Atlas › Modular forms › Weight 2 › Level 608 › 608.2.a › 608.2.a.d

Newform orbit 608.2.a.d

$q$ -expansion

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

Coefficient field

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

$x + 67$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 608$ )
19	$-1$

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 608.d (Cremona's 608e). Its optimal curve 608e1 is

$y^2 = x^3 - 56x - 4848$

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	0	3	−5	−5	−4	−3	1	0	0	10	8	0	−5	5	−6	−10	−5	−10	10	−11	−10	0	−10	−12

$\operatorname{tr} a_n$ 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 608, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.397 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, 19, 32, 38, 76, 152, 304, 608]: [0, 1, 2, 3, 4, 2, 6, 7, 13, 24, 44, 84] (certified at ell = 2147482873)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (23, 7), (29, 8)]; h of degree 18 = 1 x 18 (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(608, 2)[3]   # 608.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 608:2 1000