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

Newform orbit 98.6.a.h

Trace form

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

$2q + 8q^{2} + 14q^{3} + 32q^{4} + 70q^{5} + 56q^{6} + 128q^{8} + 244q^{9} + 280q^{10} + 62q^{11} + 224q^{12} + 1820q^{13} - 2038q^{15} + 512q^{16} + 1694q^{17} + 976q^{18} + 826q^{19} + 1120q^{20} + 248q^{22} - 2734q^{23} + 896q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{79})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 84x - 1080$

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	70	0	62	1820	1694	826	−2734	−2852	−2674	−9146	6132	−16040	−25326	14958	−1106	−28042	−102642	−11056	35070	101762	−44632	−75474	−8316

$\operatorname{tr} a_n$ 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)[7]   # 98.6.a.h

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

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