Atlas › Modular forms › Weight 4 › Level 116 › 116.4.a › 116.4.a.c

Newform orbit 116.4.a.c

Trace form

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

$3q - 10q^{3} - 20q^{5} + 8q^{7} + 93q^{9} + 86q^{11} + 124q^{13} + 54q^{15} + 14q^{17} + 88q^{19} + 280q^{21} + 68q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} + 30x^{2} + 66x - 36$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 116$ )
29	$-1$

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
$\operatorname{tr} a_p$	0	−10	−20	8	86	124	14	88	68	87	326	166	34	−946	234	−1144	488	450	−52	1196	2434	742	464	1986	−406

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 29, 58, 116]: [1, 2, 3, 9, 25, 48] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(116, 4)[2]   # 116.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 116:4 1000