Atlas › Modular forms › Weight 6 › Level 37 › 37.6.a › 37.6.a.a

Newform orbit 37.6.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 8q^{2} - 47q^{3} + 106q^{4} - 64q^{5} - 141q^{6} - 293q^{7} - 474q^{8} + 292q^{9} - 2017q^{10} - 1457q^{11} - 3917q^{12} - 536q^{13} - 488q^{14} - 254q^{15} + 2714q^{16} - 3068q^{17} + 4107q^{18} - 1900q^{19} + 1453q^{20} + 2425q^{21} + 4467q^{22} - 3986q^{23} + 11523q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 55x^{6} + 492x^{5} - 13632x^{4} - 251662x^{3} - 1400444x^{2} - 2475561x + 351451$

Atkin–Lehner signs

$p$	$w_p$
37	$+1$

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
$\operatorname{tr} a_p$	−8	−47	−64	−293	−1457	−536	−3068	−1900	−3986	−7436	5776	−9583	−25089	−22538	−60861	−15681	−54536	48694	−39724	−92187	73251	78604	−82223	181680	39092

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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 engine's checks

dim M^+ at levels [1, 37]: [1, 17] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (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(37, 6)[0]   # 37.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 37:6 1000