Atlas › Modular forms › Weight 2 › Level 458 › 458.2.a › 458.2.a.e

Newform orbit 458.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 9q^{2} + 2q^{3} + 9q^{4} + 4q^{5} + 2q^{6} + 6q^{7} + 9q^{8} + 17q^{9} + 4q^{10} - 6q^{11} + 2q^{12} + 11q^{13} + 6q^{14} - q^{15} + 9q^{16} + q^{17} + 17q^{18} - 6q^{19} + 4q^{20} - 6q^{22} + 8q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{9} - 14x^{8} - 227x^{7} + 3758x^{6} + 8467x^{5} - 280303x^{4} + 522980x^{3} + 5364982x^{2} - 21477851x + 21382184$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
229	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	9	2	4	6	−6	11	1	−6	8	5	6	3	0	8	−12	5	−39	−8	21	−3	0	−2	−40	12	11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.202 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, 229, 458]: [0, 1, 19, 59] (certified at ell = 2147478649)
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(458, 2)[4]   # 458.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 458:2 1000