Atlas › Modular forms › Weight 4 › Level 59 › 59.4.a › 59.4.a.b

Newform orbit 59.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{2} - q^{3} - 7q^{4} - 31q^{5} - 26q^{6} - 8q^{7} - 3q^{8} + 23q^{9} + 10q^{10} + 13q^{11} - 22q^{12} - 95q^{13} + 13q^{14} - 61q^{15} - 39q^{16} + 169q^{17} + 37q^{18} - 107q^{19} + 134q^{20} - 47q^{21} + 15q^{22} - 96q^{23} + 282q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 17$

Atkin–Lehner signs

$p$	$w_p$
59	$-1$

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
$\operatorname{tr} a_p$	1	−1	−31	−8	13	−95	169	−107	−96	−307	56	−291	452	−35	26	−641	118	−604	−1326	289	48	−224	1963	688	356

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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 engine's checks

dim M^+ at levels [1, 59]: [1, 16] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 2 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(59, 4)[1]   # 59.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 59:4 1000