Atlas › Modular forms › Weight 2 › Level 185 › 185.2.a › 185.2.a.e

Newform orbit 185.2.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 2q^{2} + 3q^{3} + 10q^{4} - 5q^{5} - 6q^{6} + 11q^{7} + 6q^{8} + 6q^{9} - 2q^{10} - 5q^{11} - 2q^{12} + 4q^{13} - 8q^{14} - 3q^{15} + 16q^{16} + 2q^{18} - 4q^{19} - 10q^{20} + 3q^{21} - 8q^{22} + 4q^{23} - 42q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{5} - 11x^{4} - 26x^{3} + 452x^{2} + 107x - 4707$

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 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
$\operatorname{tr} a_p$	2	3	−5	11	−5	4	0	−4	4	−4	8	5	−5	10	7	−1	−30	−14	24	−7	5	28	27	6	−26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 37, 185]: [0, 1, 3, 20] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 13 = 1 x 13 (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(185, 2)[4]   # 185.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 185:2 1000