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

Newform orbit 457.2.a.c

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 6q^{2} + 9q^{3} + 20q^{4} + 11q^{5} + 3q^{6} + 8q^{7} + 18q^{8} + 21q^{9} - q^{10} + 9q^{11} + 13q^{12} + 3q^{13} + q^{14} + 18q^{15} + 12q^{16} + 12q^{17} - 2q^{18} + 2q^{19} + 20q^{20} - 4q^{21} - 18q^{22} + 41q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{20} - 33x^{19} + 141x^{18} + 6292x^{17} - 67885x^{16} - 375661x^{15} + 7801935x^{14} - 1633251x^{13} - 430160707x^{12} + 1189095187x^{11} + 12525762421x^{10} - 58048638457x^{9} - 176412964960x^{8} + 1273446526194x^{7} + 551230564192x^{6} - 13055160661257x^{5} + 11105864081886x^{4} + 46524964965279x^{3} - 73575290241975x^{2} + 19328340071067x - 364705139833$

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 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
$\operatorname{tr} a_p$	6	9	11	8	9	3	12	2	41	2	22	−8	15	−12	96	20	40	−22	2	58	−29	−16	33	19	−15

$\operatorname{tr} a_n$ 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)[2]   # 457.2.a.c

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

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