Atlas › Modular forms › Weight 2 › Level 331 › 331.2.a › 331.2.a.c

Newform orbit 331.2.a.c

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{7} + 2x^{6} - 93x^{5} - 50x^{4} + 596x^{3} + 351x^{2} - 869x - 467$

Atkin–Lehner signs

$p$	$w_p$
331	$+1$

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
$\operatorname{tr} a_p$	−2	0	−13	−8	−10	5	−17	−19	15	−24	4	2	−18	−8	10	−21	−17	−12	−16	4	11	−2	−4	−28	−32

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 331]: [0, 28] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 4 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(331, 2)[2]   # 331.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 331:2 1000