Atlas › Modular forms › Weight 2 › Level 475 › 475.2.a › 475.2.a.b

Newform orbit 475.2.a.b

$q$ -expansion

$q + 2q^{3} - 2q^{4} + q^{7} + q^{9} + 3q^{11} - 4q^{12} + 4q^{13} + 4q^{16} + 3q^{17} + q^{19} + 2q^{21} + O(q^{25})$

Coefficient field

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

$x - 2$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 475$ )
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 475.b (Cremona's 475a). Its optimal curve 475a1 is

$y^2 + y = x^3 - x^2 + 17x - 7$

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	0	1	3	4	3	1	0	6	−4	−2	−6	1	3	−12	−6	−1	4	6	7	8	−12	12	−8

$\operatorname{tr} a_n$ 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 475, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 19, 25, 95, 475]: [0, 1, 2, 3, 12, 52] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 3 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(475, 2)[1]   # 475.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 475:2 1000