Atlas › Modular forms › Weight 2 › Level 931 › 931.2.a › 931.2.a.q

Newform orbit 931.2.a.q

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 2q^{2} + 4q^{3} + 10q^{4} + 16q^{5} + 8q^{6} - 6q^{8} + 10q^{9} - 12q^{10} + 12q^{12} + 12q^{13} + 2q^{16} + 16q^{17} + 2q^{18} + 10q^{19} + 32q^{20} + 4q^{22} + 12q^{23} - 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{10} - 10x^{9} - 169x^{8} + 1644x^{7} + 8466x^{6} - 68680x^{5} - 225150x^{4} + 784600x^{3} + 3006216x^{2} + 2018704x - 104164$

Atkin–Lehner signs

$p$	$w_p$
7	— ( $7^2 \mid 931$ )
19	$-1$

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

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−2	4	16	0	0	12	16	10	12	−12	8	4	40	4	16	0	36	16	−28	−12	0	−8	0	48	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.312 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.

The engine's checks

dim M^+ at levels [1, 7, 19, 49, 133, 931]: [0, 1, 2, 5, 14, 94] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 62 = 1 x 62 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(931, 2)[16]   # 931.2.a.q

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 931:2 1000