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

Newform orbit 559.2.a.d

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{14} - 10x^{13} - 261x^{12} + 2778x^{11} + 24278x^{10} - 291168x^{9} - 929872x^{8} + 14489954x^{7} + 8927703x^{6} - 352350128x^{5} + 273372515x^{4} + 3790500438x^{3} - 5923339868x^{2} - 12916545368x + 25080071216$

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
$\operatorname{tr} a_p$	7	1	12	1	10	14	8	10	−5	24	−6	−2	39	−14	12	−18	19	5	−22	22	−3	−5	−11	64	−3

$\operatorname{tr} a_n$ 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)[3]   # 559.2.a.d

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

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