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

Newform orbit 59.4.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 3q^{2} + 3q^{3} + 55q^{4} + 15q^{5} + 26q^{6} + 40q^{7} + 57q^{8} + 133q^{9} + 78q^{10} + 75q^{11} + 48q^{12} + 159q^{13} + 121q^{14} - 19q^{15} + 163q^{16} - 157q^{17} - 491q^{18} + 149q^{19} - 708q^{20} + 77q^{21} - 553q^{22} - 36q^{23} - 600q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 6x^{9} - 277x^{8} + 2120x^{7} + 19917x^{6} - 208716x^{5} - 56866x^{4} + 4919894x^{3} - 11511581x^{2} - 2961280x + 8601826$

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
$\operatorname{tr} a_p$	3	3	15	40	75	159	−157	149	−36	139	216	663	402	803	210	837	−590	2128	−1278	783	1204	−1488	−3303	−2592	1584

$\operatorname{tr} a_n$ 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)[2]   # 59.4.a.c

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

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