Atlas › Modular forms › Weight 2 › Level 660 › 660.2.a › 660.2.a.a

Newform orbit 660.2.a.a

$q$ -expansion

$q - q^{3} - q^{5} - 2q^{7} + q^{9} - q^{11} + 2q^{13} + q^{15} + 8q^{17} - 2q^{19} + 2q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{7} + 3T_{13} + 5T_{17} + 7T_{19} + 9T_{23} + 11T_{29}$ is a root of

$x - 102$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 660$ )
3	$+1$
5	$+1$
11	$+1$

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 660.a (Cremona's 660a). Its optimal curve 660a1 is

$y^2 = x^3 - x^2 - 21x - 54$

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	−1	−1	−2	−1	2	8	−2	8	0	0	2	0	6	8	6	−4	10	−12	8	10	−14	4	10	−18

$\operatorname{tr} a_n$ 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 660, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.631 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, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330, 660]: [0, 1, 1, 2, 1, 3, 3, 2, 5, 4, 6, 5, 10, 6, 9, 8, 18, 16, 22, 30, 28, 42, 80, 156] (certified at ell = 2147482921)
T = [(7, 1), (13, 3), (17, 5), (19, 7), (23, 9), (29, 11)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(660, 2)[0]   # 660.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 660:2 1000