Atlas › Modular forms › Weight 2 › Level 693 › 693.2.a › 693.2.a.c

Newform orbit 693.2.a.c

$q$ -expansion

$q - 2q^{4} + q^{5} - q^{7} + q^{11} - 4q^{13} + 4q^{16} - 2q^{17} - 6q^{19} - 2q^{20} + 5q^{23} + O(q^{25})$

Coefficient field

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

$x + 298$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 693$ )
7	$+1$
11	$-1$

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 693.c (Cremona's 693b). Its optimal curve 693b1 is

$y^2 + y = x^3 + 18x - 7$

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
$\operatorname{tr} a_p$	0	0	1	−1	1	−4	−2	−6	5	−10	1	−5	2	−8	−8	6	−3	−2	−3	−1	10	6	−12	15	−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 693, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.922 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, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693]: [0, 1, 1, 2, 2, 4, 6, 10, 10, 14, 36, 100] (certified at ell = 2147482951)
T = [(2, 1), (5, 2), (17, 3), (19, 4), (23, 5), (13, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 24 = 1 x 24 (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(693, 2)[2]   # 693.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 693:2 1000