Atlas › Modular forms › Weight 2 › Level 632 › 632.2.a › 632.2.a.e

Newform orbit 632.2.a.e

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 2q^{3} + 9q^{5} + 6q^{7} + 8q^{9} - q^{11} + 11q^{13} - 2q^{15} + 4q^{17} + 3q^{19} + 2q^{21} - 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} - 29x^{5} + 211x^{4} + 902x^{3} - 15260x^{2} + 41232x + 1696$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 632$ )
79	$+1$

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	2	9	6	−1	11	4	3	−3	16	−1	18	4	4	−2	2	−12	22	−7	−2	9	−6	−20	−25	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.333 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, 8, 79, 158, 316, 632]: [0, 1, 2, 3, 7, 22, 43, 84] (certified at ell = 2147483053)
T = [(3, 1), (5, 3)]; 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(632, 2)[4]   # 632.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 632:2 1000