecke relations; it is presented as the motivating conjecture for the paper's study of degree-$2$ $L$-functions and higher order modular forms.