Atlas › Modular forms › Weight 8 › Level 28 › 28.8.a › 28.8.a.a

Newform orbit 28.8.a.a

Trace form

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

$2q - 14q^{3} + 42q^{5} + 686q^{7} + 2782q^{9} + 7428q^{11} + 11830q^{13} + 20880q^{15} + 15792q^{17} + 26614q^{19} - 4802q^{21} + 32640q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3529})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 28x - 56268$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 28$ )
7	$-1$

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	−14	42	686	7428	11830	15792	26614	32640	−158016	−180740	−45824	−321720	1023868	1665972	−410628	1702134	−547526	−2590616	4129272	−8008868	2470456	−9900786	15423492	−17377472

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 7, 14, 28]: [1, 3, 5, 5, 16, 31] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (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(28, 8)[0]   # 28.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 28:8 1000