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

Newform orbit 726.2.a.k

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ , whose minimal polynomial is

$x^{2} - 2x - 23804$

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 $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	−2	2	5	−1	0	2	8	2	2	7	−9	0	2	−14	8	5	5	6	−8	12	9	−7	1	18	−17

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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)[10]   # 726.2.a.k

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

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