Atlas › Modular forms › Weight 2 › Level 975 › 975.2.a › 975.2.a.k

Newform orbit 975.2.a.k

$q$ -expansion

$q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{7} - 3q^{8} + q^{9} - q^{11} - q^{12} + q^{13} - 3q^{14} - q^{16} - 5q^{17} + q^{18} - 8q^{19} - 3q^{21} - q^{22} - 3q^{24} + O(q^{25})$

Coefficient field

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

$x + 37$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
5	— ( $5^2 \mid 975$ )
13	$-1$

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 975.k (Cremona's 975h). Its optimal curve 975h1 is

$y^2 + xy + y = x^3 - 46x - 127$

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	1	0	−3	−1	1	−5	−8	0	1	3	−8	−2	8	−11	−11	5	1	3	16	−4	12	−3	0	−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 975, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.111 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, 13, 15, 25, 39, 65, 75, 195, 325, 975]: [0, 1, 1, 1, 4, 3, 6, 8, 12, 32, 36, 144] (certified at ell = 2147482921)
T = [(2, 1), (7, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 38 = 1 x 38 (phi(ord eps) x dim S_k^new); 11 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(975, 2)[10]   # 975.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 975:2 1000