Atlas › Modular forms › Weight 2 › Level 758 › 758.2.a › 758.2.a.e

Newform orbit 758.2.a.e

Trace form

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

$10q - 10q^{2} + 6q^{3} + 10q^{4} + 5q^{5} - 6q^{6} + 6q^{7} - 10q^{8} + 14q^{9} - 5q^{10} - 4q^{11} + 6q^{12} + 19q^{13} - 6q^{14} + 8q^{15} + 10q^{16} + 8q^{17} - 14q^{18} + q^{19} + 5q^{20} + 11q^{21} + 4q^{22} + q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{10} - 11x^{9} - 7x^{8} + 366x^{7} - 439x^{6} - 4039x^{5} + 6340x^{4} + 16611x^{3} - 22147x^{2} - 22443x + 10039$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
379	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	−10	6	5	6	−4	19	8	1	1	8	16	19	−16	21	−10	17	−11	31	26	1	35	14	−17	2	31

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 379, 758]: [0, 1, 32, 97] (certified at ell = 2147481883)
T = [(3, 1), (5, 1)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 4 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(758, 2)[4]   # 758.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 758:2 1000