Atlas › Modular forms › Weight 2 › Level 504 › 504.2.a › 504.2.a.b

Newform orbit 504.2.a.b

$q$ -expansion

$q - 2q^{5} - q^{7} - 2q^{13} - 6q^{17} - 4q^{19} + 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ is a root of

$x - 104$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 504$ )
3	— ( $3^2 \mid 504$ )
7	$+1$

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 504.b (Cremona's 504f). Its optimal curve 504f1 is

$y^2 = x^3 - 66x + 205$

of rank 1 with torsion of order 4. 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	0	−2	−1	0	−2	−6	−4	4	−6	−8	−10	10	12	8	−6	−4	−10	12	−4	2	8	−4	−6	10

$\operatorname{tr} a_n$ 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 504, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.005 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, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504]: [0, 1, 1, 2, 3, 1, 3, 2, 5, 4, 5, 4, 8, 7, 9, 12, 12, 10, 16, 22, 28, 40, 54, 104] (certified at ell = 2147483647)
T = [(5, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 8 = 1 x 8 (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(504, 2)[1]   # 504.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 504:2 1000