Atlas › Modular forms › Weight 4 › Level 76 › 76.4.a › 76.4.a.b

Newform orbit 76.4.a.b

Trace form

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

$3q + q^{3} + 9q^{5} + 44q^{7} + 60q^{9} + 79q^{11} - 11q^{13} + 90q^{15} + 82q^{17} + 57q^{19} - 213q^{21} + 103q^{23} + O(q^{25})$

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

$x^{3} - 10x^{2} - 193x + 2026$

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	1	9	44	79	−11	82	57	103	−93	−116	−466	−188	−11	163	197	1381	−405	943	1052	−580	−1402	1802	1966	48

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.103 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, 3, 6, 17, 33] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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, 4)[1]   # 76.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 76:4 1000