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

Newform orbit 88.4.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 2q^{3} + 8q^{5} + 24q^{7} + 89q^{9} + 33q^{11} + 66q^{13} - 10q^{15} + 210q^{17} - 72q^{19} + 200q^{21} - 50q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - 10x^{2} - 212x + 72$

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	8	24	33	66	210	−72	−50	−50	−298	−4	254	−112	40	−550	1630	−906	−482	242	998	324	1360	1200	180

$\operatorname{tr} a_n$ 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)[3]   # 88.4.a.d

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

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