Atlas › Modular forms › Weight 2 › Level 337 › 337.2.a › 337.2.a.a

Newform orbit 337.2.a.a

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q - 6q^{2} - 11q^{3} + 10q^{4} - 10q^{5} + q^{6} - 13q^{7} - 12q^{8} + 9q^{9} - 3q^{10} - 7q^{11} - 23q^{12} - 6q^{13} - 9q^{14} - 4q^{15} + 6q^{16} - 19q^{17} - 4q^{18} + q^{19} - 18q^{20} + 12q^{21} - 4q^{22} - 41q^{23} + 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} + 17x^{11} + 104x^{10} + 241x^{9} - 96x^{8} - 1347x^{7} - 1705x^{6} + 774x^{5} + 3017x^{4} + 1928x^{3} + 82x^{2} - 270x - 61$

Atkin–Lehner signs

$p$	$w_p$
337	$+1$

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
$\operatorname{tr} a_p$	−6	−11	−10	−13	−7	−6	−19	1	−41	−7	−11	−5	−13	−8	−62	−11	−18	16	13	−9	−4	9	−38	23	−1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.173 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, 337]: [0, 28] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (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(337, 2)[0]   # 337.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 337:2 1000