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

Newform orbit 316.2.a.a

$q$ -expansion

$q - 3q^{3} + q^{5} + q^{7} + 6q^{9} - 6q^{11} - q^{13} - 3q^{15} - 4q^{17} - 6q^{19} - 3q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ is a root of

$x + 73$

Atkin–Lehner signs

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

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Elliptic curve

By modularity this newform is the one attached to the isogeny class 316.a (Cremona's 316b). Its optimal curve 316b1 is

$y^2 = x^3 - 7x - 2$

of rank 1 with torsion of order 1. Its $a_p$ , computed by Sagebrush's point-counting engine, equal the newform's for every prime $p < 1000$ .

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	1	−6	−1	−4	−6	2	−8	4	4	−6	4	−3	14	−9	6	−10	5	6	1	4	1	−11

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

Sato–Tate

The normalized $a_p / 2p^{1/2}$ for the primes $p < 1000$ not dividing 316, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 316.2.a.a

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

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