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

Newform orbit 458.2.a.d

Trace form

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

$7q - 7q^{2} + 4q^{3} + 7q^{4} + q^{5} - 4q^{6} + 7q^{7} - 7q^{8} + 7q^{9} - q^{10} + 6q^{11} + 4q^{12} + 9q^{13} - 7q^{14} + 3q^{15} + 7q^{16} - 7q^{18} + 12q^{19} + q^{20} - 2q^{21} - 6q^{22} + 11q^{23} - 4q^{24} + O(q^{25})$

Coefficient field

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

$x^{7} - 7x^{6} - 273x^{5} + 2206x^{4} + 15894x^{3} - 142679x^{2} - 69859x + 1451996$

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
$\operatorname{tr} a_p$	−7	4	1	7	6	9	0	12	11	−15	1	18	−18	30	1	8	6	7	14	−13	−2	1	6	−32	0

$\operatorname{tr} a_n$ 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)[3]   # 458.2.a.d

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

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