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

Newform orbit 763.2.a.c

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{13} + 21x^{12} - 6x^{11} - 2401x^{10} - 7279x^{9} + 96439x^{8} + 317322x^{7} - 1931985x^{6} - 4267492x^{5} + 19408640x^{4} + 11844645x^{3} - 56190228x^{2} - 10840459x + 36543532$

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
$\operatorname{tr} a_p$	−3	−6	−14	−13	12	−15	−27	−8	−10	−17	0	1	−14	−21	−10	−55	−16	8	−11	26	−16	37	−49	−35	−9

$\operatorname{tr} a_n$ 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)[2]   # 763.2.a.c

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

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