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

Newform orbit 931.2.a.n

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 2q^{2} - 2q^{3} + 10q^{4} - 2q^{5} - 4q^{6} + 12q^{8} + 15q^{9} + 7q^{11} - 22q^{12} + 6q^{13} + 2q^{15} + 24q^{16} + 19q^{17} - 12q^{18} + 7q^{19} - 8q^{20} - 6q^{22} - q^{23} + 20q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{7} + 4x^{6} - 154x^{5} - 278x^{4} + 6693x^{3} + 3774x^{2} - 87588x + 36744$

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	−2	−2	0	7	6	19	7	−1	24	0	8	−4	4	5	20	−16	5	−4	12	3	−20	−11	−10	−4

$\operatorname{tr} a_n$ 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)[13]   # 931.2.a.n

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

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