Atlas › Modular forms › Weight 2 › Level 964 › 964.2.a › 964.2.a.c

Newform orbit 964.2.a.c

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 5q^{3} + 11q^{5} + 27q^{9} + 20q^{11} + 3q^{13} + q^{15} + 4q^{17} + 5q^{19} + 8q^{21} - 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{12} - 16x^{11} + 50x^{10} + 416x^{9} - 2862x^{8} + 1349x^{7} + 29852x^{6} - 77539x^{5} + 7854x^{4} + 216718x^{3} - 324543x^{2} + 185504x - 37632$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 964$ )
241	$+1$

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	5	11	0	20	3	4	5	−9	25	12	4	28	7	−16	12	30	3	−2	−21	−16	−23	−16	21	26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.256 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, 2, 4, 241, 482, 964]: [0, 1, 2, 20, 62, 124] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 3 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(964, 2)[2]   # 964.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 964:2 1000