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

Newform orbit 88.6.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 5q^{3} + 93q^{5} - 94q^{7} + 591q^{9} - 484q^{11} + 230q^{13} + 3227q^{15} + 1856q^{17} - 40q^{19} + 6114q^{21} + 1515q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} - 88x^{3} - 13960x^{2} + 1343648x - 23071856$

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	−5	93	−94	−484	230	1856	−40	1515	11390	1339	8003	26070	8022	−31984	9148	−101047	9618	−101403	−18367	73354	−92026	62246	138439	181869

$\operatorname{tr} a_n$ 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)[3]   # 88.6.a.d

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

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