Atlas › Modular forms › Weight 2 › Level 456 › 456.2.a › 456.2.a.e

Newform orbit 456.2.a.e

Trace form

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

$2q - 2q^{3} - q^{5} - 3q^{7} + 2q^{9} + 3q^{11} + 12q^{13} + q^{15} + q^{17} + 2q^{19} + 3q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{23} + 11T_{29}$ , whose minimal polynomial is

$x^{2} - 173x + 6652$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 456$ )
3	$+1$
19	$-1$

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	−1	−3	3	12	1	2	8	4	6	6	6	1	5	−12	−8	9	−24	0	7	4	4	0	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.362 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, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456]: [0, 1, 1, 2, 3, 3, 5, 2, 8, 7, 8, 13, 24, 24, 46, 88] (certified at ell = 2147483647)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (23, 9), (29, 11)]; h of degree 8 = 1 x 8 (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(456, 2)[4]   # 456.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 456:2 1000