Atlas › Modular forms › Weight 4 › Level 117 › 117.4.a › 117.4.a.f

Newform orbit 117.4.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 2q^{2} + 10q^{4} - 4q^{5} + 30q^{7} + 6q^{8} - 4q^{10} + 16q^{11} + 39q^{13} + 176q^{14} - 110q^{16} + 146q^{17} + 94q^{19} + 244q^{20} - 56q^{22} + 48q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{3} + 14x^{2} - 2247x + 27696$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 117$ )
13	$-1$

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
$\operatorname{tr} a_p$	−2	0	−4	30	16	39	146	94	48	2	302	374	−480	−260	24	678	1788	230	74	948	−222	−24	796	−1436	3242

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.118 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, 13, 39, 117]: [1, 2, 4, 5, 16, 44] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(117, 4)[5]   # 117.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 117:4 1000