Atlas › Modular forms › Weight 2 › Level 730 › 730.2.a › 730.2.a.l

Newform orbit 730.2.a.l

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{2} + 60x + 68$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$-1$
73	$-1$

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
$\operatorname{tr} a_p$	−2	−1	2	−8	7	−1	−7	−12	−1	4	3	0	−9	−2	−2	−1	−16	3	−7	−26	2	−6	−12	−3	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.795 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, 5, 10, 73, 146, 365, 730]: [0, 1, 1, 3, 6, 20, 38, 114] (certified at ell = 2147482873)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 23 = 1 x 23 (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(730, 2)[11]   # 730.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 730:2 1000