Atlas › Modular forms › Weight 2 › Level 763 › 763.2.a › 763.2.a.d

Newform orbit 763.2.a.d

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{14} - 10x^{13} - 244x^{12} + 2351x^{11} + 22102x^{10} - 195939x^{9} - 931947x^{8} + 6837506x^{7} + 18768456x^{6} - 89211968x^{5} - 160053241x^{4} + 349853539x^{3} + 334988628x^{2} - 65696228x - 65012356$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
109	$-1$

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
$\operatorname{tr} a_p$	4	2	14	−14	−10	19	33	4	6	17	4	13	22	−7	18	53	14	32	−13	−36	8	−1	51	1	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 0.251 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, 109, 763]: [0, 1, 9, 74] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (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(763, 2)[3]   # 763.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 763:2 1000