Atlas › Modular forms › Weight 6 › Level 98 › 98.6.a › 98.6.a.f

Newform orbit 98.6.a.f

Trace form

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

$2q - 8q^{2} + 14q^{3} + 32q^{4} + 42q^{5} - 56q^{6} - 128q^{8} + 652q^{9} - 168q^{10} - 294q^{11} + 224q^{12} + 140q^{13} + 294q^{15} + 512q^{16} - 1302q^{17} - 2608q^{18} + 1442q^{19} + 672q^{20} + 1176q^{22} + 2646q^{23} - 896q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{130})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 56x + 264$

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
$\operatorname{tr} a_p$	−8	14	42	0	−294	140	−1302	1442	2646	1668	14798	5182	5124	−4520	14994	24006	−38850	23618	32002	−89376	47138	−40970	−68376	−123102	43652

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 5, 12, 25, 72] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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, 6)[5]   # 98.6.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 98:6 1000