Atlas › Modular forms › Weight 2 › Level 940 › 940.2.a › 940.2.a.b

Newform orbit 940.2.a.b

$q$ -expansion

$q - q^{3} + q^{5} + q^{7} - 2q^{9} - q^{11} - 5q^{13} - q^{15} + 2q^{17} - 5q^{19} - q^{21} - 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{17} + 9T_{19} + 11T_{23}$ is a root of

$x + 78$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 940$ )
5	$-1$
47	$+1$

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 940.b (Cremona's 940d). Its optimal curve 940d1 is

$y^2 = x^3 - x^2 - 20x + 40$

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	−1	−5	2	−5	−4	8	−7	−2	−10	2	−1	−4	−12	13	−8	−12	11	−12	−1	−2	10

$\operatorname{tr} a_n$ 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 940, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.642 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, 10, 20, 47, 94, 188, 235, 470, 940]: [0, 1, 2, 1, 3, 6, 5, 14, 27, 26, 76, 150] (certified at ell = 2147483137)
T = [(3, 1), (7, 3), (11, 5), (17, 7), (19, 9), (23, 11)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(940, 2)[1]   # 940.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 940:2 1000