Atlas › Modular forms › Weight 2 › Level 925 › 925.2.a › 925.2.a.e

Newform orbit 925.2.a.e

$q$ -expansion

$q + 2q^{2} + 3q^{3} + 2q^{4} + 6q^{6} + q^{7} + 6q^{9} - 5q^{11} + 6q^{12} + 2q^{13} + 2q^{14} - 4q^{16} + 12q^{18} + 3q^{21} - 10q^{22} - 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11}$ is a root of

$x + 19$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 925$ )
37	$-1$

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 925.e (Cremona's 925e). Its optimal curve 925e1 is

$y^2 + y = x^3 - 25x + 31$

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	3	0	1	−5	2	0	0	−2	6	−4	1	−9	−2	9	−1	8	−8	−8	9	1	4	15	4	−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 925, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.457 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, 5, 25, 37, 185, 925]: [0, 1, 3, 3, 20, 96] (certified at ell = 2147482801)
T = [(2, 1), (3, 3), (7, 5), (11, 7)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 13 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(925, 2)[4]   # 925.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 925:2 1000