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

Newform orbit 166.4.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 12q^{2} - q^{3} + 24q^{4} + 3q^{5} + 2q^{6} - 14q^{7} - 48q^{8} + 69q^{9} - 6q^{10} + 55q^{11} - 4q^{12} + 27q^{13} + 28q^{14} - 50q^{15} + 96q^{16} + 112q^{17} - 138q^{18} - 81q^{19} + 12q^{20} + 241q^{21} - 110q^{22} + 441q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} - 8x^{5} - 4411x^{4} + 53695x^{3} + 3411428x^{2} - 32839283x - 221044262$

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
$\operatorname{tr} a_p$	−12	−1	3	−14	55	27	112	−81	441	969	312	533	889	−583	−88	−167	185	−219	−1867	718	−2186	−946	−498	−506	−1524

$\operatorname{tr} a_n$ 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)[3]   # 166.4.a.d

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

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