Atlas › Modular forms › Weight 4 › Level 187 › 187.4.a › 187.4.a.d

Newform orbit 187.4.a.d

Trace form

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

$10q - 8q^{2} - 9q^{3} + 40q^{4} - 41q^{5} + 31q^{6} - 63q^{7} - 96q^{8} + 61q^{9} - 47q^{10} + 110q^{11} - 171q^{12} - 99q^{13} - 95q^{14} - 150q^{15} + 180q^{16} - 170q^{17} - 207q^{18} - 73q^{19} - 343q^{20} - 152q^{21} - 88q^{22} - 58q^{23} + 157q^{24} + O(q^{25})$

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

$x^{10} + 17x^{9} - 112x^{8} - 2557x^{7} - 1011x^{6} + 70527x^{5} + 14443x^{4} - 548647x^{3} - 12582x^{2} + 1194836x - 59171$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
17	$+1$

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
$\operatorname{tr} a_p$	−8	−9	−41	−63	110	−99	−170	−73	−58	−754	−228	−1177	−327	−487	−573	−2022	−602	−570	−12	636	−201	−1097	−477	−195	−994

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.131 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, 11, 17, 187]: [1, 4, 6, 56] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 6 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(187, 4)[3]   # 187.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 187:4 1000