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

Newform orbit 99.6.a.e

Trace form

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

$2q - q^{2} + 93q^{4} + 38q^{5} - 18q^{7} - 171q^{8} + 1546q^{10} - 242q^{11} - 66q^{13} - 304q^{14} - 543q^{16} + 920q^{17} - 2932q^{19} + 202q^{20} + 121q^{22} - 5246q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{313})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 37x - 9126$

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
$\operatorname{tr} a_p$	−1	0	38	−18	−242	−66	920	−2932	−5246	12600	9936	5996	−24244	20360	5806	−40770	−18212	−11398	65368	−61446	53412	17122	14304	58140	−183056

$\operatorname{tr} a_n$ 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)[4]   # 99.6.a.e

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

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