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

Newform orbit 779.2.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} + 11x^{7} - 93x^{6} - 993x^{5} + 3170x^{4} + 24514x^{3} - 44126x^{2} - 106569x - 41726$

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
$\operatorname{tr} a_p$	−2	−3	−6	−7	−6	−10	0	8	−10	−5	−24	−15	8	20	8	4	7	−58	6	−19	7	−31	27	−11	−39

$\operatorname{tr} a_n$ 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)[1]   # 779.2.a.b

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

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