Atlas › Modular forms › Weight 2 › Level 756 › 756.2.a › 756.2.a.g

Newform orbit 756.2.a.g

Trace form

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

$2q - 2q^{7} + 12q^{13} - 2q^{19} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{11}$ , whose minimal polynomial is

$x^{2} - 52$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 756$ )
3	— ( $3^2 \mid 756$ )
7	$+1$

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	0	0	−2	0	12	0	−2	0	0	18	−2	0	16	0	0	0	0	−4	0	8	0	0	0	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.507 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, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756]: [0, 1, 1, 2, 3, 1, 2, 5, 4, 5, 4, 4, 7, 9, 12, 11, 10, 22, 21, 28, 26, 54, 76, 150] (certified at ell = 2147483647)
T = [(5, 1), (11, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(756, 2)[6]   # 756.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 756:2 1000