# Properties

 Label 637.2.g.j Level $637$ Weight $2$ Character orbit 637.g Analytic conductor $5.086$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 7 x^{6} + 38 x^{4} - 16 x^{3} + 15 x^{2} + 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{6} q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{4} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{5} + ( -\beta_{1} + \beta_{4} ) q^{6} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{8} + ( 2 + \beta_{2} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{6} q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{4} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{5} + ( -\beta_{1} + \beta_{4} ) q^{6} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{8} + ( 2 + \beta_{2} - \beta_{6} ) q^{9} + ( 3 - \beta_{2} + \beta_{3} ) q^{10} -\beta_{6} q^{11} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{12} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{13} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{15} + ( -4 \beta_{1} + 4 \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( -2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{18} + ( \beta_{2} + 3 \beta_{3} + 2 \beta_{6} ) q^{19} + ( 3 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{20} + ( -\beta_{1} + \beta_{4} ) q^{22} + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{23} + ( -2 \beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{24} + ( 3 \beta_{1} + 2 \beta_{4} + \beta_{7} ) q^{25} + ( -\beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{26} + ( 7 - \beta_{3} - 3 \beta_{6} ) q^{27} + ( -\beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{29} + ( -1 + \beta_{2} ) q^{30} + ( \beta_{4} + \beta_{7} ) q^{31} + ( 7 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{32} + ( 5 + \beta_{2} - \beta_{6} ) q^{33} + ( \beta_{2} + 4 \beta_{3} + \beta_{6} ) q^{34} + ( 9 + 9 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{36} + ( \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{37} + ( -4 \beta_{1} + 5 \beta_{4} + \beta_{5} + 3 \beta_{7} ) q^{38} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{39} + ( -4 + \beta_{2} - 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{40} + ( -6 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + \beta_{7} ) q^{41} + ( \beta_{1} + \beta_{4} ) q^{43} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{44} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{45} + ( -6 - \beta_{1} + \beta_{2} - \beta_{3} - 6 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{46} + ( 1 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{47} + ( 5 \beta_{1} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{48} + ( -11 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 11 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{50} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{51} + ( -1 - \beta_{2} - 4 \beta_{3} - 8 \beta_{4} + \beta_{6} - \beta_{7} ) q^{52} + ( -2 \beta_{1} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{53} + ( 5 \beta_{1} - \beta_{7} ) q^{54} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{55} + ( -5 - 3 \beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{57} + ( -3 - 2 \beta_{3} - \beta_{6} ) q^{58} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( -2 \beta_{1} - 4 \beta_{4} - 3 \beta_{5} ) q^{60} + ( -2 + \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{61} + ( -2 - 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{62} + ( 1 - 2 \beta_{2} - 10 \beta_{3} ) q^{64} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{65} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{66} + ( -1 + \beta_{2} + 6 \beta_{3} + 4 \beta_{6} ) q^{67} + ( -4 \beta_{1} + 7 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{68} + ( 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{69} + ( -4 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{71} + ( -3 \beta_{3} + 2 \beta_{6} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{73} + ( -4 + 4 \beta_{1} + \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - \beta_{7} ) q^{74} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{75} + ( 5 - 11 \beta_{1} - 2 \beta_{2} - 11 \beta_{3} + 5 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{76} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{78} + ( 6 + \beta_{1} - \beta_{2} + \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{79} + ( -1 + 2 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{80} + ( 8 + \beta_{3} - 7 \beta_{6} ) q^{81} + ( 4 - 2 \beta_{2} + \beta_{3} - \beta_{6} ) q^{82} + ( 1 + \beta_{2} + 4 \beta_{3} ) q^{83} + ( 2 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{85} + ( -3 + \beta_{2} - 3 \beta_{4} - \beta_{7} ) q^{86} + ( 3 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{87} + ( -2 \beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{88} + ( \beta_{1} - 2 \beta_{5} + \beta_{7} ) q^{89} + ( -7 + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{6} ) q^{90} + ( 4 + \beta_{2} + 7 \beta_{3} + 3 \beta_{6} ) q^{92} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{93} + ( -8 + 3 \beta_{2} + \beta_{3} ) q^{94} + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{95} + ( -13 + 6 \beta_{1} + \beta_{2} + 6 \beta_{3} - 13 \beta_{4} - \beta_{7} ) q^{96} + ( 2 \beta_{1} + \beta_{4} + 5 \beta_{5} + \beta_{7} ) q^{97} + ( 7 - \beta_{3} - 6 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} - 2 q^{3} - 5 q^{4} - 7 q^{5} - 5 q^{6} - 12 q^{8} + 14 q^{9} + O(q^{10})$$ $$8 q + q^{2} - 2 q^{3} - 5 q^{4} - 7 q^{5} - 5 q^{6} - 12 q^{8} + 14 q^{9} + 22 q^{10} - 2 q^{11} + 12 q^{12} - 4 q^{13} - 3 q^{15} - 19 q^{16} - 4 q^{17} + 3 q^{18} - 2 q^{19} - 2 q^{20} - 5 q^{22} + 2 q^{23} + 6 q^{24} - 5 q^{25} - 3 q^{26} + 52 q^{27} - q^{29} - 8 q^{30} - 4 q^{31} + 33 q^{32} + 38 q^{33} - 6 q^{34} + 34 q^{36} + 10 q^{37} - 23 q^{38} - 19 q^{39} - 17 q^{40} - 22 q^{41} - 3 q^{43} + 12 q^{44} - 11 q^{45} - 24 q^{46} + 2 q^{47} + 11 q^{48} - 43 q^{50} - 7 q^{51} + 34 q^{52} - 2 q^{53} + 5 q^{54} - 3 q^{55} - 34 q^{57} - 22 q^{58} - 8 q^{59} + 11 q^{60} - 16 q^{61} - 5 q^{62} + 28 q^{64} + 4 q^{65} + 6 q^{66} - 12 q^{67} - 33 q^{68} - 18 q^{69} + 14 q^{71} + 10 q^{72} - 8 q^{73} - 20 q^{74} - 7 q^{75} + 32 q^{76} - q^{78} + 26 q^{79} - 14 q^{80} + 48 q^{81} + 28 q^{82} - 5 q^{85} - 12 q^{86} + 13 q^{87} + 6 q^{88} - q^{89} - 52 q^{90} + 24 q^{92} + 7 q^{93} - 66 q^{94} - 21 q^{95} - 58 q^{96} + 3 q^{97} + 46 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 7 x^{6} + 38 x^{4} - 16 x^{3} + 15 x^{2} + 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-12 \nu^{7} - 7 \nu^{6} - 76 \nu^{5} - 44 \nu^{4} - 602 \nu^{3} - 36 \nu^{2} - 8 \nu + 1249$$$$)/458$$ $$\beta_{3}$$ $$=$$ $$($$$$57 \nu^{7} - 24 \nu^{6} + 361 \nu^{5} + 209 \nu^{4} + 2287 \nu^{3} + 171 \nu^{2} + 38 \nu + 193$$$$)/916$$ $$\beta_{4}$$ $$=$$ $$($$$$193 \nu^{7} - 250 \nu^{6} + 1375 \nu^{5} - 361 \nu^{4} + 7125 \nu^{3} - 5375 \nu^{2} + 2724 \nu - 375$$$$)/916$$ $$\beta_{5}$$ $$=$$ $$($$$$174 \nu^{7} - 242 \nu^{6} + 1331 \nu^{5} - 507 \nu^{4} + 6897 \nu^{3} - 5203 \nu^{2} + 5383 \nu - 363$$$$)/458$$ $$\beta_{6}$$ $$=$$ $$($$$$-375 \nu^{7} + 182 \nu^{6} - 2375 \nu^{5} - 1375 \nu^{4} - 13889 \nu^{3} - 1125 \nu^{2} - 250 \nu - 2933$$$$)/916$$ $$\beta_{7}$$ $$=$$ $$($$$$-273 \nu^{7} + 356 \nu^{6} - 1958 \nu^{5} + 602 \nu^{4} - 10146 \nu^{3} + 7654 \nu^{2} - 3617 \nu + 534$$$$)/458$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + 7 \beta_{3} + \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{7} + \beta_{5} + 18 \beta_{4} - 10 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} + 15 \beta_{4} - 48 \beta_{3} - 10 \beta_{2} - 48 \beta_{1} + 15$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{6} - 86 \beta_{3} - 48 \beta_{2} + 117$$ $$\nu^{7}$$ $$=$$ $$-86 \beta_{7} - 48 \beta_{5} - 152 \beta_{4} + 337 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$\beta_{4}$$ $$-1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
263.1
 −1.11000 + 1.92258i −0.115680 + 0.200364i 0.355143 − 0.615126i 1.37054 − 2.37385i −1.11000 − 1.92258i −0.115680 − 0.200364i 0.355143 + 0.615126i 1.37054 + 2.37385i
−1.11000 + 1.92258i −0.549551 −1.46422 2.53610i −2.11000 3.65463i 0.610004 1.05656i 0 2.06113 −2.69799 9.36845
263.2 −0.115680 + 0.200364i 3.32225 0.973236 + 1.68569i −1.11568 1.93242i −0.384320 + 0.665661i 0 −0.913059 8.03736 0.516249
263.3 0.355143 0.615126i −2.40788 0.747746 + 1.29513i −0.644857 1.11692i −0.855143 + 1.48115i 0 2.48280 2.79790 −0.916066
263.4 1.37054 2.37385i −1.36482 −2.75677 4.77486i 0.370541 + 0.641796i −1.87054 + 3.23987i 0 −9.63087 −1.13727 2.03137
373.1 −1.11000 1.92258i −0.549551 −1.46422 + 2.53610i −2.11000 + 3.65463i 0.610004 + 1.05656i 0 2.06113 −2.69799 9.36845
373.2 −0.115680 0.200364i 3.32225 0.973236 1.68569i −1.11568 + 1.93242i −0.384320 0.665661i 0 −0.913059 8.03736 0.516249
373.3 0.355143 + 0.615126i −2.40788 0.747746 1.29513i −0.644857 + 1.11692i −0.855143 1.48115i 0 2.48280 2.79790 −0.916066
373.4 1.37054 + 2.37385i −1.36482 −2.75677 + 4.77486i 0.370541 0.641796i −1.87054 3.23987i 0 −9.63087 −1.13727 2.03137
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.g.j 8
7.b odd 2 1 637.2.g.k 8
7.c even 3 1 637.2.f.i 8
7.c even 3 1 637.2.h.i 8
7.d odd 6 1 91.2.f.c 8
7.d odd 6 1 637.2.h.h 8
13.c even 3 1 637.2.h.i 8
21.g even 6 1 819.2.o.h 8
28.f even 6 1 1456.2.s.q 8
91.g even 3 1 inner 637.2.g.j 8
91.g even 3 1 8281.2.a.bp 4
91.h even 3 1 637.2.f.i 8
91.m odd 6 1 637.2.g.k 8
91.m odd 6 1 1183.2.a.k 4
91.n odd 6 1 637.2.h.h 8
91.p odd 6 1 1183.2.a.l 4
91.u even 6 1 8281.2.a.bt 4
91.v odd 6 1 91.2.f.c 8
91.w even 12 2 1183.2.c.g 8
273.r even 6 1 819.2.o.h 8
364.ba even 6 1 1456.2.s.q 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 7.d odd 6 1
91.2.f.c 8 91.v odd 6 1
637.2.f.i 8 7.c even 3 1
637.2.f.i 8 91.h even 3 1
637.2.g.j 8 1.a even 1 1 trivial
637.2.g.j 8 91.g even 3 1 inner
637.2.g.k 8 7.b odd 2 1
637.2.g.k 8 91.m odd 6 1
637.2.h.h 8 7.d odd 6 1
637.2.h.h 8 91.n odd 6 1
637.2.h.i 8 7.c even 3 1
637.2.h.i 8 13.c even 3 1
819.2.o.h 8 21.g even 6 1
819.2.o.h 8 273.r even 6 1
1183.2.a.k 4 91.m odd 6 1
1183.2.a.l 4 91.p odd 6 1
1183.2.c.g 8 91.w even 12 2
1456.2.s.q 8 28.f even 6 1
1456.2.s.q 8 364.ba even 6 1
8281.2.a.bp 4 91.g even 3 1
8281.2.a.bt 4 91.u even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{8} - T_{2}^{7} + 7 T_{2}^{6} + 38 T_{2}^{4} - 16 T_{2}^{3} + 15 T_{2}^{2} + 3 T_{2} + 1$$ $$T_{3}^{4} + T_{3}^{3} - 9 T_{3}^{2} - 16 T_{3} - 6$$ $$T_{5}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 15 T^{2} - 16 T^{3} + 38 T^{4} + 7 T^{6} - T^{7} + T^{8}$$
$3$ $$( -6 - 16 T - 9 T^{2} + T^{3} + T^{4} )^{2}$$
$5$ $$81 + 9 T + 109 T^{2} + 114 T^{3} + 160 T^{4} + 86 T^{5} + 37 T^{6} + 7 T^{7} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( -6 - 16 T - 9 T^{2} + T^{3} + T^{4} )^{2}$$
$13$ $$28561 + 8788 T + 2704 T^{2} + 416 T^{3} + 17 T^{4} + 32 T^{5} + 16 T^{6} + 4 T^{7} + T^{8}$$
$17$ $$2809 + 3180 T + 2964 T^{2} + 1144 T^{3} + 437 T^{4} + 72 T^{5} + 28 T^{6} + 4 T^{7} + T^{8}$$
$19$ $$( 500 - 55 T^{2} + T^{3} + T^{4} )^{2}$$
$23$ $$5184 + 4032 T + 5872 T^{2} - 1840 T^{3} + 1484 T^{4} - 36 T^{5} + 42 T^{6} - 2 T^{7} + T^{8}$$
$29$ $$25 - 105 T + 331 T^{2} - 452 T^{3} + 468 T^{4} - 64 T^{5} + 23 T^{6} + T^{7} + T^{8}$$
$31$ $$2916 - 1404 T + 1378 T^{2} - 94 T^{3} + 219 T^{4} + 29 T^{6} + 4 T^{7} + T^{8}$$
$37$ $$256 - 704 T + 2208 T^{2} + 428 T^{3} + 745 T^{4} - 258 T^{5} + 83 T^{6} - 10 T^{7} + T^{8}$$
$41$ $$318096 - 127464 T + 135112 T^{2} + 58490 T^{3} + 17793 T^{4} + 2826 T^{5} + 335 T^{6} + 22 T^{7} + T^{8}$$
$43$ $$4 + 16 T + 58 T^{2} + 36 T^{3} + 35 T^{4} + 7 T^{5} + 12 T^{6} + 3 T^{7} + T^{8}$$
$47$ $$10000 - 1200 T + 5244 T^{2} + 1012 T^{3} + 2477 T^{4} + 126 T^{5} + 55 T^{6} - 2 T^{7} + T^{8}$$
$53$ $$1929321 + 1236210 T + 597640 T^{2} + 130156 T^{3} + 22769 T^{4} + 1500 T^{5} + 144 T^{6} + 2 T^{7} + T^{8}$$
$59$ $$498436 + 388300 T + 248138 T^{2} + 53646 T^{3} + 11035 T^{4} + 484 T^{5} + 141 T^{6} + 8 T^{7} + T^{8}$$
$61$ $$( -100 - 176 T - 15 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$67$ $$( 11010 - 634 T - 211 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$71$ $$4129024 - 1446784 T + 474432 T^{2} - 68288 T^{3} + 12256 T^{4} - 1200 T^{5} + 212 T^{6} - 14 T^{7} + T^{8}$$
$73$ $$3139984 - 155936 T + 218612 T^{2} - 17880 T^{3} + 13093 T^{4} - 776 T^{5} + 183 T^{6} + 8 T^{7} + T^{8}$$
$79$ $$58982400 - 7864320 T + 2077696 T^{2} - 262144 T^{3} + 52260 T^{4} - 5532 T^{5} + 542 T^{6} - 26 T^{7} + T^{8}$$
$83$ $$( -426 + 442 T - 97 T^{2} + T^{4} )^{2}$$
$89$ $$11664 + 19872 T + 26188 T^{2} + 13280 T^{3} + 5333 T^{4} + 297 T^{5} + 72 T^{6} + T^{7} + T^{8}$$
$97$ $$246238864 + 10482256 T + 5232284 T^{2} - 109588 T^{3} + 79337 T^{4} - 421 T^{5} + 314 T^{6} - 3 T^{7} + T^{8}$$