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

Newform orbit 37.10.a.a

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 32q^{2} - 251q^{3} + 2730q^{4} - 2159q^{5} - 4401q^{6} - 12576q^{7} - 20394q^{8} + 69112q^{9} - 106605q^{10} - 112451q^{11} + 220963q^{12} + 7129q^{13} + 294824q^{14} - 63644q^{15} + 145178q^{16} - 890862q^{17} - 3066225q^{18} - 1435874q^{19} - 3193339q^{20} - 4745036q^{21} - 4350913q^{22} - 2565799q^{23} - 15286101q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} + 283x^{12} - 122743x^{11} - 31998132x^{10} + 6162080219x^{9} + 1305251766605x^{8} - 156366686581706x^{7} - 22991892854515203x^{6} + 1893737441952789659x^{5} + 173141238255369566084x^{4} - 9110126287645159038079x^{3} - 431085697660864288503321x^{2} + 10455196139397586329153065x - 54227391723920659885835900$

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
$\operatorname{tr} a_p$	−32	−251	−2159	−12576	−112451	7129	−890862	−1435874	−2565799	−2992323	−8242245	−24364093	−50975109	−18142836	−14353596	74438872	−251964328	202847323	−12257509	−310979094	−249752015	30429049	−2559788658	−3063565514	−429307758

$\operatorname{tr} a_n$ 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)[0]   # 37.10.a.a

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

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