Atlas › Modular forms › Weight 2 › Level 397 › 397.2.a › 397.2.a.e

Newform orbit 397.2.a.e

Trace form

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

$13q - 7q^{2} - 11q^{3} + 13q^{4} - 5q^{5} - q^{6} - 17q^{7} - 21q^{8} + 20q^{9} - 3q^{10} - 31q^{11} - 17q^{12} + q^{13} - 2q^{14} - 11q^{15} + 21q^{16} - 3q^{17} - 19q^{18} - 17q^{19} - 12q^{20} + 3q^{21} + 11q^{22} - 36q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

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

$x^{13} + 40x^{12} + 518x^{11} + 843x^{10} - 33730x^{9} - 221300x^{8} + 310432x^{7} + 6226961x^{6} + 11212927x^{5} - 36930628x^{4} - 158098473x^{3} - 210784573x^{2} - 115379355x - 21696887$

Atkin–Lehner signs

$p$	$w_p$
397	$+1$

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
$\operatorname{tr} a_p$	−7	−11	−5	−17	−31	1	−3	−17	−36	−2	0	−5	9	−27	−19	−20	−4	9	−27	−27	7	−6	−71	8	24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 397]: [0, 33] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(397, 2)[4]   # 397.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 397:2 1000