Atlas › Modular forms › Weight 6 › Level 76 › 76.6.a › 76.6.a.a

Newform orbit 76.6.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 8q^{3} - 9q^{5} - 13q^{7} + 189q^{9} - 1229q^{11} - 56q^{13} - 1698q^{15} - 3149q^{17} + 1083q^{19} - 6186q^{21} - 3212q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} + 17x^{2} - 1093x + 2611$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 76$ )
19	$-1$

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	−8	−9	−13	−1229	−56	−3149	1083	−3212	2514	−10784	−526	−14246	−77	2893	29600	−2612	59895	−3050	−33562	44027	24944	−220156	−116120	171204

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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 engine's checks

dim M^+ at levels [1, 2, 4, 19, 38, 76]: [1, 2, 4, 10, 27, 53] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(76, 6)[0]   # 76.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 76:6 1000