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

Newform orbit 76.6.a.b

Trace form

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

$4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + 706q^{11} + 788q^{13} + 2792q^{15} + 240q^{17} - 1444q^{19} + 7128q^{21} + 5884q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 100x^{3} - 10854x^{2} - 955876x - 17355931$

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	10	−110	30	706	788	240	−1444	5884	5240	−860	−20732	−10204	−12554	−4826	−76484	23898	−32482	5022	121300	−104700	117128	92832	5988	22972

$\operatorname{tr} a_n$ 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)[1]   # 76.6.a.b

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

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