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

Newform orbit 337.2.a.b

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 3q^{2} + 9q^{3} + 15q^{4} + 10q^{5} + q^{6} + 7q^{7} + 9q^{8} + 16q^{9} - 3q^{10} + 9q^{11} + 9q^{12} - 4q^{13} - q^{14} + 8q^{15} + 3q^{16} + 15q^{17} - 7q^{18} + q^{19} + 18q^{20} - 4q^{21} - 18q^{22} + 45q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 12x^{14} + 18x^{13} + 297x^{12} - 1192x^{11} - 1356x^{10} + 13591x^{9} - 11498x^{8} - 46643x^{7} + 80176x^{6} + 34247x^{5} - 124275x^{4} + 18463x^{3} + 57441x^{2} - 11485x - 8029$

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 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
$\operatorname{tr} a_p$	3	9	10	7	9	−4	15	1	45	1	11	−11	1	−12	70	1	6	−12	−11	41	−18	−23	2	3	−27

$\operatorname{tr} a_n$ 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)[1]   # 337.2.a.b

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

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