Atlas › Modular forms › Weight 4 › Level 88 › 88.4.a › 88.4.a.c

Newform orbit 88.4.a.c

Trace form

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

$2q - 2q^{3} - 6q^{5} - 56q^{7} - 12q^{9} + 22q^{11} - 44q^{13} - 154q^{15} - 8q^{17} - 188q^{19} + 96q^{21} - 66q^{23} + O(q^{25})$

Coefficient field

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

$x^{2} + 8x - 164$

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	−2	−6	−56	22	−44	−8	−188	−66	−56	206	142	252	−372	−200	−1060	−782	−240	282	754	68	52	4	278	306

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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, 3, 5, 4, 11, 21, 40] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(88, 4)[2]   # 88.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 88:4 1000