Atlas › Modular forms › Weight 6 › Level 87 › 87.6.a › 87.6.a.d

Newform orbit 87.6.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - q^{2} - 72q^{3} + 149q^{4} + 14q^{5} + 9q^{6} + 128q^{7} - 258q^{8} + 648q^{9} - 299q^{10} - 388q^{11} - 1341q^{12} + 1520q^{13} + 664q^{14} - 126q^{15} + 6561q^{16} + 2604q^{17} - 81q^{18} + 806q^{19} + 2871q^{20} - 1152q^{21} + 17645q^{22} + 4108q^{23} + 2322q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} - 13x^{7} - 20128x^{6} + 91711x^{5} + 126406235x^{4} + 1118232378x^{3} - 241243275704x^{2} - 7031974605920x - 51959124898560$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
29	$-1$

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
$\operatorname{tr} a_p$	−1	−72	14	128	−388	1520	2604	806	4108	6728	10076	4148	−24760	11770	−15306	65632	−43360	−24484	98248	17704	190280	8284	−162352	161536	506164

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 3, 29, 87]: [1, 3, 13, 52] (certified at ell = 2147483549)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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(87, 6)[3]   # 87.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 87:6 1000