Atlas › Modular forms › Weight 2 › Level 425 › 425.2.a › 425.2.a.a

Newform orbit 425.2.a.a

$q$ -expansion

$q - q^{2} - 2q^{3} - q^{4} + 2q^{6} + 2q^{7} + 3q^{8} + q^{9} + 2q^{11} + 2q^{12} - 2q^{13} - 2q^{14} - q^{16} - q^{17} - q^{18} - 4q^{21} - 2q^{22} - 6q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

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

$x + 7$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 425$ )
17	$+1$

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 425.a (Cremona's 425d). Its optimal curve 425d1 is

$y^2 + xy = x^3 - 213x - 1208$

of rank 1 with torsion of order 2. 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
$\operatorname{tr} a_p$	−1	−2	0	2	2	−2	−1	0	−6	−6	−10	−2	10	−4	−12	10	8	−14	−8	−2	14	−14	−4	6	−2

$\operatorname{tr} a_n$ 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 425, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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, 17, 25, 85, 425]: [0, 1, 2, 3, 10, 46] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 26 = 1 x 26 (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(425, 2)[0]   # 425.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 425:2 1000