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

Newform orbit 100.6.a.d

Trace form

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

$2q - 238q^{9} - 200q^{11} - 4488q^{19} - 2728q^{21} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{31})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 12400$

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	0	0	0	−200	0	0	−4488	0	−15708	−4288	0	−14828	0	0	0	51944	−6116	0	75216	0	159456	0	−1652	0

$\operatorname{tr} a_n$ 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)[3]   # 100.6.a.d

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

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