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

Newform orbit 779.2.a.c

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{11} + 9x^{10} - 109x^{9} - 1041x^{8} + 2758x^{7} + 29254x^{6} - 36354x^{5} - 304815x^{4} + 223266x^{3} + 1291048x^{2} - 487080x - 1802132$

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 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
$\operatorname{tr} a_p$	0	−3	−5	−8	−9	−22	−5	−11	6	−19	−8	−1	−11	−19	−5	−8	5	−53	−8	3	−30	−23	−1	5	11

$\operatorname{tr} a_n$ 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)[2]   # 779.2.a.c

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

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