Atlas › Modular forms › Weight 4 › Level 164 › 164.4.a › 164.4.a.b

Newform orbit 164.4.a.b

Trace form

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

$7q + 4q^{3} + 10q^{5} + 99q^{9} + 84q^{11} + 170q^{13} + 184q^{15} + 182q^{17} + 116q^{19} + 304q^{21} + 168q^{23} + O(q^{25})$

Coefficient field

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

$x^{7} - 14x^{6} - 900x^{5} + 14760x^{4} + 146444x^{3} - 3175120x^{2} + 11251500x + 8619000$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 164$ )
41	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	4	10	0	84	170	182	116	168	370	64	514	287	−228	−984	−366	−1204	250	−548	−2208	1046	−1224	−2332	206	3054

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 4, 41, 82, 164]: [1, 2, 3, 12, 34, 66] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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(164, 4)[1]   # 164.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 164:4 1000