Atlas › Modular forms › Weight 2 › Level 566 › 566.2.a › 566.2.a.g

Newform orbit 566.2.a.g

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{9} - 3x^{8} - 286x^{7} + 1854x^{6} + 20897x^{5} - 211878x^{4} - 69568x^{3} + 6405804x^{2} - 23208940x + 24293744$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
283	$+1$

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 even 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
$\operatorname{tr} a_p$	9	3	0	4	11	−3	−6	−1	19	1	20	−3	−1	11	−4	−37	−3	−7	10	2	6	16	−20	−9	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 283, 566]: [0, 1, 24, 73] (certified at ell = 2147480197)
T = [(3, 1), (5, 3)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 4 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(566, 2)[6]   # 566.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 566:2 1000