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

Newform orbit 37.6.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 12q^{2} + 25q^{3} + 154q^{4} + 136q^{5} + 75q^{6} + 99q^{7} + 486q^{8} + 697q^{9} + 1455q^{10} + 1555q^{11} + 1283q^{12} + 1196q^{13} + 1234q^{14} + 836q^{15} - 326q^{16} + 760q^{17} - 2013q^{18} + 122q^{19} + 1131q^{20} - 4467q^{21} - 4409q^{22} + 1354q^{23} - 8307q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 37x^{7} - 916x^{6} + 42708x^{5} + 66376x^{4} - 13653342x^{3} + 74415411x^{2} + 1078175133x - 8412796410$

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 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
$\operatorname{tr} a_p$	12	25	136	99	1555	1196	760	122	1354	−3398	−4912	10952	29529	49192	53327	11697	64944	−9608	−10932	−43803	817	−47618	268701	182696	−36598

$\operatorname{tr} a_n$ 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)[1]   # 37.6.a.b

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

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