Atlas › Modular forms › Weight 2 › Level 778 › 778.2.a › 778.2.a.c

Newform orbit 778.2.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{9} + 30x^{8} + 186x^{7} - 1868x^{6} - 22216x^{5} - 14969x^{4} + 450929x^{3} + 1069793x^{2} - 1015510x - 1662199$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
389	$+1$

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
$\operatorname{tr} a_p$	−9	−6	−8	−3	−5	1	−8	−7	−17	−3	4	−10	−15	−11	−47	−20	−45	1	−12	−21	−1	9	−59	−23	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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, 2, 389, 778]: [0, 1, 33, 99] (certified at ell = 2147479897)
T = [(3, 1), (5, 3)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(778, 2)[2]   # 778.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 778:2 1000