Atlas › Modular forms › Weight 6 › Level 59 › 59.6.a › 59.6.a.b

Newform orbit 59.6.a.b

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 3q^{2} + 18q^{3} + 303q^{4} + 128q^{5} + 14q^{6} + 282q^{7} + 249q^{8} + 1621q^{9} + 598q^{10} + 34q^{11} + 864q^{12} + 1790q^{13} + 1087q^{14} + 2900q^{15} + 12155q^{16} + 6130q^{17} + 10417q^{18} + 5342q^{19} + 11772q^{20} + 7976q^{21} + 7657q^{22} + 1552q^{23} + 612q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 21x^{14} - 2818x^{13} + 45592x^{12} + 3256636x^{11} - 36054946x^{10} - 1958286259x^{9} + 12683401959x^{8} + 642288327357x^{7} - 1891527333549x^{6} - 113103468686273x^{5} + 37981626453747x^{4} + 9749892991189600x^{3} + 16196441633250386x^{2} - 306086191132525860x - 1023863300749578656$

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	18	128	282	34	1790	6130	5342	1552	7476	−3468	22158	4670	20134	−23192	22148	52215	156158	166884	−3954	32606	352558	153906	20806	45926

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 26] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 5 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, 6)[1]   # 59.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 59:6 1000