Atlas › Modular forms › Weight 4 › Level 128 › 128.4.a › 128.4.a.e

Newform orbit 128.4.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{3} - 4q^{5} - 8q^{7} + 50q^{9} - 92q^{11} - 100q^{13} - 184q^{15} + 92q^{17} - 4q^{19} - 176q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 16x - 1136$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 128$ )

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	−4	−8	−92	−100	92	−4	8	−84	−384	172	−300	−300	−16	12	644	−292	172	408	412	−400	−948	572	2204

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.123 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, 8, 16, 32, 64, 128]: [1, 2, 3, 5, 8, 14, 26, 50] (certified at ell = 2147483489)
T = [(3, 1), (5, 3)]; 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(128, 4)[4]   # 128.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 128:4 1000