Atlas › Modular forms › Weight 2 › Level 968 › 968.2.a › 968.2.a.d

Newform orbit 968.2.a.d

$q$ -expansion

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

Coefficient field

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

$x + 106$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 968$ )
11	— ( $11^2 \mid 968$ )

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 968.d (Cremona's 968a). Its optimal curve 968a1 is

$y^2 = x^3 + x^2 + 15x - 13$

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	1	1	−4	0	−4	−4	4	−3	−8	9	−5	12	−8	4	−10	7	8	11	−9	−4	−8	0	−1	1

$\operatorname{tr} a_n$ 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 968, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.771 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, 11, 22, 44, 88, 121, 242, 484, 968]: [0, 1, 2, 3, 2, 5, 9, 16, 12, 35, 69, 136] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 8 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(968, 2)[3]   # 968.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 968:2 1000