Atlas › Modular forms › Weight 4 › Level 166 › 166.4.a › 166.4.a.c

Newform orbit 166.4.a.c

Trace form

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

$5q - 10q^{2} + 2q^{3} + 20q^{4} - 22q^{5} - 4q^{6} + 14q^{7} - 40q^{8} - 3q^{9} + 44q^{10} - 66q^{11} + 8q^{12} - 38q^{13} - 28q^{14} - 50q^{15} + 80q^{16} - 58q^{17} + 6q^{18} - 100q^{19} - 88q^{20} - 197q^{21} + 132q^{22} - 283q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} + 64x^{4} - 514x^{3} - 52581x^{2} - 18637x + 5910115$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
83	$-1$

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
$\operatorname{tr} a_p$	−10	2	−22	14	−66	−38	−58	−100	−283	−678	−42	−278	−983	−116	−326	48	−14	194	1334	1052	1852	912	415	−1478	−54

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.13 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, 2, 83, 166]: [1, 2, 22, 65] (certified at ell = 2147483323)
T = [(3, 1), (5, 3)]; h of degree 21 = 1 x 21 (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(166, 4)[2]   # 166.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 166:4 1000