Atlas › Modular forms › Weight 4 › Level 86 › 86.4.a › 86.4.a.d

Newform orbit 86.4.a.d

Trace form

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

$3q - 6q^{2} - 9q^{3} + 12q^{4} - 5q^{5} + 18q^{6} - 8q^{7} - 24q^{8} + 10q^{9} + 10q^{10} + 96q^{11} - 36q^{12} + 174q^{13} + 16q^{14} + 148q^{15} + 48q^{16} + 113q^{17} - 20q^{18} + 133q^{19} - 20q^{20} + 142q^{21} - 192q^{22} - 175q^{23} + 72q^{24} + O(q^{25})$

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

$x^{3} + 14x^{2} - 461x - 6990$

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 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
$\operatorname{tr} a_p$	−6	−9	−5	−8	96	174	113	133	−175	−97	215	−183	21	−129	−551	812	−524	1106	952	−312	−1226	709	−106	372	−1185

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 12, 35] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 2 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, 4)[3]   # 86.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 86:4 1000