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

Newform orbit 87.6.a.b

Trace form

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

$5q - 9q^{2} - 45q^{3} + 53q^{4} + 14q^{5} + 81q^{6} - 68q^{7} - 15q^{8} + 405q^{9} + 11q^{10} + 348q^{11} - 477q^{12} - 316q^{13} + 513q^{14} - 126q^{15} - 1231q^{16} - 1104q^{17} - 729q^{18} - 3344q^{19} - 3315q^{20} + 612q^{21} - 11588q^{22} - 2198q^{23} + 135q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 5x^{4} - 2990x^{3} - 54620x^{2} + 351880x + 8066816$

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
$\operatorname{tr} a_p$	−9	−45	14	−68	348	−316	−1104	−3344	−2198	−4205	18	−19332	990	−10530	−11882	5924	30784	−51432	−39792	114146	−70428	−117622	40660	−143048	−314356

$\operatorname{tr} a_n$ 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)[1]   # 87.6.a.b

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

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