Atlas › Modular forms › Weight 4 › Level 157 › 157.4.a › 157.4.a.b

Newform orbit 157.4.a.b

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + 11q^{2} + 25q^{3} + 95q^{4} + 15q^{5} + 9q^{6} + 29q^{7} + 156q^{8} + 242q^{9} + 92q^{10} + 223q^{11} + 243q^{12} + 100q^{13} + 153q^{14} + 150q^{15} + 435q^{16} + 313q^{17} + 418q^{18} + 133q^{19} + 105q^{20} + 104q^{21} + 97q^{22} + 513q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{21} - 36x^{20} + 103x^{19} + 9878x^{18} - 92125x^{17} - 999898x^{16} + 14458373x^{15} + 40602847x^{14} - 1074332817x^{13} + 63480815x^{12} + 43959513911x^{11} - 61419049406x^{10} - 1019488111238x^{9} + 2093104121666x^{8} + 12933713966875x^{7} - 30437731639289x^{6} - 79578057589126x^{5} + 204565834404879x^{4} + 149500996778515x^{3} - 534638519740816x^{2} + 219035053957980x + 62554906820512$

Atkin–Lehner signs

$p$	$w_p$
157	$+1$

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
$\operatorname{tr} a_p$	11	25	15	29	223	100	313	133	513	367	518	366	340	1048	688	1337	206	−2017	785	224	−1624	1115	3652	721	−1868

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.12 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 engine's checks

dim M^+ at levels [1, 157]: [1, 41] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 6 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(157, 4)[1]   # 157.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 157:4 1000