Atlas › Modular forms › Weight 2 › Level 956 › 956.2.a › 956.2.a.b

Newform orbit 956.2.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 4q^{3} + 9q^{7} + 11q^{9} - 3q^{11} + 11q^{13} + 11q^{15} - 6q^{17} + 24q^{19} + 7q^{21} + 5q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 4x^{8} - 49x^{7} + 163x^{6} + 742x^{5} - 1489x^{4} - 3942x^{3} - 303x^{2} + 568x - 64$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 956$ )
239	$+1$

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
$\operatorname{tr} a_p$	0	4	0	9	−3	11	−6	24	5	−4	28	17	2	15	2	0	15	18	38	3	6	2	−4	−29	−23

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.254 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, 2, 4, 239, 478, 956]: [0, 1, 2, 21, 62, 123] (certified at ell = 2147482093)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 3 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(956, 2)[1]   # 956.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 956:2 1000