# Properties

 Label 847.2.a.p Level $847$ Weight $2$ Character orbit 847.a Self dual yes Analytic conductor $6.763$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 35 x^{4} - 30 x^{3} - 30 x^{2} + 30 x - 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 - \beta_{4} - \beta_{7} ) q^{3} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( 2 + \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{6} + q^{7} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + ( 3 + \beta_{2} - \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 - \beta_{4} - \beta_{7} ) q^{3} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( 2 + \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{6} + q^{7} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + ( 3 + \beta_{2} - \beta_{6} - \beta_{7} ) q^{9} + ( 1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{10} + ( -\beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{12} + ( -1 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{13} + \beta_{1} q^{14} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{15} + ( \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{17} + ( 1 + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{18} + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{19} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{20} + ( 1 - \beta_{4} - \beta_{7} ) q^{21} + ( -\beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{23} + ( -3 + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{7} ) q^{24} + ( 2 + \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{25} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{26} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{27} + ( 1 + \beta_{5} + \beta_{6} ) q^{28} + ( -\beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( -4 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{30} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( 3 + \beta_{2} + 2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{32} + ( -2 - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{34} + ( 2 + \beta_{2} + \beta_{3} ) q^{35} + ( -5 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{36} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{37} + ( -2 - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{38} + ( 4 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{39} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{40} + ( -1 - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{41} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{42} + ( -2 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{43} + ( 5 - \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} - 4 \beta_{6} ) q^{45} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{46} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{47} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{48} + q^{49} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} + 4 \beta_{7} ) q^{50} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{51} + ( -4 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{52} + ( 4 + \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{53} + ( 5 - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{54} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{56} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + 4 \beta_{6} + \beta_{7} ) q^{57} + ( -2 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{58} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{59} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{60} + ( 3 + 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + ( -3 - \beta_{2} + \beta_{3} + \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{62} + ( 3 + \beta_{2} - \beta_{6} - \beta_{7} ) q^{63} + ( 1 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{64} + ( 3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{65} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{67} + ( 1 - 5 \beta_{1} + \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{68} + ( -\beta_{2} - 4 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{69} + ( 1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{70} + ( -1 - \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{71} + ( -7 - 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{72} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 5 \beta_{5} - 2 \beta_{6} ) q^{73} + ( 4 + 2 \beta_{2} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{74} + ( -3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{75} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{76} + ( -2 - \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{78} + ( 2 \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( 3 + 4 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 10 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{80} + ( 2 + 4 \beta_{1} + \beta_{2} - 3 \beta_{4} - 3 \beta_{6} ) q^{81} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{82} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{83} + ( -\beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{84} + ( -1 - 3 \beta_{1} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{85} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} + \beta_{6} - \beta_{7} ) q^{86} + ( -3 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{87} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{89} + ( -2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{90} + ( -1 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{91} + ( 7 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{92} + ( 4 + \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - \beta_{7} ) q^{93} + ( -5 + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{94} + ( -2 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 5 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{95} + ( 4 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{96} + ( 5 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9} + O(q^{10})$$ $$8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9} + 6 q^{10} + 9 q^{12} - 6 q^{13} + q^{14} + 11 q^{15} + q^{16} - 5 q^{17} + 8 q^{18} - 13 q^{19} + 23 q^{20} + 4 q^{21} + 16 q^{23} - 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} + 7 q^{28} + 9 q^{29} - 36 q^{30} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} + 13 q^{39} + 5 q^{40} - 10 q^{41} - q^{42} - 4 q^{43} + 35 q^{45} + 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} + 6 q^{50} + 13 q^{51} - 41 q^{52} + 37 q^{53} + 30 q^{54} + 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} + 19 q^{61} - 18 q^{62} + 14 q^{63} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} - 35 q^{72} - 25 q^{73} + 33 q^{74} - 13 q^{75} + 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} - 25 q^{83} + 9 q^{84} + 3 q^{85} + 4 q^{86} - 36 q^{87} + 37 q^{89} - 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} - 42 q^{94} + 21 q^{95} - 6 q^{96} + 15 q^{97} + q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 35 x^{4} - 30 x^{3} - 30 x^{2} + 30 x - 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} - 5 \nu^{5} - 30 \nu^{4} - 30 \nu^{3} + 130 \nu^{2} + 135 \nu - 140$$$$)/25$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} - 4 \nu^{6} + 35 \nu^{5} + 35 \nu^{4} - 165 \nu^{3} - 60 \nu^{2} + 180 \nu - 45$$$$)/25$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} + 6 \nu^{6} - 40 \nu^{5} - 65 \nu^{4} + 160 \nu^{3} + 165 \nu^{2} - 170 \nu - 20$$$$)/25$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{7} + 7 \nu^{6} + 95 \nu^{5} - 55 \nu^{4} - 280 \nu^{3} + 105 \nu^{2} + 210 \nu - 90$$$$)/25$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} - 7 \nu^{6} - 95 \nu^{5} + 55 \nu^{4} + 280 \nu^{3} - 80 \nu^{2} - 210 \nu + 15$$$$)/25$$ $$\beta_{7}$$ $$=$$ $$($$$$12 \nu^{7} - \nu^{6} - 135 \nu^{5} + 15 \nu^{4} + 440 \nu^{3} - 90 \nu^{2} - 405 \nu + 170$$$$)/25$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 6 \beta_{6} + 7 \beta_{5} - \beta_{4} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 7 \beta_{6} + 8 \beta_{5} + 8 \beta_{4} + 10 \beta_{3} - 7 \beta_{2} + 28 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{7} + 35 \beta_{6} + 47 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + \beta_{2} + 10 \beta_{1} + 77$$ $$\nu^{7}$$ $$=$$ $$13 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} + 54 \beta_{4} + 76 \beta_{3} - 42 \beta_{2} + 165 \beta_{1} + 31$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.43045 −1.70716 −1.40927 0.226211 0.669744 1.11447 1.98451 2.55194
−2.43045 1.43421 3.90710 1.25863 −3.48577 1.00000 −4.63512 −0.943053 −3.05904
1.2 −1.70716 2.29155 0.914391 4.06637 −3.91205 1.00000 1.85331 2.25122 −6.94194
1.3 −1.40927 −2.16338 −0.0139645 −1.83139 3.04878 1.00000 2.83822 1.68022 2.58091
1.4 0.226211 −0.219130 −1.94883 −2.49552 −0.0495696 1.00000 −0.893270 −2.95198 −0.564516
1.5 0.669744 3.13977 −1.55144 2.14378 2.10284 1.00000 −2.37856 6.85818 1.43578
1.6 1.11447 −2.85882 −0.757964 3.45608 −3.18606 1.00000 −3.07366 5.17284 3.85168
1.7 1.98451 2.78639 1.93830 −0.0269243 5.52964 1.00000 −0.122446 4.76399 −0.0534317
1.8 2.55194 −0.410598 4.51241 3.42898 −1.04782 1.00000 6.41153 −2.83141 8.75055
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.a.p 8
3.b odd 2 1 7623.2.a.ct 8
7.b odd 2 1 5929.2.a.bt 8
11.b odd 2 1 847.2.a.o 8
11.c even 5 2 77.2.f.b 16
11.c even 5 2 847.2.f.w 16
11.d odd 10 2 847.2.f.v 16
11.d odd 10 2 847.2.f.x 16
33.d even 2 1 7623.2.a.cw 8
33.h odd 10 2 693.2.m.i 16
77.b even 2 1 5929.2.a.bs 8
77.j odd 10 2 539.2.f.e 16
77.m even 15 4 539.2.q.g 32
77.p odd 30 4 539.2.q.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 11.c even 5 2
539.2.f.e 16 77.j odd 10 2
539.2.q.f 32 77.p odd 30 4
539.2.q.g 32 77.m even 15 4
693.2.m.i 16 33.h odd 10 2
847.2.a.o 8 11.b odd 2 1
847.2.a.p 8 1.a even 1 1 trivial
847.2.f.v 16 11.d odd 10 2
847.2.f.w 16 11.c even 5 2
847.2.f.x 16 11.d odd 10 2
5929.2.a.bs 8 77.b even 2 1
5929.2.a.bt 8 7.b odd 2 1
7623.2.a.ct 8 3.b odd 2 1
7623.2.a.cw 8 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(847))$$:

 $$T_{2}^{8} - \cdots$$ $$T_{3}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 + 30 T - 30 T^{2} - 30 T^{3} + 35 T^{4} + 10 T^{5} - 11 T^{6} - T^{7} + T^{8}$$
$3$ $$16 + 96 T + 64 T^{2} - 186 T^{3} + 15 T^{4} + 54 T^{5} - 11 T^{6} - 4 T^{7} + T^{8}$$
$5$ $$-16 - 576 T + 680 T^{2} + 62 T^{3} - 285 T^{4} + 67 T^{5} + 22 T^{6} - 10 T^{7} + T^{8}$$
$7$ $$( -1 + T )^{8}$$
$11$ $$T^{8}$$
$13$ $$-13744 - 10784 T + 2104 T^{2} + 3114 T^{3} + 195 T^{4} - 261 T^{5} - 36 T^{6} + 6 T^{7} + T^{8}$$
$17$ $$-1616 - 1832 T + 660 T^{2} + 1206 T^{3} + 65 T^{4} - 211 T^{5} - 37 T^{6} + 5 T^{7} + T^{8}$$
$19$ $$7920 - 9960 T + 344 T^{2} + 3028 T^{3} - 415 T^{4} - 305 T^{5} + 15 T^{6} + 13 T^{7} + T^{8}$$
$23$ $$859 + 1310 T - 1702 T^{2} - 3298 T^{3} - 765 T^{4} + 342 T^{5} + 40 T^{6} - 16 T^{7} + T^{8}$$
$29$ $$-495 - 3210 T - 5599 T^{2} - 3049 T^{3} + 310 T^{4} + 445 T^{5} - 45 T^{6} - 9 T^{7} + T^{8}$$
$31$ $$-51280 + 53800 T + 34100 T^{2} - 27410 T^{3} + 2135 T^{4} + 990 T^{5} - 106 T^{6} - 9 T^{7} + T^{8}$$
$37$ $$-461 + 2540 T - 2433 T^{2} - 2041 T^{3} + 1060 T^{4} + 261 T^{5} - 65 T^{6} - 7 T^{7} + T^{8}$$
$41$ $$3664 + 18552 T + 25420 T^{2} + 11214 T^{3} + 305 T^{4} - 694 T^{5} - 67 T^{6} + 10 T^{7} + T^{8}$$
$43$ $$-971 - 8740 T - 10617 T^{2} + 812 T^{3} + 2780 T^{4} - 268 T^{5} - 105 T^{6} + 4 T^{7} + T^{8}$$
$47$ $$-2312080 + 522080 T + 221680 T^{2} - 59580 T^{3} - 4355 T^{4} + 1945 T^{5} - 56 T^{6} - 16 T^{7} + T^{8}$$
$53$ $$557531 - 944112 T + 116873 T^{2} + 80581 T^{3} - 16950 T^{4} - 769 T^{5} + 433 T^{6} - 37 T^{7} + T^{8}$$
$59$ $$-13680 - 51000 T - 35684 T^{2} + 7274 T^{3} + 5385 T^{4} - 50 T^{5} - 140 T^{6} - T^{7} + T^{8}$$
$61$ $$15920 + 27640 T + 1920 T^{2} - 12220 T^{3} - 1585 T^{4} + 1070 T^{5} + 4 T^{6} - 19 T^{7} + T^{8}$$
$67$ $$-27395 - 35255 T + 5380 T^{2} + 12660 T^{3} - 255 T^{4} - 1160 T^{5} - 16 T^{6} + 19 T^{7} + T^{8}$$
$71$ $$-19471 - 151555 T + 130762 T^{2} - 13964 T^{3} - 10945 T^{4} + 3004 T^{5} - 170 T^{6} - 13 T^{7} + T^{8}$$
$73$ $$324144 - 744504 T + 303200 T^{2} + 89828 T^{3} - 11135 T^{4} - 2812 T^{5} + 32 T^{6} + 25 T^{7} + T^{8}$$
$79$ $$-3982275 + 2765925 T - 306995 T^{2} - 121275 T^{3} + 22330 T^{4} + 1005 T^{5} - 295 T^{6} + T^{8}$$
$83$ $$27504 - 46944 T + 15080 T^{2} + 7078 T^{3} - 2785 T^{4} - 542 T^{5} + 122 T^{6} + 25 T^{7} + T^{8}$$
$89$ $$952400 - 1415520 T + 512284 T^{2} + 45228 T^{3} - 34845 T^{4} + 2150 T^{5} + 320 T^{6} - 37 T^{7} + T^{8}$$
$97$ $$-2155696 + 977888 T + 283960 T^{2} - 165414 T^{3} + 7665 T^{4} + 3144 T^{5} - 222 T^{6} - 15 T^{7} + T^{8}$$