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

Newform orbit 185.4.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + q^{2} + 6q^{3} + 73q^{4} + 60q^{5} + 20q^{6} + 77q^{7} + 33q^{8} + 154q^{9} + 5q^{10} + 141q^{11} - 232q^{12} + 34q^{13} - 119q^{14} + 30q^{15} + 469q^{16} + 3q^{17} - 83q^{18} + 330q^{19} + 365q^{20} + 200q^{21} + 271q^{22} + 124q^{23} - 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} - 7x^{11} - 319x^{10} + 2475x^{9} + 28857x^{8} - 230363x^{7} - 870655x^{6} + 6677881x^{5} + 9870410x^{4} - 60087166x^{3} - 49870498x^{2} + 149122556x + 87557596$

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
$\operatorname{tr} a_p$	1	6	60	77	141	34	3	330	124	−9	289	444	439	399	−250	−521	838	−985	994	−312	−472	2406	−352	52	1981

$\operatorname{tr} a_n$ 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)[3]   # 185.4.a.d

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

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