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

Newform orbit 758.2.a.f

Trace form

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

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

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

$x^{10} - 3x^{9} - 37x^{8} + 138x^{7} + 239x^{6} - 1635x^{5} + 1998x^{4} + 605x^{3} - 2223x^{2} + 487x + 455$

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	−3	10	8	15	6	11	−3	−14	26	5	4	9	−4	−3	−5	−7	18	−15	3	−8	−13	−14	3

$\operatorname{tr} a_n$ 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)[5]   # 758.2.a.f

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

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