Atlas › Modular forms › Weight 2 › Level 482 › 482.2.a › 482.2.a.e

Newform orbit 482.2.a.e

Trace form

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

$9q + 9q^{2} + 4q^{3} + 9q^{4} + 5q^{5} + 4q^{6} + 6q^{7} + 9q^{8} + 13q^{9} + 5q^{10} + 2q^{11} + 4q^{12} + 5q^{13} + 6q^{14} - 9q^{15} + 9q^{16} + 3q^{17} + 13q^{18} + 2q^{19} + 5q^{20} - 4q^{21} + 2q^{22} - 3q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

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

$x^{9} - 9x^{8} - x^{7} + 224x^{6} - 674x^{5} + 30x^{4} + 2485x^{3} - 3577x^{2} + 1599x - 122$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
241	$+1$

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	4	5	6	2	5	3	2	−3	11	−4	4	−4	9	−21	−5	−17	6	7	−32	7	4	−2	−12	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 241, 482]: [0, 1, 20, 62] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 21 = 1 x 21 (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(482, 2)[4]   # 482.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 482:2 1000