Atlas › Modular forms › Weight 2 › Level 280 › 280.2.a › 280.2.a.b

Newform orbit 280.2.a.b

$q$ -expansion

$q - q^{3} - q^{5} - q^{7} - 2q^{9} - 5q^{11} + q^{13} + q^{15} + 3q^{17} - 6q^{19} + q^{21} - 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{11}$ is a root of

$x + 16$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 280$ )
5	$+1$
7	$+1$

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 280.b (Cremona's 280a). Its optimal curve 280a1 is

$y^2 = x^3 - x^2 - x + 5$

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	−1	−5	1	3	−6	−6	−9	0	6	8	6	3	−12	8	−4	−4	8	10	−3	−12	−16	7

$\operatorname{tr} a_n$ 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 280, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.348 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, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, 280]: [0, 1, 2, 1, 1, 3, 3, 4, 6, 7, 6, 10, 12, 16, 30, 56] (certified at ell = 2147483629)
T = [(3, 1), (11, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 1 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(280, 2)[1]   # 280.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 280:2 1000