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

Newform orbit 379.2.a.b

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{18} - 4x^{17} - 64x^{16} + 278x^{15} + 1448x^{14} - 7045x^{13} - 14011x^{12} + 82429x^{11} + 58010x^{10} - 494437x^{9} - 49143x^{8} + 1529365x^{7} - 295872x^{6} - 2184929x^{5} + 631519x^{4} + 990787x^{3} - 103546x^{2} - 144136x - 12488$

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 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	1	22	1	10	9	14	2	12	47	0	5	52	−12	3	−4	10	28	−37	−8	−3	−27	16	28	−2

$\operatorname{tr} a_n$ 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)[1]   # 379.2.a.b

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

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