Atlas › Modular forms › Weight 2 › Level 620 › 620.2.a › 620.2.a.a

Newform orbit 620.2.a.a

$q$ -expansion

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

Coefficient field

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

$x + 9$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 620$ )
5	$-1$
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 620.a (Cremona's 620c). Its optimal curve 620c1 is

$y^2 = x^3 + 8x + 4$

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	−2	2	2	−3	−3	−4	−4	−1	−7	−3	−5	−6	1	−11	8	4	15	−13	−12	7	−18	10

$\operatorname{tr} a_n$ 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 620, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.358 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, 5, 10, 20, 31, 62, 124, 155, 310, 620]: [0, 1, 2, 1, 3, 6, 3, 10, 19, 18, 52, 102] (certified at ell = 2147482921)
T = [(3, 1), (7, 3)]; h of degree 10 = 1 x 10 (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(620, 2)[0]   # 620.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 620:2 1000