Atlas › Modular forms › Weight 2 › Level 131 › 131.2.a › 131.2.a.b

Newform orbit 131.2.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + q^{3} + 16q^{4} + 4q^{5} - 4q^{6} + q^{7} - 6q^{8} + 15q^{9} + 2q^{11} - 8q^{12} + 11q^{13} - 6q^{14} - 2q^{15} + 28q^{16} - 2q^{17} - 24q^{18} - 14q^{20} + 9q^{21} - 4q^{22} - 10q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - x^{9} - 36x^{8} + 70x^{7} + 313x^{6} - 924x^{5} + 401x^{4} + 387x^{3} - 270x^{2} + 36x + 1$

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
$\operatorname{tr} a_p$	0	1	4	1	2	11	−2	0	−10	16	6	34	−13	9	−6	30	−5	51	−10	0	−14	24	−22	14	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 0.161 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, 131]: [0, 12] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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, 2)[1]   # 131.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 131:2 1000