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

Newform orbit 778.2.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{12} - 40x^{11} + 412x^{10} + 3380x^{9} - 88960x^{8} + 338007x^{7} + 3537033x^{6} - 30529913x^{5} + 22780072x^{4} + 424531629x^{3} - 1341679812x^{2} + 1422796884x - 500793320$

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 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
$\operatorname{tr} a_p$	12	4	12	5	0	9	15	−10	8	6	1	1	8	−15	32	5	4	14	−27	0	4	−15	17	9	−19

$\operatorname{tr} a_n$ 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)[3]   # 778.2.a.d

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

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