# Properties

 Label 288.2.v.b Level $288$ Weight $2$ Character orbit 288.v Analytic conductor $2.300$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.v (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( \beta_{2} - \beta_{3} + \beta_{6} ) q^{4} + ( \beta_{6} + \beta_{7} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{7} + ( \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( \beta_{2} - \beta_{3} + \beta_{6} ) q^{4} + ( \beta_{6} + \beta_{7} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{7} + ( \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{8} + ( \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{10} + ( -1 + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{13} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{14} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{16} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{19} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{20} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{22} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{23} + ( -1 + 3 \beta_{5} + \beta_{7} ) q^{25} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{26} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{6} - \beta_{7} ) q^{28} + ( 2 - 4 \beta_{2} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{29} + ( 4 - 2 \beta_{5} - 2 \beta_{6} ) q^{31} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} ) q^{32} + ( -2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{34} + ( -1 - 2 \beta_{2} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{35} + ( -2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{37} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{4} + 3 \beta_{5} ) q^{38} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{40} + ( -3 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{41} + ( -1 - \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{43} + ( -2 + 2 \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{7} ) q^{44} + ( 2 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{46} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{7} ) q^{47} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{49} + ( 3 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{50} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} ) q^{52} + ( 2 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{53} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{55} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{56} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 6 \beta_{6} + \beta_{7} ) q^{58} + ( 3 - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{59} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{61} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{62} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{64} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( -6 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{67} + ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{68} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 5 \beta_{6} - \beta_{7} ) q^{70} + ( -3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{71} + ( -5 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - 3 \beta_{7} ) q^{73} + ( -2 \beta_{2} + 5 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 7 \beta_{7} ) q^{74} + ( -2 + \beta_{1} - \beta_{3} + 5 \beta_{5} + \beta_{6} ) q^{76} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{77} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} ) q^{79} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} ) q^{80} + ( -4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{82} + ( -4 - 3 \beta_{1} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{83} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{85} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{86} + ( 4 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{88} + ( -1 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{89} + ( 5 + 6 \beta_{3} - 5 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{91} + ( 2 + \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{92} + ( -4 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{94} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{95} + ( 4 + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{97} + ( 4 - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} + 4q^{4} - 8q^{7} + 4q^{8} + O(q^{10})$$ $$8q + 4q^{2} + 4q^{4} - 8q^{7} + 4q^{8} - 4q^{11} - 8q^{13} - 12q^{14} + 4q^{19} - 4q^{20} + 4q^{22} + 8q^{23} - 8q^{25} + 20q^{26} - 16q^{28} + 32q^{31} + 24q^{32} - 16q^{35} - 8q^{37} - 8q^{38} + 16q^{40} - 8q^{41} - 12q^{43} - 20q^{44} + 12q^{46} - 16q^{50} + 12q^{52} - 8q^{53} - 16q^{55} - 8q^{56} - 12q^{58} + 20q^{59} + 24q^{61} + 24q^{62} - 8q^{64} - 36q^{67} - 16q^{68} - 8q^{70} + 24q^{71} - 32q^{73} - 8q^{74} - 20q^{76} - 16q^{77} - 8q^{80} - 20q^{82} - 20q^{83} + 8q^{85} - 4q^{86} + 8q^{88} + 16q^{89} + 40q^{91} + 16q^{92} - 24q^{94} + 8q^{95} + 32q^{97} + 24q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{7} - 7 \nu^{6} + 24 \nu^{5} - 42 \nu^{4} + 59 \nu^{3} - 48 \nu^{2} + 24 \nu - 5$$ $$\beta_{2}$$ $$=$$ $$-2 \nu^{7} + 7 \nu^{6} - 24 \nu^{5} + 43 \nu^{4} - 61 \nu^{3} + 54 \nu^{2} - 29 \nu + 8$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{7} + 10 \nu^{6} - 35 \nu^{5} + 60 \nu^{4} - 87 \nu^{3} + 73 \nu^{2} - 42 \nu + 11$$ $$\beta_{4}$$ $$=$$ $$-3 \nu^{7} + 11 \nu^{6} - 38 \nu^{5} + 70 \nu^{4} - 102 \nu^{3} + 91 \nu^{2} - 53 \nu + 13$$ $$\beta_{5}$$ $$=$$ $$5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19$$ $$\beta_{6}$$ $$=$$ $$-5 \nu^{7} + 18 \nu^{6} - 63 \nu^{5} + 115 \nu^{4} - 170 \nu^{3} + 152 \nu^{2} - 89 \nu + 23$$ $$\beta_{7}$$ $$=$$ $$-8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 3 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} - \beta_{6} + 7 \beta_{5} - \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 3 \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} + 11 \beta_{4} - \beta_{3} - 5 \beta_{2} + 9 \beta_{1} + 14$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{7} - 11 \beta_{6} - 29 \beta_{5} + 17 \beta_{4} - 23 \beta_{3} + 13 \beta_{2} - 3 \beta_{1} + 18$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-67 \beta_{7} + 59 \beta_{6} - 41 \beta_{5} - 29 \beta_{4} - 15 \beta_{3} + 37 \beta_{2} - 47 \beta_{1} - 48$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} + 113 \beta_{6} + 97 \beta_{5} - 105 \beta_{4} + 91 \beta_{3} - 31 \beta_{2} - 39 \beta_{1} - 122$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-\beta_{5}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 − 0.691860i 0.5 + 2.10607i 0.5 + 0.691860i 0.5 − 2.10607i 0.5 − 1.44392i 0.5 + 0.0297061i 0.5 + 1.44392i 0.5 − 0.0297061i
0.443806 1.34277i 0 −1.60607 1.19186i −0.707107 0.292893i 0 −2.27133 2.27133i −2.31318 + 1.62764i 0 −0.707107 + 0.819496i
37.2 1.26330 + 0.635665i 0 1.19186 + 1.60607i −0.707107 0.292893i 0 1.68554 + 1.68554i 0.484753 + 2.78658i 0 −0.707107 0.819496i
109.1 0.443806 + 1.34277i 0 −1.60607 + 1.19186i −0.707107 + 0.292893i 0 −2.27133 + 2.27133i −2.31318 1.62764i 0 −0.707107 0.819496i
109.2 1.26330 0.635665i 0 1.19186 1.60607i −0.707107 + 0.292893i 0 1.68554 1.68554i 0.484753 2.78658i 0 −0.707107 + 0.819496i
181.1 −1.11137 + 0.874559i 0 0.470294 1.94392i 0.707107 1.70711i 0 −0.665096 0.665096i 1.17740 + 2.57172i 0 0.707107 + 2.51564i
181.2 1.40426 0.167452i 0 1.94392 0.470294i 0.707107 1.70711i 0 −2.74912 2.74912i 2.65103 0.985930i 0 0.707107 2.51564i
253.1 −1.11137 0.874559i 0 0.470294 + 1.94392i 0.707107 + 1.70711i 0 −0.665096 + 0.665096i 1.17740 2.57172i 0 0.707107 2.51564i
253.2 1.40426 + 0.167452i 0 1.94392 + 0.470294i 0.707107 + 1.70711i 0 −2.74912 + 2.74912i 2.65103 + 0.985930i 0 0.707107 + 2.51564i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.v.b 8
3.b odd 2 1 32.2.g.b 8
4.b odd 2 1 1152.2.v.b 8
12.b even 2 1 128.2.g.b 8
15.d odd 2 1 800.2.y.b 8
15.e even 4 1 800.2.ba.c 8
15.e even 4 1 800.2.ba.d 8
24.f even 2 1 256.2.g.c 8
24.h odd 2 1 256.2.g.d 8
32.g even 8 1 inner 288.2.v.b 8
32.h odd 8 1 1152.2.v.b 8
48.i odd 4 1 512.2.g.e 8
48.i odd 4 1 512.2.g.h 8
48.k even 4 1 512.2.g.f 8
48.k even 4 1 512.2.g.g 8
96.o even 8 1 128.2.g.b 8
96.o even 8 1 256.2.g.c 8
96.o even 8 1 512.2.g.f 8
96.o even 8 1 512.2.g.g 8
96.p odd 8 1 32.2.g.b 8
96.p odd 8 1 256.2.g.d 8
96.p odd 8 1 512.2.g.e 8
96.p odd 8 1 512.2.g.h 8
192.q odd 16 2 4096.2.a.k 8
192.s even 16 2 4096.2.a.q 8
480.br even 8 1 800.2.ba.c 8
480.bu odd 8 1 800.2.y.b 8
480.cb even 8 1 800.2.ba.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b 8 3.b odd 2 1
32.2.g.b 8 96.p odd 8 1
128.2.g.b 8 12.b even 2 1
128.2.g.b 8 96.o even 8 1
256.2.g.c 8 24.f even 2 1
256.2.g.c 8 96.o even 8 1
256.2.g.d 8 24.h odd 2 1
256.2.g.d 8 96.p odd 8 1
288.2.v.b 8 1.a even 1 1 trivial
288.2.v.b 8 32.g even 8 1 inner
512.2.g.e 8 48.i odd 4 1
512.2.g.e 8 96.p odd 8 1
512.2.g.f 8 48.k even 4 1
512.2.g.f 8 96.o even 8 1
512.2.g.g 8 48.k even 4 1
512.2.g.g 8 96.o even 8 1
512.2.g.h 8 48.i odd 4 1
512.2.g.h 8 96.p odd 8 1
800.2.y.b 8 15.d odd 2 1
800.2.y.b 8 480.bu odd 8 1
800.2.ba.c 8 15.e even 4 1
800.2.ba.c 8 480.br even 8 1
800.2.ba.d 8 15.e even 4 1
800.2.ba.d 8 480.cb even 8 1
1152.2.v.b 8 4.b odd 2 1
1152.2.v.b 8 32.h odd 8 1
4096.2.a.k 8 192.q odd 16 2
4096.2.a.q 8 192.s even 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 2 T_{5}^{2} + 4 T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 32 T + 24 T^{2} - 8 T^{3} + 2 T^{4} - 4 T^{5} + 6 T^{6} - 4 T^{7} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 2 + 4 T + 2 T^{2} + T^{4} )^{2}$$
$7$ $$784 + 1344 T + 1152 T^{2} + 224 T^{3} + 56 T^{4} + 48 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$11$ $$4 + 48 T + 160 T^{2} - 56 T^{3} + 224 T^{4} + 64 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$13$ $$6724 - 8528 T + 2520 T^{2} + 448 T^{3} + 200 T^{4} + 104 T^{5} + 36 T^{6} + 8 T^{7} + T^{8}$$
$17$ $$256 + 5120 T^{2} + 1056 T^{4} + 64 T^{6} + T^{8}$$
$19$ $$196 - 336 T + 832 T^{2} - 168 T^{3} + 32 T^{4} + 48 T^{5} - 8 T^{6} - 4 T^{7} + T^{8}$$
$23$ $$16 + 64 T + 128 T^{2} + 32 T^{3} - 8 T^{4} - 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$29$ $$188356 - 17360 T + 17272 T^{2} + 6256 T^{3} + 72 T^{4} - 168 T^{5} - 12 T^{6} + T^{8}$$
$31$ $$( 8 - 8 T + T^{2} )^{4}$$
$37$ $$64516 - 67056 T + 59320 T^{2} - 20640 T^{3} + 3464 T^{4} - 168 T^{5} - 44 T^{6} + 8 T^{7} + T^{8}$$
$41$ $$26896 - 7872 T + 1152 T^{2} + 416 T^{3} + 968 T^{4} - 240 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$43$ $$31684 + 12816 T + 5888 T^{2} + 4856 T^{3} + 1760 T^{4} + 256 T^{5} + 56 T^{6} + 12 T^{7} + T^{8}$$
$47$ $$256 + 1024 T^{2} + 544 T^{4} + 64 T^{6} + T^{8}$$
$53$ $$158404 + 238800 T + 147160 T^{2} + 47328 T^{3} + 9800 T^{4} + 1272 T^{5} + 100 T^{6} + 8 T^{7} + T^{8}$$
$59$ $$643204 - 211728 T + 44896 T^{2} - 20712 T^{3} + 5408 T^{4} - 528 T^{5} + 136 T^{6} - 20 T^{7} + T^{8}$$
$61$ $$42436 - 24720 T + 34968 T^{2} + 6336 T^{3} + 72 T^{4} + 648 T^{5} + 132 T^{6} - 24 T^{7} + T^{8}$$
$67$ $$1285956 + 734832 T + 233280 T^{2} + 68040 T^{3} + 18144 T^{4} + 3456 T^{5} + 504 T^{6} + 36 T^{7} + T^{8}$$
$71$ $$21196816 - 9060672 T + 1936512 T^{2} - 173472 T^{3} + 10232 T^{4} - 1200 T^{5} + 288 T^{6} - 24 T^{7} + T^{8}$$
$73$ $$38416 + 112896 T + 165888 T^{2} + 88192 T^{3} + 26504 T^{4} + 4672 T^{5} + 512 T^{6} + 32 T^{7} + T^{8}$$
$79$ $$99361024 + 4775936 T^{2} + 78880 T^{4} + 512 T^{6} + T^{8}$$
$83$ $$138250564 + 16837456 T + 1548544 T^{2} - 131864 T^{3} - 22752 T^{4} - 304 T^{5} + 184 T^{6} + 20 T^{7} + T^{8}$$
$89$ $$17007376 - 7786112 T + 1782272 T^{2} - 209472 T^{3} + 14024 T^{4} - 672 T^{5} + 128 T^{6} - 16 T^{7} + T^{8}$$
$97$ $$( -992 + 288 T + 40 T^{2} - 16 T^{3} + T^{4} )^{2}$$