Atlas › Modular forms › Weight 2 › Level 569 › 569.2.a › 569.2.a.a

Newform orbit 569.2.a.a

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} + 11x^{15} + 23x^{14} - 126x^{13} - 477x^{12} + 349x^{11} + 2681x^{10} + 555x^{9} - 5924x^{8} - 3667x^{7} + 5121x^{6} + 5026x^{5} - 798x^{4} - 2141x^{3} - 618x^{2} + 32x + 23$

Atkin–Lehner signs

$p$	$w_p$
569	$+1$

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
$\operatorname{tr} a_p$	−3	−8	−6	−16	−7	−11	−8	−33	−10	−10	−61	−4	−12	−10	−9	8	−9	−42	1	−4	−3	−59	13	−3	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 569]: [0, 48] (certified at ell = 2147481337)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 5 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(569, 2)[0]   # 569.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 569:2 1000