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

Newform orbit 763.2.a.b

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{10} + 11x^{9} - 107x^{8} - 1194x^{7} + 3648x^{6} + 39417x^{5} - 55764x^{4} - 460568x^{3} + 348522x^{2} + 1738674x - 655065$

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 odd 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
$\operatorname{tr} a_p$	−5	−2	−10	10	−20	−12	−18	0	−24	−34	8	−25	10	−12	−9	−51	7	−24	13	−9	28	−13	−13	41	43

$\operatorname{tr} a_n$ 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)[1]   # 763.2.a.b

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

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