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

Newform orbit 59.4.a.a

Trace form

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

$2q - 4q^{2} - 2q^{3} + 2q^{4} + 6q^{5} + 4q^{6} - 22q^{7} - 12q^{8} - 52q^{9} - 32q^{10} - 48q^{11} - 2q^{12} - 32q^{13} + 24q^{14} - 6q^{15} + 18q^{16} - 84q^{17} + 104q^{18} - 2q^{19} + 86q^{20} + 22q^{21} + 116q^{22} + 164q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 6x + 4$

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
$\operatorname{tr} a_p$	−4	−2	6	−22	−48	−32	−84	−2	164	114	−124	−292	−418	−292	48	326	118	−268	920	656	232	−114	8	−852	−480

$\operatorname{tr} a_n$ 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)[0]   # 59.4.a.a

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

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