Atlas › Modular forms › Weight 6 › Level 89 › 89.6.a › 89.6.a.a

Newform orbit 89.6.a.a

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 9q^{2} - 9q^{3} + 127q^{4} - 39q^{5} - 177q^{6} - 540q^{7} - 417q^{8} + 758q^{9} - 1165q^{10} - 1450q^{11} - 681q^{12} - 778q^{13} - 452q^{14} - 2859q^{15} + 87q^{16} - 2135q^{17} - 4544q^{18} - 6085q^{19} - 2113q^{20} - 6532q^{21} - 5358q^{22} - 9461q^{23} - 24297q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} + 18x^{14} - 2166x^{13} - 24858x^{12} + 1783246x^{11} + 7679362x^{10} - 694004752x^{9} + 958301962x^{8} + 122270879657x^{7} - 571483618796x^{6} - 9075967223946x^{5} + 57345279796360x^{4} + 274859616415624x^{3} - 1951942201184432x^{2} - 3053797921584448x + 20565700276547968$

Atkin–Lehner signs

$p$	$w_p$
89	$+1$

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	−9	−39	−540	−1450	−778	−2135	−6085	−9461	−6128	−26917	−8222	−4168	−20479	5444	35139	−30468	39346	−41164	−48730	−110811	−292854	107492	−118815	−378853

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 89]: [1, 38] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 7 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(89, 6)[0]   # 89.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 89:6 1000