Atlas › Modular forms › Weight 2 › Level 916 › 916.2.a › 916.2.a.g

Newform orbit 916.2.a.g

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - q^{3} + 10q^{5} + 3q^{7} + 14q^{9} + 3q^{11} + 11q^{13} - q^{15} + 19q^{17} - 12q^{19} - 3q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ , whose minimal polynomial is

$x^{9} - 164x^{8} + 5388x^{7} + 316579x^{6} - 21519791x^{5} + 262400139x^{4} + 5703458214x^{3} - 148799161253x^{2} + 1084873795924x - 2573290549681$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 916$ )
229	$+1$

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	−1	10	3	3	11	19	−12	8	11	9	18	18	20	6	17	12	31	−12	27	12	−11	−13	42	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.451 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 229, 458, 916]: [0, 1, 2, 19, 59, 118] (certified at ell = 2147478649)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 19 = 1 x 19 (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(916, 2)[6]   # 916.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 916:2 1000