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

Newform orbit 166.4.a.e

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 14q^{2} + 8q^{3} + 28q^{4} + 9q^{5} + 16q^{6} + 33q^{7} + 56q^{8} + 115q^{9} + 18q^{10} + 124q^{11} + 32q^{12} + 17q^{13} + 66q^{14} + 208q^{15} + 112q^{16} + 157q^{17} + 230q^{18} + 149q^{19} + 36q^{20} + 115q^{21} + 248q^{22} - 170q^{23} + 64q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{7} - 35x^{6} - 5613x^{5} + 213609x^{4} + 4672368x^{3} - 191170071x^{2} + 1404175262x - 207299713$

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 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
$\operatorname{tr} a_p$	14	8	9	33	124	17	157	149	−170	−102	127	−354	94	55	276	−481	408	−678	351	−100	−1526	−1110	581	−470	−408

$\operatorname{tr} a_n$ 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)[4]   # 166.4.a.e

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

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