Atlas › Modular forms › Weight 2 › Level 722 › 722.2.a › 722.2.a.a

Newform orbit 722.2.a.a

$q$ -expansion

$q - q^{2} - 3q^{3} + q^{4} + 2q^{5} + 3q^{6} - 3q^{7} - q^{8} + 6q^{9} - 2q^{10} - 2q^{11} - 3q^{12} + 3q^{13} + 3q^{14} - 6q^{15} + q^{16} - q^{17} - 6q^{18} + 2q^{20} + 9q^{21} + 2q^{22} + 5q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ is a root of

$x + 26$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
19	— ( $19^2 \mid 722$ )

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 722.a (Cremona's 722b). Its optimal curve 722b1 is

$y^2 + xy = x^3 - x^2 - x - 11$

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	−3	2	−3	−2	3	−1	0	5	3	6	−6	−12	−10	−8	3	−3	0	−15	0	−11	12	2	−6	−12

$\operatorname{tr} a_n$ 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 722, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.403 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, 19, 38, 361, 722]: [0, 1, 2, 7, 32, 97] (certified at ell = 2147483323)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 7 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(722, 2)[0]   # 722.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 722:2 1000