Atlas › Modular forms › Weight 2 › Level 825 › 825.2.a › 825.2.a.b

Newform orbit 825.2.a.b

$q$ -expansion

$q - q^{3} - 2q^{4} + q^{7} + q^{9} - q^{11} + 2q^{12} + q^{13} + 4q^{16} + 6q^{17} - 7q^{19} - q^{21} - 6q^{23} + O(q^{25})$

Coefficient field

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

$x - 3$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	— ( $5^2 \mid 825$ )
11	$+1$

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 825.b (Cremona's 825a). Its optimal curve 825a1 is

$y^2 + y = x^3 - x^2 - 23x + 53$

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	0	1	−1	1	6	−7	−6	−6	−7	−2	−6	1	0	6	0	5	−5	−12	−14	−4	6	6	−17

$\operatorname{tr} a_n$ 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 825, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.45 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, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825]: [0, 1, 1, 2, 4, 3, 6, 8, 12, 28, 32, 124] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(825, 2)[1]   # 825.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 825:2 1000