Atlas › Modular forms › Weight 2 › Level 726 › 726.2.a › 726.2.a.f

Newform orbit 726.2.a.f

$q$ -expansion

$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} + q^{8} + q^{9} - q^{10} - q^{12} + 3q^{13} - 4q^{14} + q^{15} + q^{16} - q^{17} + q^{18} - 8q^{19} - q^{20} + 4q^{21} - 8q^{23} - q^{24} + O(q^{25})$

Coefficient field

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

$x + 42$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
11	— ( $11^2 \mid 726$ )

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 726.f (Cremona's 726g). Its optimal curve 726g1 is

$y^2 + xy + y = x^3 + x^2 + 179x + 1475$

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	−1	−4	0	3	−1	−8	−8	−9	0	3	3	−8	12	11	0	−2	4	0	−6	−4	16	7	−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 726, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.52 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, 6, 11, 22, 33, 66, 121, 242, 363, 726]: [0, 1, 1, 3, 2, 5, 6, 16, 12, 35, 46, 136] (certified at ell = 2147482921)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 6 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(726, 2)[5]   # 726.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 726:2 1000