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

Newform orbit 779.2.a.e

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + q^{2} + q^{3} + 29q^{4} + 11q^{5} + 13q^{6} + 6q^{7} + 3q^{8} + 32q^{9} + 18q^{10} + q^{11} - 2q^{12} + 26q^{13} - 10q^{14} - 13q^{15} + 49q^{16} + 7q^{17} - 27q^{18} - 21q^{19} + 24q^{20} + 23q^{21} - q^{22} + 8q^{23} + 42q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{21} - 4x^{20} - 494x^{19} + 2202x^{18} + 98314x^{17} - 464203x^{16} - 10328586x^{15} + 49197482x^{14} + 637460319x^{13} - 2909802824x^{12} - 24377349180x^{11} + 99555366948x^{10} + 594505031441x^{9} - 1950764145046x^{8} - 9337021082703x^{7} + 20533323818210x^{6} + 91705599146252x^{5} - 93348268175541x^{4} - 505027930998024x^{3} - 10977444797968x^{2} + 1139733849460520x + 871734439705900$

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
$\operatorname{tr} a_p$	1	1	11	6	1	26	7	−21	8	11	4	17	21	15	−5	−23	3	107	−22	−39	30	−21	29	−7	6

$\operatorname{tr} a_n$ 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)[4]   # 779.2.a.e

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

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