Atlas › Modular forms › Weight 6 › Level 99 › 99.6.a › 99.6.a.f

Newform orbit 99.6.a.f

Trace form

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

$2q + 5q^{2} + 37q^{4} - 58q^{5} - 286q^{7} + 375q^{8} - 1030q^{10} + 242q^{11} - 166q^{13} - 1600q^{14} - 335q^{16} + 800q^{17} - 1476q^{19} - 5498q^{20} + 605q^{22} + 3370q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{177})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 53x - 2882$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 99$ )
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
$\operatorname{tr} a_p$	5	0	−58	−286	242	−166	800	−1476	3370	−6600	−7528	−29916	5780	−16656	−7850	−14178	−17300	−2946	31336	33810	60644	1870	58296	−92388	7120

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.283 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, 3, 9, 11, 33, 99]: [1, 3, 6, 6, 22, 62] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (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(99, 6)[5]   # 99.6.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 99:6 1000