# Properties

 Label 2646.2.d.f Level $2646$ Weight $2$ Character orbit 2646.d Analytic conductor $21.128$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{48}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{48}^{12} q^{2} - q^{4} + ( -2 \zeta_{48}^{4} - \zeta_{48}^{9} + \zeta_{48}^{12} + \zeta_{48}^{15} ) q^{5} -\zeta_{48}^{12} q^{8} +O(q^{10})$$ $$q + \zeta_{48}^{12} q^{2} - q^{4} + ( -2 \zeta_{48}^{4} - \zeta_{48}^{9} + \zeta_{48}^{12} + \zeta_{48}^{15} ) q^{5} -\zeta_{48}^{12} q^{8} + ( 1 - \zeta_{48}^{3} + \zeta_{48}^{5} - 2 \zeta_{48}^{8} - \zeta_{48}^{13} ) q^{10} + ( 2 \zeta_{48} - \zeta_{48}^{3} + \zeta_{48}^{5} - 2 \zeta_{48}^{7} - \zeta_{48}^{9} + 2 \zeta_{48}^{11} - \zeta_{48}^{12} + \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{11} + ( \zeta_{48}^{2} - 2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} - \zeta_{48}^{6} + \zeta_{48}^{9} + \zeta_{48}^{10} - 2 \zeta_{48}^{13} + 2 \zeta_{48}^{14} + \zeta_{48}^{15} ) q^{13} + q^{16} + ( \zeta_{48}^{2} + \zeta_{48}^{6} + \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{17} + ( -1 - 2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{8} - 2 \zeta_{48}^{9} - 2 \zeta_{48}^{10} - 4 \zeta_{48}^{14} - 2 \zeta_{48}^{15} ) q^{19} + ( 2 \zeta_{48}^{4} + \zeta_{48}^{9} - \zeta_{48}^{12} - \zeta_{48}^{15} ) q^{20} + ( 1 - 2 \zeta_{48} + \zeta_{48}^{3} + \zeta_{48}^{5} - 2 \zeta_{48}^{7} + \zeta_{48}^{9} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{22} + ( -\zeta_{48}^{2} + \zeta_{48}^{3} - \zeta_{48}^{5} + \zeta_{48}^{6} + \zeta_{48}^{10} - 2 \zeta_{48}^{11} - 3 \zeta_{48}^{12} - \zeta_{48}^{13} ) q^{23} + ( -\zeta_{48}^{2} + 2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} - \zeta_{48}^{6} + \zeta_{48}^{10} - 4 \zeta_{48}^{11} + 2 \zeta_{48}^{13} ) q^{25} + ( -\zeta_{48}^{2} - \zeta_{48}^{3} - \zeta_{48}^{5} - \zeta_{48}^{6} + 2 \zeta_{48}^{9} - \zeta_{48}^{10} + \zeta_{48}^{13} + 2 \zeta_{48}^{14} - 2 \zeta_{48}^{15} ) q^{26} + ( -4 \zeta_{48} - \zeta_{48}^{2} - \zeta_{48}^{3} + \zeta_{48}^{5} + \zeta_{48}^{6} + 4 \zeta_{48}^{7} + 2 \zeta_{48}^{9} + \zeta_{48}^{10} + 2 \zeta_{48}^{11} + 2 \zeta_{48}^{12} + \zeta_{48}^{13} - 2 \zeta_{48}^{15} ) q^{29} + ( 1 - \zeta_{48}^{2} - \zeta_{48}^{3} + \zeta_{48}^{5} + \zeta_{48}^{6} - 2 \zeta_{48}^{8} - 2 \zeta_{48}^{9} - \zeta_{48}^{10} - \zeta_{48}^{13} - 2 \zeta_{48}^{14} - 2 \zeta_{48}^{15} ) q^{31} + \zeta_{48}^{12} q^{32} + ( \zeta_{48}^{2} - \zeta_{48}^{6} + \zeta_{48}^{10} + 2 \zeta_{48}^{14} ) q^{34} + ( 1 + 4 \zeta_{48} - 3 \zeta_{48}^{2} + \zeta_{48}^{3} + \zeta_{48}^{5} - 3 \zeta_{48}^{6} + 4 \zeta_{48}^{7} - 2 \zeta_{48}^{9} + 3 \zeta_{48}^{10} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} - 2 \zeta_{48}^{15} ) q^{37} + ( 2 \zeta_{48}^{2} + 2 \zeta_{48}^{3} - 2 \zeta_{48}^{4} + 2 \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{10} + \zeta_{48}^{12} - 2 \zeta_{48}^{13} - 4 \zeta_{48}^{14} ) q^{38} + ( -1 + \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{8} + \zeta_{48}^{13} ) q^{40} + ( \zeta_{48}^{2} - \zeta_{48}^{3} + 2 \zeta_{48}^{4} - \zeta_{48}^{5} + \zeta_{48}^{6} - 3 \zeta_{48}^{9} + \zeta_{48}^{10} - \zeta_{48}^{12} + \zeta_{48}^{13} - 2 \zeta_{48}^{14} + 3 \zeta_{48}^{15} ) q^{41} + ( -2 + 2 \zeta_{48} + 2 \zeta_{48}^{2} + \zeta_{48}^{3} + \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{7} - \zeta_{48}^{9} - 2 \zeta_{48}^{10} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{43} + ( -2 \zeta_{48} + \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{7} + \zeta_{48}^{9} - 2 \zeta_{48}^{11} + \zeta_{48}^{12} - \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{44} + ( 3 + 2 \zeta_{48} - \zeta_{48}^{2} - \zeta_{48}^{6} + 2 \zeta_{48}^{7} - \zeta_{48}^{9} + \zeta_{48}^{10} - \zeta_{48}^{15} ) q^{46} + ( -2 \zeta_{48}^{2} + 2 \zeta_{48}^{3} - 4 \zeta_{48}^{4} + 2 \zeta_{48}^{5} - 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} + 2 \zeta_{48}^{12} - 2 \zeta_{48}^{13} + 4 \zeta_{48}^{14} ) q^{47} + ( -4 \zeta_{48} + \zeta_{48}^{2} - \zeta_{48}^{6} + 4 \zeta_{48}^{7} + 2 \zeta_{48}^{9} - \zeta_{48}^{10} - 2 \zeta_{48}^{15} ) q^{50} + ( -\zeta_{48}^{2} + 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} + \zeta_{48}^{6} - \zeta_{48}^{9} - \zeta_{48}^{10} + 2 \zeta_{48}^{13} - 2 \zeta_{48}^{14} - \zeta_{48}^{15} ) q^{52} + ( 2 \zeta_{48} - 3 \zeta_{48}^{3} + 3 \zeta_{48}^{5} - 2 \zeta_{48}^{7} - \zeta_{48}^{9} + 6 \zeta_{48}^{11} + 2 \zeta_{48}^{12} + 3 \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{53} + ( -3 - 2 \zeta_{48}^{2} + 4 \zeta_{48}^{3} - 4 \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 6 \zeta_{48}^{8} - 3 \zeta_{48}^{9} - 2 \zeta_{48}^{10} + 4 \zeta_{48}^{13} - 4 \zeta_{48}^{14} - 3 \zeta_{48}^{15} ) q^{55} + ( -2 - 2 \zeta_{48} - \zeta_{48}^{2} - 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} - \zeta_{48}^{6} - 2 \zeta_{48}^{7} + \zeta_{48}^{9} + \zeta_{48}^{10} + 4 \zeta_{48}^{11} - 2 \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{58} + ( 2 \zeta_{48}^{2} - 5 \zeta_{48}^{3} - 4 \zeta_{48}^{4} - 5 \zeta_{48}^{5} + 2 \zeta_{48}^{6} - \zeta_{48}^{9} + 2 \zeta_{48}^{10} + 2 \zeta_{48}^{12} + 5 \zeta_{48}^{13} - 4 \zeta_{48}^{14} + \zeta_{48}^{15} ) q^{59} + ( -2 \zeta_{48}^{2} + 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{9} - 2 \zeta_{48}^{10} + 2 \zeta_{48}^{13} - 4 \zeta_{48}^{14} + 2 \zeta_{48}^{15} ) q^{61} + ( \zeta_{48}^{2} + 2 \zeta_{48}^{3} + 2 \zeta_{48}^{4} + 2 \zeta_{48}^{5} + \zeta_{48}^{6} + \zeta_{48}^{9} + \zeta_{48}^{10} - \zeta_{48}^{12} - 2 \zeta_{48}^{13} - 2 \zeta_{48}^{14} - \zeta_{48}^{15} ) q^{62} - q^{64} + ( -6 \zeta_{48} + 6 \zeta_{48}^{2} + 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} - 6 \zeta_{48}^{6} + 6 \zeta_{48}^{7} + 3 \zeta_{48}^{9} - 6 \zeta_{48}^{10} - 4 \zeta_{48}^{11} + 4 \zeta_{48}^{12} - 2 \zeta_{48}^{13} - 3 \zeta_{48}^{15} ) q^{65} + ( -2 + 4 \zeta_{48} + \zeta_{48}^{3} + \zeta_{48}^{5} + 4 \zeta_{48}^{7} - 2 \zeta_{48}^{9} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} - 2 \zeta_{48}^{15} ) q^{67} + ( -\zeta_{48}^{2} - \zeta_{48}^{6} - \zeta_{48}^{10} + 2 \zeta_{48}^{14} ) q^{68} + ( -6 \zeta_{48} + 2 \zeta_{48}^{2} - 2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} - 2 \zeta_{48}^{6} + 6 \zeta_{48}^{7} + 3 \zeta_{48}^{9} - 2 \zeta_{48}^{10} + 4 \zeta_{48}^{11} - \zeta_{48}^{12} + 2 \zeta_{48}^{13} - 3 \zeta_{48}^{15} ) q^{71} + ( -2 - \zeta_{48}^{3} + \zeta_{48}^{5} + 4 \zeta_{48}^{8} + 3 \zeta_{48}^{9} - \zeta_{48}^{13} + 3 \zeta_{48}^{15} ) q^{73} + ( -2 \zeta_{48} + 3 \zeta_{48}^{2} - 2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} - 3 \zeta_{48}^{6} + 2 \zeta_{48}^{7} + \zeta_{48}^{9} - 3 \zeta_{48}^{10} + 4 \zeta_{48}^{11} + \zeta_{48}^{12} + 2 \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{74} + ( 1 + 2 \zeta_{48}^{2} - 2 \zeta_{48}^{6} - 2 \zeta_{48}^{8} + 2 \zeta_{48}^{9} + 2 \zeta_{48}^{10} + 4 \zeta_{48}^{14} + 2 \zeta_{48}^{15} ) q^{76} + ( 6 \zeta_{48} + 4 \zeta_{48}^{2} + \zeta_{48}^{3} + \zeta_{48}^{5} + 4 \zeta_{48}^{6} + 6 \zeta_{48}^{7} - 3 \zeta_{48}^{9} - 4 \zeta_{48}^{10} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} - 3 \zeta_{48}^{15} ) q^{79} + ( -2 \zeta_{48}^{4} - \zeta_{48}^{9} + \zeta_{48}^{12} + \zeta_{48}^{15} ) q^{80} + ( -1 + \zeta_{48}^{2} - 3 \zeta_{48}^{3} + 3 \zeta_{48}^{5} - \zeta_{48}^{6} + 2 \zeta_{48}^{8} - \zeta_{48}^{9} + \zeta_{48}^{10} - 3 \zeta_{48}^{13} + 2 \zeta_{48}^{14} - \zeta_{48}^{15} ) q^{82} + ( 4 \zeta_{48}^{2} - \zeta_{48}^{3} - \zeta_{48}^{5} + 4 \zeta_{48}^{6} + 4 \zeta_{48}^{9} + 4 \zeta_{48}^{10} + \zeta_{48}^{13} - 8 \zeta_{48}^{14} - 4 \zeta_{48}^{15} ) q^{83} + ( -2 \zeta_{48} - 3 \zeta_{48}^{2} + \zeta_{48}^{3} + \zeta_{48}^{5} - 3 \zeta_{48}^{6} - 2 \zeta_{48}^{7} + \zeta_{48}^{9} + 3 \zeta_{48}^{10} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{85} + ( -2 \zeta_{48} - 2 \zeta_{48}^{2} - \zeta_{48}^{3} + \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{7} + \zeta_{48}^{9} + 2 \zeta_{48}^{10} + 2 \zeta_{48}^{11} - 2 \zeta_{48}^{12} + \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{86} + ( -1 + 2 \zeta_{48} - \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{7} - \zeta_{48}^{9} + 2 \zeta_{48}^{11} - \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{88} + ( \zeta_{48}^{2} - \zeta_{48}^{3} - 10 \zeta_{48}^{4} - \zeta_{48}^{5} + \zeta_{48}^{6} - \zeta_{48}^{9} + \zeta_{48}^{10} + 5 \zeta_{48}^{12} + \zeta_{48}^{13} - 2 \zeta_{48}^{14} + \zeta_{48}^{15} ) q^{89} + ( \zeta_{48}^{2} - \zeta_{48}^{3} + \zeta_{48}^{5} - \zeta_{48}^{6} - \zeta_{48}^{10} + 2 \zeta_{48}^{11} + 3 \zeta_{48}^{12} + \zeta_{48}^{13} ) q^{92} + ( 2 - 2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} - 4 \zeta_{48}^{8} + 2 \zeta_{48}^{9} - 2 \zeta_{48}^{10} - 4 \zeta_{48}^{14} + 2 \zeta_{48}^{15} ) q^{94} + ( 6 \zeta_{48} - 8 \zeta_{48}^{2} - 4 \zeta_{48}^{3} + 4 \zeta_{48}^{5} + 8 \zeta_{48}^{6} - 6 \zeta_{48}^{7} - 3 \zeta_{48}^{9} + 8 \zeta_{48}^{10} + 8 \zeta_{48}^{11} - 3 \zeta_{48}^{12} + 4 \zeta_{48}^{13} + 3 \zeta_{48}^{15} ) q^{95} + ( -4 - 2 \zeta_{48}^{2} + \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 8 \zeta_{48}^{8} + 3 \zeta_{48}^{9} - 2 \zeta_{48}^{10} + \zeta_{48}^{13} - 4 \zeta_{48}^{14} + 3 \zeta_{48}^{15} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{4} + O(q^{10})$$ $$16q - 16q^{4} + 16q^{16} + 16q^{22} + 16q^{37} - 32q^{43} + 48q^{46} - 32q^{58} - 16q^{64} - 32q^{67} - 16q^{88} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2645.1
 0.130526 + 0.991445i 0.991445 − 0.130526i −0.991445 + 0.130526i 0.793353 − 0.608761i −0.130526 − 0.991445i −0.608761 − 0.793353i 0.608761 + 0.793353i −0.793353 + 0.608761i 0.130526 − 0.991445i 0.991445 + 0.130526i −0.991445 − 0.130526i 0.793353 + 0.608761i −0.130526 + 0.991445i −0.608761 + 0.793353i 0.608761 − 0.793353i −0.793353 − 0.608761i
1.00000i 0 −1.00000 −3.57981 0 0 1.00000i 0 3.57981i
2645.2 1.00000i 0 −1.00000 −2.49742 0 0 1.00000i 0 2.49742i
2645.3 1.00000i 0 −1.00000 −0.966684 0 0 1.00000i 0 0.966684i
2645.4 1.00000i 0 −1.00000 −0.115708 0 0 1.00000i 0 0.115708i
2645.5 1.00000i 0 −1.00000 0.115708 0 0 1.00000i 0 0.115708i
2645.6 1.00000i 0 −1.00000 0.966684 0 0 1.00000i 0 0.966684i
2645.7 1.00000i 0 −1.00000 2.49742 0 0 1.00000i 0 2.49742i
2645.8 1.00000i 0 −1.00000 3.57981 0 0 1.00000i 0 3.57981i
2645.9 1.00000i 0 −1.00000 −3.57981 0 0 1.00000i 0 3.57981i
2645.10 1.00000i 0 −1.00000 −2.49742 0 0 1.00000i 0 2.49742i
2645.11 1.00000i 0 −1.00000 −0.966684 0 0 1.00000i 0 0.966684i
2645.12 1.00000i 0 −1.00000 −0.115708 0 0 1.00000i 0 0.115708i
2645.13 1.00000i 0 −1.00000 0.115708 0 0 1.00000i 0 0.115708i
2645.14 1.00000i 0 −1.00000 0.966684 0 0 1.00000i 0 0.966684i
2645.15 1.00000i 0 −1.00000 2.49742 0 0 1.00000i 0 2.49742i
2645.16 1.00000i 0 −1.00000 3.57981 0 0 1.00000i 0 3.57981i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2645.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.d.f 16
3.b odd 2 1 inner 2646.2.d.f 16
7.b odd 2 1 inner 2646.2.d.f 16
21.c even 2 1 inner 2646.2.d.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2646.2.d.f 16 1.a even 1 1 trivial
2646.2.d.f 16 3.b odd 2 1 inner
2646.2.d.f 16 7.b odd 2 1 inner
2646.2.d.f 16 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 20 T_{5}^{6} + 98 T_{5}^{4} - 76 T_{5}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(2646, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$T^{16}$$
$5$ $$( 1 - 76 T^{2} + 98 T^{4} - 20 T^{6} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$( 2401 + 3700 T^{2} + 774 T^{4} + 52 T^{6} + T^{8} )^{2}$$
$13$ $$( 6724 + 4160 T^{2} + 860 T^{4} + 64 T^{6} + T^{8} )^{2}$$
$17$ $$( -6 + T^{2} )^{8}$$
$19$ $$( 22801 + 83308 T^{2} + 6134 T^{4} + 140 T^{6} + T^{8} )^{2}$$
$23$ $$( 18769 + 10556 T^{2} + 1458 T^{4} + 68 T^{6} + T^{8} )^{2}$$
$29$ $$( 329476 + 85920 T^{2} + 5948 T^{4} + 144 T^{6} + T^{8} )^{2}$$
$31$ $$( 1 + 20 T^{2} + 98 T^{4} + 76 T^{6} + T^{8} )^{2}$$
$37$ $$( -1337 + 692 T - 90 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$41$ $$( 49 - 34516 T^{2} + 3734 T^{4} - 116 T^{6} + T^{8} )^{2}$$
$43$ $$( -392 - 224 T - 16 T^{2} + 8 T^{3} + T^{4} )^{4}$$
$47$ $$( 256 - 27904 T^{2} + 4640 T^{4} - 176 T^{6} + T^{8} )^{2}$$
$53$ $$( 9388096 + 788992 T^{2} + 22512 T^{4} + 256 T^{6} + T^{8} )^{2}$$
$59$ $$( 41886784 - 2309888 T^{2} + 44144 T^{4} - 352 T^{6} + T^{8} )^{2}$$
$61$ $$( 4096 + 38912 T^{2} + 6272 T^{4} + 160 T^{6} + T^{8} )^{2}$$
$67$ $$( -206 - 208 T - 36 T^{2} + 8 T^{3} + T^{4} )^{4}$$
$71$ $$( 17497489 + 1388772 T^{2} + 35954 T^{4} + 348 T^{6} + T^{8} )^{2}$$
$73$ $$( 3136 + 8704 T^{2} + 4208 T^{4} + 128 T^{6} + T^{8} )^{2}$$
$79$ $$( -2744 - 1344 T - 184 T^{2} + T^{4} )^{4}$$
$83$ $$( 7761796 - 3073808 T^{2} + 73172 T^{4} - 520 T^{6} + T^{8} )^{2}$$
$89$ $$( 15062161 - 1608308 T^{2} + 38582 T^{4} - 340 T^{6} + T^{8} )^{2}$$
$97$ $$( 24840256 + 2070400 T^{2} + 45392 T^{4} + 368 T^{6} + T^{8} )^{2}$$