Atlas › Modular forms › Weight 2 › Level 775 › 775.2.a › 775.2.a.g

Newform orbit 775.2.a.g

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{3} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ , whose minimal polynomial is

$x^{4} + 72x^{3} - 11422x^{2} - 520400x + 22092517$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 775$ )
31	$-1$

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
$\operatorname{tr} a_p$	1	1	0	0	−6	−16	−1	5	0	6	4	−9	13	−7	14	−11	13	22	10	3	−9	−16	17	−12	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.451 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, 5, 25, 31, 155, 775]: [0, 1, 3, 3, 18, 82] (certified at ell = 2147482921)
T = [(2, 1), (3, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 13 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(775, 2)[6]   # 775.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 775:2 1000