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

Newform orbit 763.2.a.e

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{17} - 13x^{16} - 271x^{15} + 4012x^{14} + 24795x^{13} - 465410x^{12} - 887923x^{11} + 27166434x^{10} + 3120560x^{9} - 884143405x^{8} + 634590485x^{7} + 16451214844x^{6} - 17327754368x^{5} - 169655041208x^{4} + 177175878693x^{3} + 881317448234x^{2} - 610938044208x - 1835217971620$

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
$\operatorname{tr} a_p$	7	2	12	17	20	11	19	−4	22	37	−10	13	−6	9	−4	45	−12	6	−3	36	−4	5	7	13	−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 0.251 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)[4]   # 763.2.a.e

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

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