Atlas › Modular forms › Weight 4 › Level 131 › 131.4.a › 131.4.a.b

Newform orbit 131.4.a.b

Trace form

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

$21q + 3q^{2} + 7q^{3} + 111q^{4} + 33q^{5} + 26q^{6} + 38q^{7} + 57q^{8} + 284q^{9} + 82q^{10} + 51q^{11} + 76q^{12} + 205q^{13} + 133q^{14} + 136q^{15} + 663q^{16} + 62q^{17} + 135q^{18} + 255q^{19} + 316q^{20} + 472q^{21} + 490q^{22} + 306q^{23} + 402q^{24} + O(q^{25})$

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

$x^{21} - 10x^{20} - 541x^{19} + 5099x^{18} + 123175x^{17} - 1057095x^{16} - 15733206x^{15} + 117705933x^{14} + 1254427992x^{13} - 7773513722x^{12} - 65153242023x^{11} + 315470225012x^{10} + 2225169017642x^{9} - 7820379611531x^{8} - 49330633034639x^{7} + 111365682870920x^{6} + 682586911022760x^{5} - 738157256977079x^{4} - 5376488235466859x^{3} + 27196966688368x^{2} + 18564728852327322x + 16972856964400193$

Atkin–Lehner signs

$p$	$w_p$
131	$+1$

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
$\operatorname{tr} a_p$	3	7	33	38	51	205	62	255	306	789	716	1163	518	361	−620	−704	−1063	2313	659	912	1008	3032	−285	−61	4666

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 131]: [1, 34] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 32 = 1 x 32 (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(131, 4)[1]   # 131.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 131:4 1000