Atlas › Modular forms › Weight 2 › Level 757 › 757.2.a › 757.2.a.b

Newform orbit 757.2.a.b

Trace form

The 33 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$33q + 10q^{2} + 19q^{3} + 34q^{4} + 7q^{5} + 8q^{6} + 6q^{7} + 30q^{8} + 38q^{9} + 4q^{10} + 18q^{11} + 27q^{12} + 9q^{13} + 16q^{14} + 6q^{15} + 28q^{16} + 33q^{17} + 20q^{18} + 20q^{19} + 11q^{20} - 6q^{21} - 9q^{22} + 25q^{23} + 23q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 33. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{33} - 29x^{32} + 294x^{31} - 595x^{30} - 11008x^{29} + 80396x^{28} - 17490x^{27} - 1814249x^{26} + 5741366x^{25} + 13197875x^{24} - 101672216x^{23} + 56707769x^{22} + 788554961x^{21} - 1555258587x^{20} - 2786735709x^{19} + 10864017551x^{18} + 1382114171x^{17} - 38913950742x^{16} + 22902300393x^{15} + 79533059554x^{14} - 83904939912x^{13} - 93720599504x^{12} + 140097416494x^{11} + 60498341577x^{10} - 127541424298x^{9} - 18809795286x^{8} + 63381223004x^{7} + 2641635648x^{6} - 16018185238x^{5} - 724551105x^{4} + 1813747062x^{3} + 151081785x^{2} - 62152787x - 5685974$

Atkin–Lehner signs

$p$	$w_p$
757	$-1$

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	19	7	6	18	9	33	20	25	17	0	−11	23	27	78	11	75	−17	15	19	2	−7	117	10	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.244 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, 757]: [0, 63] (certified at ell = 2147478481)
T = [(2, 1), (3, 1)]; h of degree 62 = 1 x 62 (phi(ord eps) x dim S_k^new); 6 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(757, 2)[1]   # 757.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 757:2 1000