Atlas › Modular forms › Weight 4 › Level 37 › 37.4.a › 37.4.a.a

Newform orbit 37.4.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 6q^{2} - 11q^{3} + 6q^{4} - 29q^{5} - 27q^{6} - 32q^{7} - 90q^{8} + q^{9} + 81q^{10} - 11q^{11} + 143q^{12} - 45q^{13} + 82q^{14} - 16q^{15} + 378q^{16} - 56q^{17} + 255q^{18} - 144q^{19} - 165q^{20} + 108q^{21} + 73q^{22} - 275q^{23} + 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 17x^{3} + 98x^{2} + 231x + 187$

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
$\operatorname{tr} a_p$	−6	−11	−29	−32	−11	−45	−56	−144	−275	−29	111	148	−9	−190	−472	318	−1530	−31	−5	390	−967	17	−3138	224	−2532

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.092 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, 11] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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, 4)[0]   # 37.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 37:4 1000