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

Newform orbit 559.2.a.c

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{7} + 11x^{6} - 21x^{5} - 387x^{4} + 116x^{3} + 3208x^{2} - 1400x - 4912$

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 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
$\operatorname{tr} a_p$	−2	−3	0	−1	−8	−7	−18	−2	−5	−18	−8	8	−29	−7	−2	−20	3	−1	18	4	9	−21	−5	−30	29

$\operatorname{tr} a_n$ 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)[2]   # 559.2.a.c

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

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