Atlas › Modular forms › Weight 10 › Level 37 › 37.10.a › 37.10.a.b

Newform orbit 37.10.a.b

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 48q^{2} + 397q^{3} + 3498q^{4} + 2841q^{5} + 3375q^{6} + 6632q^{7} + 41046q^{8} + 101917q^{9} + 129003q^{10} + 44949q^{11} - 123661q^{12} + 38913q^{13} + 16434q^{14} + 119816q^{15} + 859962q^{16} + 893196q^{17} + 1833339q^{18} + 1532124q^{19} + 4974963q^{20} + 1851132q^{21} + 3195323q^{22} + 5911773q^{23} + 7885413q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 445x^{13} - 98435x^{12} + 60711620x^{11} + 2095424449x^{10} - 3061811325289x^{9} + 58838973622328x^{8} + 69130440034581907x^{7} - 2361980083380473937x^{6} - 676670439183284956320x^{5} + 15748170964979492905215x^{4} + 2574707559298497716523141x^{3} + 2046694429691111350155051x^{2} - 603091703766392998410384054x - 531641677049297363230589232$

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
$\operatorname{tr} a_p$	48	397	2841	6632	44949	38913	893196	1532124	5911773	8764377	13188927	26238254	22153785	1779790	40080072	−102088122	56191266	−178507397	287062499	224382678	271440727	379128625	1664083206	3293434692	2385468336

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 29] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 8 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, 10)[1]   # 37.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 37:10 1000