Atlas › Modular forms › Weight 2 › Level 980 › 980.2.a › 980.2.a.b

Newform orbit 980.2.a.b

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 2T_{11} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{13} + 7T_{17} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ is a root of

$x - 661$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 980$ )
5	$-1$
7	— ( $7^2 \mid 980$ )

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 980.b (Cremona's 980i). Its optimal curve 980i1 is

$y^2 = x^3 + 1568x - 72716$

of rank 0 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	0	−5	3	1	−6	6	−9	4	2	4	10	1	4	8	8	12	8	−2	13	4	−4	13

$\operatorname{tr} a_n$ 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 980, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.602 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, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980]: [0, 1, 2, 1, 1, 3, 4, 6, 7, 6, 5, 16, 16, 30, 31, 30, 88, 174] (certified at ell = 2147483353)
T = [(3, 1), (11, 2), (19, 3), (23, 4), (29, 5), (13, 6), (17, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 13 = 1 x 13 (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(980, 2)[1]   # 980.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 980:2 1000