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

Newform orbit 87.6.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 3q^{2} + 36q^{3} + 49q^{4} - 136q^{5} - 27q^{6} - 28q^{7} - 807q^{8} + 324q^{9} - 311q^{10} - 872q^{11} + 441q^{12} - 1090q^{13} - 675q^{14} - 1224q^{15} + 3105q^{16} - 2602q^{17} - 243q^{18} + 1436q^{19} - 869q^{20} - 252q^{21} - 1668q^{22} - 4886q^{23} - 7263q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} + 139x^{3} + 4704x^{2} - 8612x - 1295008$

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 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
$\operatorname{tr} a_p$	−3	36	−136	−28	−872	−1090	−2602	1436	−4886	3364	−10638	−4610	−20720	−15354	9894	−49774	−52280	4034	−21460	18042	149030	68218	−143684	−45618	5086

$\operatorname{tr} a_n$ 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)[0]   # 87.6.a.a

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

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