Atlas › Modular forms › Weight 2 › Level 931 › 931.2.a › 931.2.a.b

Newform orbit 931.2.a.b

$q$ -expansion

$q + 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} - 4q^{6} + q^{9} - 6q^{10} + 4q^{11} - 4q^{12} + 6q^{13} + 6q^{15} - 4q^{16} + 7q^{17} + 2q^{18} - q^{19} - 6q^{20} + 8q^{22} + 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3}$ is a root of

$x + 4$

Atkin–Lehner signs

$p$	$w_p$
7	— ( $7^2 \mid 931$ )
19	$+1$

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 931.b (Cremona's 931c). Its optimal curve 931c1 is

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

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
$\operatorname{tr} a_p$	2	−2	−3	0	4	6	7	−1	3	0	0	−2	4	5	−4	6	−8	2	8	−2	−10	12	3	8	−4

$\operatorname{tr} a_n$ 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 931, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.312 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, 7, 19, 49, 133, 931]: [0, 1, 2, 5, 14, 94] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 62 = 1 x 62 (phi(ord eps) x dim S_k^new); 8 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(931, 2)[1]   # 931.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 931:2 1000