Atlas › Modular forms › Weight 2 › Level 371 › 371.2.a › 371.2.a.e

Newform orbit 371.2.a.e

Trace form

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

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

Coefficient field

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

$x^{9} - 9x^{8} - 153x^{7} + 1285x^{6} + 7772x^{5} - 54264x^{4} - 169764x^{3} + 564444x^{2} + 2133496x + 1640048$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
53	$-1$

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
$\operatorname{tr} a_p$	0	3	9	−9	0	13	0	10	−6	8	15	21	24	−31	3	9	−8	42	−20	−14	22	5	−7	−32	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.192 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, 7, 53, 371]: [0, 1, 5, 38] (certified at ell = 2147483053)
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(371, 2)[4]   # 371.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 371:2 1000