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

Newform orbit 331.2.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} + 5x^{2} - 13x - 1$

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	−1	−6	3	−7	−13	−3	6	−2	−2	−14	11	−20	4	−13	8	−11	−3	13	−3	−6	27	14	−11	28

$\operatorname{tr} a_n$ 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)[1]   # 331.2.a.b

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

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