Atlas › Modular forms › Weight 2 › Level 379 › 379.2.a › 379.2.a.a

Newform orbit 379.2.a.a

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 5q^{2} - 5q^{3} + 9q^{4} - 20q^{5} - 8q^{6} - 3q^{7} - 12q^{8} + 4q^{9} - 20q^{11} - 8q^{12} - 11q^{13} - 14q^{14} + 5q^{16} - 20q^{17} - 7q^{18} - 12q^{19} - 31q^{20} - 27q^{21} + 14q^{22} + 2q^{23} - 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} + 10x^{12} + 19x^{11} - 82x^{10} - 253x^{9} + 207x^{8} + 878x^{7} - 318x^{6} - 1180x^{5} + 383x^{4} + 499x^{3} - 130x^{2} - 37x + 1$

Atkin–Lehner signs

$p$	$w_p$
379	$+1$

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
$\operatorname{tr} a_p$	−5	−5	−20	−3	−20	−11	−20	−12	2	−41	−12	−1	−50	10	−17	−12	−20	−34	9	−16	−13	11	−2	−30	33

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 379]: [0, 32] (certified at ell = 2147481883)
T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 3 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(379, 2)[0]   # 379.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 379:2 1000