Atlas › Modular forms › Weight 6 › Level 100 › 100.6.a › 100.6.a.e

Newform orbit 100.6.a.e

Trace form

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

$2q + 20q^{3} - 40q^{7} + 532q^{9} - 60q^{11} + 920q^{13} + 2910q^{17} + 2092q^{19} - 5308q^{21} + 120q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{409})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} + 20x - 10125$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 100$ )
5	— ( $5^2 \mid 100$ )

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
$\operatorname{tr} a_p$	0	20	0	−40	−60	920	2910	2092	120	3552	−8888	12140	−12438	1160	−1200	26340	−36696	19204	−90460	2736	12770	−16184	−30300	−47322	−2980

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.29 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, 4, 5, 10, 20, 25, 50, 100]: [1, 2, 4, 3, 9, 18, 13, 39, 78] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 8 = 1 x 8 (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(100, 6)[4]   # 100.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 100:6 1000