Atlas › Modular forms › Weight 8 › Level 37 › 37.8.a › 37.8.a.b

Newform orbit 37.8.a.b

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 16q^{2} + 121q^{3} + 794q^{4} + 376q^{5} + 519q^{6} + 2243q^{7} + 3870q^{8} + 9826q^{9} - 12629q^{10} + 9415q^{11} + 27955q^{12} + 12512q^{13} + 18260q^{14} + 25714q^{15} + 167866q^{16} + 54312q^{17} + 163911q^{18} + 97192q^{19} + 85625q^{20} + 97795q^{21} - 12345q^{22} + 107342q^{23} + 163119q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{11} - 137x^{10} - 9177x^{9} + 1544077x^{8} + 28512590x^{7} - 5745282384x^{6} - 49814019603x^{5} + 7581539205123x^{4} + 76274251969145x^{3} - 2199792138785719x^{2} - 10850964339267476x + 169022040552063416$

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
$\operatorname{tr} a_p$	16	121	376	2243	9415	12512	54312	97192	107342	41748	−272248	−557183	525465	−1376086	2269179	−346415	4598828	6208418	2199016	4653285	−1080699	−1336084	28551309	−8994788	−3968264

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 23] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 5 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, 8)[1]   # 37.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 37:8 1000