Atlas › Modular forms › Weight 2 › Level 963 › 963.2.a › 963.2.a.f

Newform orbit 963.2.a.f

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + q^{2} + 7q^{4} - 5q^{5} + 4q^{7} + 6q^{8} - q^{10} + 2q^{11} + 18q^{13} + 9q^{14} - q^{16} + q^{17} - 4q^{19} + 10q^{20} - 5q^{22} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{7} + 14x^{6} - 103x^{5} - 1730x^{4} + 1328x^{3} + 52848x^{2} + 80992x - 27616$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 963$ )
107	$+1$

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
$\operatorname{tr} a_p$	1	0	−5	4	2	18	1	−4	0	3	4	25	0	11	9	−8	19	25	−24	19	30	−21	−12	22	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.393 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, 3, 9, 107, 321, 963]: [0, 1, 2, 10, 38, 110] (certified at ell = 2147482621)
T = [(2, 1), (5, 3)]; h of degree 44 = 1 x 44 (phi(ord eps) x dim S_k^new); 7 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(963, 2)[5]   # 963.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 963:2 1000