Atlas › Modular forms › Weight 2 › Level 656 › 656.2.a › 656.2.a.a

Newform orbit 656.2.a.a

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ is a root of

$x - 10$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 656$ )
41	$-1$

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 656.a (Cremona's 656b). Its optimal curve 656b1 is

$y^2 = x^3 + x^2 - 12x - 20$

of rank 0 with torsion of order 2. 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	−2	2	2	−2	6	−6	2	0	6	8	10	1	0	6	−2	4	−2	10	2	−2	2	−12	10	−6

$\operatorname{tr} a_n$ 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 656, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.434 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, 8, 16, 41, 82, 164, 328, 656]: [0, 1, 2, 3, 4, 4, 12, 24, 46, 88] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 6 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(656, 2)[0]   # 656.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 656:2 1000