Atlas › Modular forms › Weight 2 › Level 920 › 920.2.a › 920.2.a.a

Newform orbit 920.2.a.a

$q$ -expansion

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{29} + 9T_{31} + 11T_{37}$ is a root of

$x + 91$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 920$ )
5	$-1$
23	$-1$

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 920.a (Cremona's 920b). Its optimal curve 920b1 is

$y^2 = x^3 - 187x + 991$

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	0	1	0	0	1	−3	3	−8	3	−2	−11	−14	−8	−4	−4	7	−9	0	4	−2	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 920, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.942 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, 8, 10, 20, 23, 40, 46, 92, 115, 184, 230, 460, 920]: [0, 1, 2, 1, 3, 3, 6, 3, 10, 8, 15, 14, 28, 40, 78, 152] (certified at ell = 2147483053)
T = [(3, 1), (7, 3), (11, 5), (29, 7), (31, 9), (37, 11)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 7 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(920, 2)[0]   # 920.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 920:2 1000