Atlas › Modular forms › Weight 4 › Level 185 › 185.4.a › 185.4.a.a

Newform orbit 185.4.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 5q^{2} - 6q^{3} + 13q^{4} + 35q^{5} - 40q^{6} - 49q^{7} - 39q^{8} + q^{9} - 25q^{10} - 123q^{11} + 48q^{12} - 48q^{13} + 7q^{14} - 30q^{15} - 91q^{16} - 13q^{17} - 11q^{18} - 172q^{19} + 65q^{20} - 220q^{21} + 175q^{22} - 234q^{23} - 222q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 11x^{6} - 29x^{5} - 353x^{4} + 217x^{3} + 2803x^{2} - 1329x - 4681$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
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
$\operatorname{tr} a_p$	−5	−6	35	−49	−123	−48	−13	−172	−234	−599	−889	−259	−231	−281	−294	1123	−552	329	−70	−516	1018	−4164	848	−718	1141

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.127 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, 5, 37, 185]: [1, 3, 11, 60] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 36 = 1 x 36 (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(185, 4)[0]   # 185.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 185:4 1000