Atlas › Modular forms › Weight 2 › Level 377 › 377.2.a › 377.2.a.f

Newform orbit 377.2.a.f

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - x^{8} - 30x^{7} + 25x^{6} + 222x^{5} - 64x^{4} - 338x^{3} + 193x^{2} + 17x - 13$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
29	$+1$

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
$\operatorname{tr} a_p$	1	0	2	17	3	9	−2	14	13	−9	13	17	−3	8	−7	3	3	−7	39	4	12	16	10	−11	−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 0.187 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, 13, 29, 377]: [0, 1, 3, 36] (certified at ell = 2147483353)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (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(377, 2)[5]   # 377.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 377:2 1000