Atlas › Modular forms › Weight 6 › Level 88 › 88.6.a › 88.6.a.a

Newform orbit 88.6.a.a

Trace form

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

$2q - 14q^{3} + 18q^{5} + 48q^{7} - 92q^{9} + 242q^{11} - 708q^{13} - 126q^{15} - 1808q^{17} - 1596q^{19} - 3592q^{21} - 2454q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{37})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 4x - 144$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 88$ )
11	$-1$

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	18	48	242	−708	−1808	−1596	−2454	−5320	−2230	−10026	−7820	11812	344	−16228	18878	12672	50470	66326	14428	17260	26620	−71786	−103854

$\operatorname{tr} a_n$ 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, 8, 11, 22, 44, 88]: [1, 2, 4, 7, 6, 17, 33, 64] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 4 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(88, 6)[0]   # 88.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 88:6 1000