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

Newform orbit 37.8.a.a

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 24q^{2} - 95q^{3} + 602q^{4} - 624q^{5} - 777q^{6} - 501q^{7} - 3810q^{8} + 6181q^{9} + 8595q^{10} - 8325q^{11} - 19645q^{12} - 17108q^{13} - 65418q^{14} - 55756q^{15} - 56998q^{16} - 72924q^{17} - 156165q^{18} - 47786q^{19} - 226209q^{20} - 65313q^{21} - 138973q^{22} - 148086q^{23} - 68031q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 119x^{9} - 7109x^{8} - 996295x^{7} + 17503960x^{6} + 2689381514x^{5} - 15779310925x^{4} - 2532238654253x^{3} + 610609057119x^{2} + 620523208892907x + 2449613912536098$

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
$\operatorname{tr} a_p$	−24	−95	−624	−501	−8325	−17108	−72924	−47786	−148086	−164154	−189560	506530	814263	−590572	−1534185	−2518209	−5894748	−2569480	−6983232	−5013963	−11678449	−3853378	−15677895	−25836	4648834

$\operatorname{tr} a_n$ 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)[0]   # 37.8.a.a

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

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