Atlas › Modular forms › Weight 2 › Level 675 › 675.2.a › 675.2.a.b

Newform orbit 675.2.a.b

$q$ -expansion

$q - q^{2} - q^{4} + 3q^{8} - 5q^{11} + 5q^{13} - q^{16} - 4q^{17} - 2q^{19} + 5q^{22} + 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ is a root of

$x + 313$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 675$ )
5	— ( $5^2 \mid 675$ )

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 675.b (Cremona's 675b). Its optimal curve 675b1 is

$y^2 + xy + y = x^3 - x^2 - 5x + 2$

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
$\operatorname{tr} a_p$	−1	0	0	0	−5	5	−4	−2	3	−10	6	−5	−10	−10	5	2	−5	−11	0	5	−10	12	−12	0	−5

$\operatorname{tr} a_n$ 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 675, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.887 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, 9, 15, 25, 27, 45, 75, 135, 225, 675]: [0, 1, 1, 2, 4, 3, 4, 8, 12, 20, 32, 92] (certified at ell = 2147482801)
T = [(2, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 25 = 1 x 25 (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(675, 2)[1]   # 675.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 675:2 1000