Newform orbit 675.2.a.q

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$; the sum of their $q$-expansions is

$2q + 3q^{2} + 3q^{4} + 6q^{8} + 13q^{16} + 12q^{17} - 4q^{19} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$, whose minimal polynomial is

$x^{2} - 9x - 431$

Complex multiplication by $\mathbb{Q}(\sqrt{-15})$: $a_p = 0$ for every conjugate at the 4 primes $p \le 31$ inert in it (exactly, from $\operatorname{tr} a_{p^2} = -\dim \cdot p^{k-1}$), and $\operatorname{tr} a_p = 0$ at all 86 inert primes $p < 1000$. This is evidence from the stored coefficients, not a proof.

Atkin–Lehner signs

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$.

Traces of Hecke eigenvalues

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.887 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

newform_orbits(675, 2)[16]   # 675.2.a.q

cargo run --release -p sagebrush-web --example atlas -- 675:2 1000