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

Newform orbit 940.2.a.d

$q$ -expansion

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

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

$x + 41$

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.d (Cremona's 940e). Its optimal curve 940e1 is

$y^2 = x^3 - x^2 - 45x - 103$

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	2	1	2	4	1	0	−4	−3	0	−2	10	2	1	1	−12	5	−5	8	9	3	−15	2	−1	0

$\operatorname{tr} a_n$ 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)[3]   # 940.2.a.d

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

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