Atlas › Modular forms › Weight 2 › Level 960 › 960.2.a › 960.2.a.n

Newform orbit 960.2.a.n

$q$ -expansion

$q + q^{3} + q^{5} - 4q^{7} + q^{9} + 6q^{13} + q^{15} - 2q^{17} + 4q^{19} - 4q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 3171T_{7} + 4390T_{11} + 6706T_{13} + 9950T_{17} + 7802T_{19} + 8602T_{23} + 4058T_{29} + 9130T_{31}$ is a root of

$x - 132024$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 960$ )
3	$-1$
5	$-1$

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 960.n (Cremona's 960p). Its optimal curve 960p1 is

$y^2 = x^3 + x^2 + 15x + 15$

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	−4	0	6	−2	4	8	6	0	6	10	−4	−8	−10	0	−6	−4	0	−14	−16	12	2	2

$\operatorname{tr} a_n$ 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 960, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 3.651 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, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960]: [0, 1, 1, 2, 1, 3, 3, 3, 5, 4, 4, 6, 8, 10, 6, 10, 12, 18, 10, 16, 20, 32, 28, 36, 56, 52, 104, 200] (certified at ell = 2147483489)
T = [(7, 3171), (11, 4390), (13, 6706), (17, 9950), (19, 7802), (23, 8602), (29, 4058), (31, 9130)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 10 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(960, 2)[13]   # 960.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 960:2 1000