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

Newform orbit 371.2.a.f

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q - q^{2} - q^{3} + 19q^{4} + 2q^{5} + q^{6} + 11q^{7} + 20q^{9} - 11q^{10} + 5q^{11} - 18q^{12} + 13q^{13} - q^{14} - q^{15} + 43q^{16} - 6q^{17} - 9q^{18} + 7q^{19} - 16q^{20} - q^{21} + q^{22} - 2q^{23} + 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{11} + 4x^{10} - 254x^{9} - 616x^{8} + 23141x^{7} + 29744x^{6} - 964668x^{5} - 306948x^{4} + 18681776x^{3} - 10675088x^{2} - 136958848x + 189864960$

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
$\operatorname{tr} a_p$	−1	−1	2	11	5	13	−6	7	−2	7	22	24	−1	21	−7	−11	−2	39	38	0	0	5	−40	10	13

$\operatorname{tr} a_n$ 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)[5]   # 371.2.a.f

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

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