Atlas › Modular forms › Weight 2 › Level 779 › 779.2.a › 779.2.a.d

Newform orbit 779.2.a.d

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{19} - 15x^{18} - 289x^{17} + 4911x^{16} + 30895x^{15} - 643790x^{14} - 1455180x^{13} + 43881228x^{12} + 23652903x^{11} - 1702106023x^{10} + 374243163x^{9} + 38566787331x^{8} - 19151027884x^{7} - 495223229072x^{6} + 271842698001x^{5} + 3189763030515x^{4} - 1964722207677x^{3} - 7029599166900x^{2} + 8118789555532x - 2264036666916$

Atkin–Lehner signs

$p$	$w_p$
19	$-1$
41	$+1$

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
$\operatorname{tr} a_p$	0	5	2	1	−2	28	8	19	−4	31	28	17	−19	12	−2	17	5	58	24	15	39	39	−31	27	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 19, 41, 779]: [0, 2, 4, 72] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 61 = 1 x 61 (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(779, 2)[3]   # 779.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 779:2 1000