Atlas › Modular forms › Weight 2 › Level 59 › 59.2.a › 59.2.a.a

Newform orbit 59.2.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 2x^{4} - 13x^{3} + 9x^{2} + 2x - 2$

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 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
$\operatorname{tr} a_p$	0	−2	2	2	−2	8	−1	6	−8	14	0	18	−10	−4	−20	−10	5	22	0	3	−8	10	6	10	−22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.098 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, 59]: [0, 6] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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, 2)[0]   # 59.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 59:2 1000