Atlas › Modular forms › Weight 2 › Level 496 › 496.2.a › 496.2.a.e

Newform orbit 496.2.a.e

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23} + 9T_{29} + 10T_{37} + 11T_{41} + 12T_{43}$ is a root of

$x - 61$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 496$ )
31	$-1$

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 496.e (Cremona's 496b). Its optimal curve 496b1 is

$y^2 = x^3 - x^2 - 1$

of rank 0 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	2	1	3	2	−2	−6	−1	6	4	1	−2	7	−4	−8	8	−3	−6	12	−3	−10	12	−2	−16	−7

$\operatorname{tr} a_n$ 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 496, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.682 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, 31, 62, 124, 248, 496]: [0, 1, 2, 3, 4, 3, 10, 19, 36, 68] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8), (29, 9), (37, 10), (41, 11), (43, 12)]; h of degree 15 = 1 x 15 (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(496, 2)[4]   # 496.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 496:2 1000