Atlas › Modular forms › Weight 6 › Level 86 › 86.6.a › 86.6.a.b

Newform orbit 86.6.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 12q^{2} - 28q^{3} + 48q^{4} - 14q^{5} - 112q^{6} - 182q^{7} + 192q^{8} - 61q^{9} - 56q^{10} - 677q^{11} - 448q^{12} - 1239q^{13} - 728q^{14} - 1028q^{15} + 768q^{16} - 1285q^{17} - 244q^{18} - 2070q^{19} - 224q^{20} - 580q^{21} - 2708q^{22} - 5753q^{23} - 1792q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{3} + 42x^{2} - 1068x + 5256$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
43	$-1$

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
$\operatorname{tr} a_p$	12	−28	−14	−182	−677	−1239	−1285	−2070	−5753	−3604	−1545	19012	4731	5547	1540	14889	3440	−5012	−2315	−16266	7948	54940	99835	67734	−89703

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 2, 43, 86]: [1, 2, 20, 57] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 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(86, 6)[1]   # 86.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 86:6 1000