Atlas › Modular forms › Weight 2 › Level 457 › 457.2.a › 457.2.a.b

Newform orbit 457.2.a.b

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 10q^{2} - 12q^{3} + 16q^{4} - 9q^{5} + q^{6} - 15q^{7} - 21q^{8} + 17q^{9} + 5q^{10} + 3q^{11} - 16q^{12} - 22q^{13} - 4q^{14} - 16q^{15} + 18q^{16} - 18q^{17} - 13q^{18} - 30q^{20} + 2q^{21} + q^{22} - 36q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{15} + 46x^{14} + 753x^{13} + 3557x^{12} - 39225x^{11} - 549127x^{10} - 1558213x^{9} + 10229180x^{8} + 78781977x^{7} + 124652164x^{6} - 363414093x^{5} - 1324451877x^{4} - 496791642x^{3} + 2410780819x^{2} + 3061684520x + 1047726751$

Atkin–Lehner signs

$p$	$w_p$
457	$+1$

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 odd 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
$\operatorname{tr} a_p$	−10	−12	−9	−15	3	−22	−18	0	−36	−6	−18	−15	−9	−10	−85	−28	−29	31	−24	−26	−21	8	−63	3	−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 0.191 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, 457]: [0, 38] (certified at ell = 2147478649)
T = [(2, 1), (3, 3)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 5 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(457, 2)[1]   # 457.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 457:2 1000