Atlas › Modular forms › Weight 4 › Level 98 › 98.4.a › 98.4.a.h

Newform orbit 98.4.a.h

Trace form

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

$2q + 4q^{2} + 8q^{4} + 16q^{8} + 122q^{9} + 40q^{11} - 176q^{15} + 32q^{16} + 244q^{18} + 80q^{22} + 96q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{22})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 352$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	— ( $7^2 \mid 98$ )

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
$\operatorname{tr} a_p$	4	0	0	0	40	0	0	0	96	−332	0	−156	0	872	0	124	0	0	1160	−1088	0	−1360	0	0	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 7, 14, 49, 98]: [1, 2, 3, 8, 15, 44] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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(98, 4)[7]   # 98.4.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 98:4 1000