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

Newform orbit 566.2.a.f

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} - 15x^{4} - 118x^{3} + 2892x^{2} - 12560x + 4236$

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
$\operatorname{tr} a_p$	−5	−3	6	11	0	−2	−6	9	−18	8	−16	25	9	15	0	19	−14	26	−3	4	7	14	−18	28	5

$\operatorname{tr} a_n$ 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)[5]   # 566.2.a.f

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

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