Atlas › Modular forms › Weight 2 › Level 559 › 559.2.a › 559.2.a.e

Newform orbit 559.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 2q^{2} + q^{3} + 18q^{4} - 2q^{5} + 7q^{6} + q^{7} + 3q^{8} + 36q^{9} - 2q^{10} + 16q^{11} - 15q^{13} + 5q^{14} + q^{15} + 16q^{16} + 22q^{17} + 12q^{18} - 4q^{19} - 6q^{20} - 4q^{21} - 18q^{22} + 19q^{23} + 28q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{15} - 5x^{14} - 397x^{13} + 1765x^{12} + 60034x^{11} - 214542x^{10} - 4569040x^{9} + 11149494x^{8} + 190969235x^{7} - 224037851x^{6} - 4298003117x^{5} - 290397285x^{4} + 44225707084x^{3} + 48788623176x^{2} - 101872914216x - 131870369552$

Atkin–Lehner signs

$p$	$w_p$
13	$+1$
43	$-1$

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
$\operatorname{tr} a_p$	2	1	−2	1	16	−15	22	−4	19	36	−4	6	31	15	−14	28	−5	−15	2	4	23	3	3	28	−27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 13, 43, 559]: [0, 1, 4, 52] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 43 = 1 x 43 (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(559, 2)[4]   # 559.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 559:2 1000