In this paper, we exhibit a noncongruence subgroup $\Gamma$ whose space of weight 3 cusp forms $S_3(\Gamma )$ admits a basis satisfying the Atkin–Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to $S_3(\Gamma )$ by Scholl and also verifies the Atkin–Swinnerton-Dyer congruence conjecture for this space.

In this paper we study the compatible family of degree- $4$ Scholl representations $\rho _{\ell }$ associated with a space $S$ of weight $\kappa > 2$ noncongruence cusp forms satisfying Quaternion Multiplication over a biquadratic extension of $\mathbb {Q}$ . It is shown that $\rho _\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $\mathrm {GL}_4$ over $\mathbb {Q}$ . Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.

This paper is concerned with a compatible family of 4-dimensional $\ell$ -adic representations $\rho _{\ell }$ of $G_{\mathbb {Q}}:=\mathrm {Gal}(\overline {\mathbb {Q}}/\mathbb {Q})$ attached to the space of weight-3 cuspforms $S_3(\Gamma )$ on a noncongruence subgroup $\Gamma \subset \mathrm {SL}_2(\mathbb Z)$ . For this representation we prove that: [1.] It is automorphic: the $L$ -function $L(s, \rho _{\ell }^{\vee })$ agrees with the $L$ -function for an automorphic form for $\text {GL}_4(\mathbb A_{{\mathbb Q}})$ , where $\rho _{\ell }^{\vee }$ is the dual of $\rho _{\ell }$ . [2.] For each prime $p \ge 5$ there is a basis $h_p = \{ h_p ^+, h_p ^- \}$ of $S_3(\Gamma )$ whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the $q$ -expansion coefficients of a newform $f$ of level 432. The structure of this basis depends on the class of $p$ modulo 12. The key point is that the representation $\rho _{\ell }$ admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.

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