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

Newform orbit 569.2.a.b

Trace form

The 31 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$31q + 8q^{3} + 44q^{4} + 4q^{5} + 5q^{6} + 20q^{7} + 37q^{9} + 11q^{10} + q^{11} + 16q^{12} + 11q^{13} + 4q^{14} + 16q^{15} + 66q^{16} - 2q^{17} - 16q^{18} + 39q^{19} - 3q^{20} - 8q^{22} + 12q^{23} + q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 31. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{31} - 8x^{30} - 91x^{29} + 859x^{28} + 3042x^{27} - 38285x^{26} - 41280x^{25} + 947677x^{24} - 76041x^{23} - 14575046x^{22} + 10619419x^{21} + 146936075x^{20} - 165342105x^{19} - 996616116x^{18} + 1373964196x^{17} + 4594601782x^{16} - 7085342868x^{15} - 14386678131x^{14} + 23657773884x^{13} + 30297928436x^{12} - 51499857359x^{11} - 42116783966x^{10} + 71978930511x^{9} + 37436276086x^{8} - 62043100066x^{7} - 20039488429x^{6} + 30310941768x^{5} + 5632082696x^{4} - 6933119138x^{3} - 536973731x^{2} + 406887583x - 5327417$

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 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
$\operatorname{tr} a_p$	0	8	4	20	1	11	−2	39	12	−10	89	0	−6	26	33	−12	31	36	5	24	−1	57	−11	−25	4

$\operatorname{tr} a_n$ 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)[1]   # 569.2.a.b

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

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