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

Newform orbit 188.4.a.b

Trace form

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

$6q + 8q^{3} + 40q^{7} + 68q^{9} + 8q^{11} + 90q^{13} + 198q^{15} + 94q^{17} + 242q^{19} + 340q^{21} + 118q^{23} + O(q^{25})$

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

$x^{6} - 8x^{5} - 821x^{4} + 6740x^{3} + 161069x^{2} - 816058x - 10480872$

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	8	0	40	8	90	94	242	118	100	212	354	66	868	282	−262	−144	−630	−48	−88	−2170	936	−112	−2542	−2254

$\operatorname{tr} a_n$ 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)[1]   # 188.4.a.b

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

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