Atkin–Swinnerton-Dyer congruence conjecture for noncongruence cusp forms

Let $\Gamma$ be a noncongruence subgroup of $SL_2(\mathbb Z)$ with finite index, let $k\geq 2$, and let $S_k(\Gamma)$ have dimension $d$. Denote by $X_{\Gamma}$ the modular curve associated with $\Gamma$, and suppose that it has a model over $\mathbb Q$. A form is $M$-integral if its Fourier coefficients at the cusp $\infty$ have the integrality property described in the setup, and write $a_{n/p}(h)=0$ when $p\nmid n$. Atkin–Swinnerton-Dyer congruences. There exist a positive integer $M$ and an $M$-integral basis $f_1,\ldots,f_d$ of $S_k(\Gamma)$ such that, for every prime $p\nmid M$, there are a nonsingular matrix $(\lambda_{i,j})$, algebraic integers $A_p(j)$ satisfying

$$|\sigma(A_p(j))|\leq 2p^{(k-1)/2}$$

for every embedding $\sigma$, and characters $\chi_j$ unramified outside $M$, such that, for $h_j=\sum_i\lambda_{i,j}f_i$ and every $n\geq 1$,

$$\operatorname{ord}_p\bigl(a_{np}(h_j)-A_p(j)a_n(h_j)+\chi_j(p)p^{k-1}a_{n/p}(h_j)\bigr)\geq (k-1)(1+\operatorname{ord}_p n).$$

Equivalently, the quotient of the expression in parentheses by $(np)^{k-1}$ is integral at all places dividing $p$. This conjecture seeks a Hecke-like system of congruence relations for cusp forms on noncongruence subgroups, generalizing the arithmetic phenomena observed by Atkin and Swinnerton-Dyer; its status is not established by the supplied source context.