Atlas › Modular forms › Weight 2 › Level 316 › 316.2.a › 316.2.a.d

Newform orbit 316.2.a.d

Trace form

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

$2q + 4q^{3} + 3q^{5} + 2q^{9} + 3q^{11} + q^{13} + 6q^{15} - 6q^{17} - 3q^{19} + 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{2} + 19x - 5373$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 316$ )
79	$+1$

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	4	3	0	3	1	−6	−3	3	−6	7	−4	6	−4	12	0	6	−6	−5	18	−15	−2	0	−15	−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 0.324 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, 79, 158, 316]: [0, 1, 2, 7, 22, 43] (certified at ell = 2147483053)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 6 = 1 x 6 (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(316, 2)[3]   # 316.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 316:2 1000