Atlas › Modular forms › Weight 2 › Level 684 › 684.2.a › 684.2.a.c

Newform orbit 684.2.a.c

$q$ -expansion

$q + 3q^{5} + q^{7} + 5q^{11} - 6q^{13} + 5q^{17} + q^{19} - 4q^{23} + O(q^{25})$

Coefficient field

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

$x - 23$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 684$ )
3	— ( $3^2 \mid 684$ )
19	$-1$

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 684.c (Cremona's 684c). Its optimal curve 684c1 is

$y^2 = x^3 + 24x - 268$

of rank 0 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	0	3	1	5	−6	5	1	−4	−6	6	−8	8	9	−1	−2	8	11	0	4	−11	−8	4	−10	−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 684, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.507 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, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, 342, 684]: [0, 1, 1, 2, 3, 2, 5, 5, 2, 9, 7, 8, 13, 24, 22, 46, 64, 126] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (23, 5), (11, 6), (29, 7), (31, 8)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(684, 2)[2]   # 684.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 684:2 1000