Atlas › Modular forms › Weight 2 › Level 716 › 716.2.a › 716.2.a.c

Newform orbit 716.2.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 3q^{3} - q^{5} + 10q^{7} + 13q^{9} + 7q^{11} + 7q^{15} + 7q^{17} + 28q^{19} + 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{10} - 2x^{9} - 62x^{8} + 163x^{7} + 1039x^{6} - 2846x^{5} - 6438x^{4} + 18346x^{3} + 9807x^{2} - 41229x + 19013$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 716$ )
179	$+1$

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	3	−1	10	7	0	7	28	3	−6	18	4	15	21	6	−18	16	−15	3	−3	13	28	−26	−9	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 179, 358, 716]: [0, 1, 2, 16, 47, 93] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(716, 2)[2]   # 716.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 716:2 1000