Atlas › Modular forms › Weight 2 › Level 88 › 88.2.a › 88.2.a.b

Newform orbit 88.2.a.b

Trace form

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

$2q + q^{3} + 3q^{5} - 2q^{7} + 3q^{9} - 2q^{11} - 2q^{13} - 7q^{15} + 4q^{17} - 8q^{19} - 18q^{21} + 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 10x + 8$

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	1	3	−2	−2	−2	4	−8	9	−2	−7	−11	6	−6	16	8	−5	−6	15	−5	2	−14	10	−7	27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.193 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 8, 11, 22, 44, 88]: [0, 1, 2, 3, 2, 5, 9, 16] (certified at ell = 2147483171)
T = [(3, 1), (5, 3)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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, 2)[1]   # 88.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 88:2 1000