Atlas › Modular forms › Weight 2 › Level 776 › 776.2.a › 776.2.a.e

Newform orbit 776.2.a.e

Trace form

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

$9q - 2q^{3} + 5q^{5} + 3q^{7} + 17q^{9} - 7q^{11} + 8q^{13} + 7q^{15} + 13q^{17} - q^{19} + 10q^{21} + 4q^{23} + O(q^{25})$

Coefficient field

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

$x^{9} - 3x^{8} - 60x^{7} + 179x^{6} + 760x^{5} - 2142x^{4} - 2712x^{3} + 6340x^{2} + 2304x + 144$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 776$ )
97	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	−2	5	3	−7	8	13	−1	4	15	18	26	17	−3	11	18	−11	11	−9	5	25	14	−40	−2	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 4, 8, 97, 194, 388, 776]: [0, 1, 2, 3, 8, 26, 52, 102] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 3 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(776, 2)[4]   # 776.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 776:2 1000