Atlas › Modular forms › Weight 6 › Level 96 › 96.6.a › 96.6.a.g

Newform orbit 96.6.a.g

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 18q^{3} + 36q^{5} - 120q^{7} + 162q^{9} - 200q^{11} + 284q^{13} - 324q^{15} + 2676q^{17} + 72q^{19} + 1080q^{21} + 3840q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{31})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{2} - 20476x - 108630012$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 96$ )
3	$+1$

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	−18	36	−120	−200	284	2676	72	3840	10212	10488	13148	4164	−5832	1520	9012	−55096	−63444	36792	37664	−37836	144888	−109272	−32556	69092

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.491 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, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96]: [1, 2, 3, 4, 7, 7, 13, 12, 24, 22, 44, 84] (certified at ell = 2147483497)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 6 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(96, 6)[6]   # 96.6.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 96:6 1000