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

Newform orbit 185.4.a.c

Trace form

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

$10q + 7q^{2} + 4q^{3} + 53q^{4} - 50q^{5} + 12q^{6} + 49q^{7} + 57q^{8} + 148q^{9} - 35q^{10} - 79q^{11} + 250q^{12} + 130q^{13} + 251q^{14} - 20q^{15} + 277q^{16} + 75q^{17} + 241q^{18} + 152q^{19} - 265q^{20} + 52q^{21} + 401q^{22} + 452q^{23} + 402q^{24} + O(q^{25})$

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

$x^{10} - 11x^{9} - 227x^{8} + 1967x^{7} + 20337x^{6} - 119423x^{5} - 837107x^{4} + 3029001x^{3} + 15453638x^{2} - 27486494x - 99709778$

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
$\operatorname{tr} a_p$	7	4	−50	49	−79	130	75	152	452	13	265	−370	−277	1703	1084	285	−2160	3233	−908	−1008	2404	1312	−54	−744	1225

$\operatorname{tr} a_n$ 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)[2]   # 185.4.a.c

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

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