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

Newform orbit 397.2.a.d

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{10} - 7x^{9} - 130x^{8} + 877x^{7} + 5880x^{6} - 38417x^{5} - 110142x^{4} + 680164x^{3} + 810830x^{2} - 3935080x - 2418033$

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 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
$\operatorname{tr} a_p$	7	0	7	10	15	−7	−3	16	11	13	−16	4	2	12	−5	28	18	−55	4	20	−32	5	56	0	−26

$\operatorname{tr} a_n$ 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)[3]   # 397.2.a.d

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

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