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

Newform orbit 166.4.a.a

$q$ -expansion

$q + 2q^{2} - 5q^{3} + 4q^{4} + 8q^{5} - 10q^{6} - 31q^{7} + 8q^{8} - 2q^{9} + 16q^{10} + 3q^{11} - 20q^{12} - 24q^{13} - 62q^{14} - 40q^{15} + 16q^{16} + 5q^{17} - 4q^{18} - 144q^{19} + 32q^{20} + 155q^{21} + 6q^{22} - 40q^{23} - 40q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5}$ is a root of

$x - 19$

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
$\operatorname{tr} a_p$	2	−5	8	−31	3	−24	5	−144	−40	−71	−215	−111	438	−180	534	738	−79	−163	−578	−138	586	−412	−83	−1036	−1016

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 166, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 166.4.a.a

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

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