Atlas › Modular forms › Weight 4 › Level 188 › 188.4.a › 188.4.a.a

Newform orbit 188.4.a.a

Trace form

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

$6q - 4q^{3} - 10q^{5} - 44q^{7} + 68q^{9} - 58q^{11} - 108q^{13} - 302q^{15} - 146q^{17} - 192q^{19} - 348q^{21} - 306q^{23} + O(q^{25})$

Coefficient field

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

$x^{6} + 14x^{5} - 123x^{4} - 2126x^{3} + 1727x^{2} + 80168x + 132970$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 188$ )
47	$+1$

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	−44	−58	−108	−146	−192	−306	−218	−352	−466	−114	−642	−282	278	800	−10	−2058	572	−322	−336	−908	1634	2186

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.125 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, 47, 94, 188]: [1, 2, 3, 13, 38, 75] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(188, 4)[0]   # 188.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 188:4 1000