Atlas › Modular forms › Weight 2 › Level 723 › 723.2.a › 723.2.a.f

Newform orbit 723.2.a.f

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 4q^{2} - 10q^{3} + 16q^{4} - 10q^{5} + 4q^{6} + 3q^{7} - 21q^{8} + 10q^{9} - q^{10} - 9q^{11} - 16q^{12} - 22q^{13} - 14q^{14} + 10q^{15} + 20q^{16} - 12q^{17} - 4q^{18} - 3q^{19} - 33q^{20} - 3q^{21} - 5q^{23} + 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ , whose minimal polynomial is

$x^{10} + 280x^{9} + 28394x^{8} + 1094661x^{7} - 6501778x^{6} - 1598022262x^{5} - 33021533003x^{4} + 156629151695x^{3} + 10777615458247x^{2} + 87416307008964x + 69051227879549$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
241	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

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
$\operatorname{tr} a_p$	−4	−10	−10	3	−9	−22	−12	−3	−5	−11	19	−33	−20	−17	−25	−27	−25	−10	−30	−18	−30	42	9	−36	−21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.402 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, 241, 723]: [0, 1, 20, 82] (certified at ell = 2147482801)
T = [(2, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 10 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(723, 2)[5]   # 723.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 723:2 1000