# Properties

 Label 845.2.e.m Level $845$ Weight $2$ Character orbit 845.e Analytic conductor $6.747$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 65) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{5} ) q^{2} + ( \beta_{4} + \beta_{6} + \beta_{7} ) q^{3} + ( 1 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} ) q^{4} - q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{6} + ( \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{7} + ( 1 + \beta_{3} + 2 \beta_{6} ) q^{8} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{5} ) q^{2} + ( \beta_{4} + \beta_{6} + \beta_{7} ) q^{3} + ( 1 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} ) q^{4} - q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{6} + ( \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{7} + ( 1 + \beta_{3} + 2 \beta_{6} ) q^{8} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{9} + ( 1 - \beta_{1} - \beta_{5} ) q^{10} + ( -2 + 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( -1 - 3 \beta_{2} - 2 \beta_{6} ) q^{12} + ( -1 + 3 \beta_{2} - \beta_{6} ) q^{14} + ( -\beta_{4} - \beta_{6} - \beta_{7} ) q^{15} + ( \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{16} + ( \beta_{3} - \beta_{4} + \beta_{7} ) q^{17} + ( 4 + 2 \beta_{2} + 2 \beta_{6} ) q^{18} + ( -4 \beta_{5} + \beta_{7} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} ) q^{20} + ( 2 - 2 \beta_{3} - 3 \beta_{6} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{22} + ( 2 + 2 \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( 8 - 4 \beta_{1} - 8 \beta_{5} + \beta_{6} + \beta_{7} ) q^{24} + q^{25} + ( -2 \beta_{2} + \beta_{3} - \beta_{6} ) q^{27} + ( -2 + \beta_{1} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{28} + ( -3 + 2 \beta_{1} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{30} + ( 2 - 2 \beta_{6} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{32} + ( \beta_{3} - \beta_{4} - 5 \beta_{5} - 4 \beta_{7} ) q^{33} + ( -2 + \beta_{2} + \beta_{3} ) q^{34} + ( -\beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{35} + ( -6 + 4 \beta_{1} - 2 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{36} + ( 3 - 2 \beta_{1} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{37} + ( -1 + 5 \beta_{2} + \beta_{3} + \beta_{6} ) q^{38} + ( -1 - \beta_{3} - 2 \beta_{6} ) q^{40} + ( -2 + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{41} + ( -\beta_{1} + 3 \beta_{4} + 5 \beta_{6} + 5 \beta_{7} ) q^{42} + ( \beta_{3} - \beta_{4} + \beta_{7} ) q^{43} + ( 2 - \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{44} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} ) q^{45} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 4 \beta_{7} ) q^{46} + ( 4 - 4 \beta_{2} - 2 \beta_{6} ) q^{47} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} + 8 \beta_{5} - \beta_{7} ) q^{48} + ( -4 - 2 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} ) q^{49} + ( -1 + \beta_{1} + \beta_{5} ) q^{50} + ( 1 - 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 - 2 \beta_{2} ) q^{53} + ( 5 - 3 \beta_{1} + \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{54} + ( 2 - 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 6 \beta_{7} ) q^{56} + ( -1 - 2 \beta_{2} - 3 \beta_{3} - 4 \beta_{6} ) q^{57} + ( 5 - 5 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 7 \beta_{5} + 4 \beta_{7} ) q^{58} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{59} + ( 1 + 3 \beta_{2} + 2 \beta_{6} ) q^{60} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} ) q^{61} + ( -2 + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{62} + ( -4 + 6 \beta_{1} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{63} + ( 3 - 4 \beta_{3} - 2 \beta_{6} ) q^{64} + ( 3 + \beta_{2} - 4 \beta_{3} - 5 \beta_{6} ) q^{66} + ( -7 - \beta_{4} + 7 \beta_{5} ) q^{67} + ( -2 - \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( -4 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} ) q^{69} + ( 1 - 3 \beta_{2} + \beta_{6} ) q^{70} + ( -6 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{71} + ( 6 - 6 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 6 \beta_{7} ) q^{72} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{73} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{7} ) q^{74} + ( \beta_{4} + \beta_{6} + \beta_{7} ) q^{75} + ( -7 + 5 \beta_{1} - \beta_{4} + 7 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{76} + ( 4 + 4 \beta_{2} - 3 \beta_{3} - 5 \beta_{6} ) q^{77} + ( -6 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{79} + ( -\beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{80} + ( 3 - 2 \beta_{1} - 3 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{81} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{82} + ( -4 + 2 \beta_{3} + 4 \beta_{6} ) q^{83} + ( -5 + 5 \beta_{1} + 5 \beta_{2} + \beta_{3} - \beta_{4} + 9 \beta_{5} - 3 \beta_{7} ) q^{84} + ( -\beta_{3} + \beta_{4} - \beta_{7} ) q^{85} + ( -2 + \beta_{2} + \beta_{3} ) q^{86} + ( -6 + 6 \beta_{1} + 6 \beta_{2} + \beta_{3} - \beta_{4} + 8 \beta_{5} - 5 \beta_{7} ) q^{87} + ( 6 - 3 \beta_{4} - 6 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{88} + ( 8 - 8 \beta_{1} - 2 \beta_{4} - 8 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{89} + ( -4 - 2 \beta_{2} - 2 \beta_{6} ) q^{90} + ( 3 + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{92} + ( -6 + 4 \beta_{1} + 4 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{93} + ( 8 - 2 \beta_{1} + 2 \beta_{4} - 8 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} ) q^{94} + ( 4 \beta_{5} - \beta_{7} ) q^{95} + ( 2 - \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{96} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{7} ) q^{97} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} - 6 \beta_{7} ) q^{98} + ( 6 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} + 2q^{3} - 2q^{4} - 8q^{5} + 4q^{6} - 10q^{7} + 12q^{8} - 4q^{9} + O(q^{10})$$ $$8q - 2q^{2} + 2q^{3} - 2q^{4} - 8q^{5} + 4q^{6} - 10q^{7} + 12q^{8} - 4q^{9} + 2q^{10} - 20q^{12} + 4q^{14} - 2q^{15} - 2q^{16} + 2q^{17} + 40q^{18} - 16q^{19} + 2q^{20} + 8q^{21} - 12q^{22} + 10q^{23} + 24q^{24} + 8q^{25} - 4q^{27} - 8q^{28} - 8q^{29} - 4q^{30} + 16q^{31} - 4q^{32} - 18q^{33} - 8q^{34} + 10q^{35} - 20q^{36} + 2q^{37} + 16q^{38} - 12q^{40} - 8q^{41} + 4q^{42} + 2q^{43} + 24q^{44} + 4q^{45} - 16q^{46} + 16q^{47} + 28q^{48} - 12q^{49} - 2q^{50} + 8q^{51} - 24q^{53} + 16q^{54} - 12q^{56} - 28q^{57} - 22q^{58} - 12q^{59} + 20q^{60} - 28q^{61} - 4q^{62} - 4q^{63} + 8q^{64} + 12q^{66} - 30q^{67} - 14q^{68} + 16q^{69} - 4q^{70} - 4q^{71} - 12q^{72} - 16q^{73} + 10q^{74} + 2q^{75} - 20q^{76} + 36q^{77} - 16q^{79} + 2q^{80} + 8q^{81} - 4q^{82} - 24q^{83} + 28q^{84} - 2q^{85} - 8q^{86} + 22q^{87} + 18q^{88} + 12q^{89} - 40q^{90} + 44q^{92} - 8q^{93} + 32q^{94} + 16q^{95} + 8q^{96} - 2q^{97} + 24q^{98} + 48q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + \nu^{5} + 4 \nu^{4} - 3 \nu^{3} - 10 \nu^{2} + 8 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} - \nu^{5} + 4 \nu^{4} - \nu^{3} - 6 \nu^{2} + 10 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{7} - 8 \nu^{6} + 3 \nu^{5} + 10 \nu^{4} - 13 \nu^{3} - 8 \nu^{2} + 32 \nu - 24$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{7} - 14 \nu^{6} + 9 \nu^{5} + 16 \nu^{4} - 27 \nu^{3} - 14 \nu^{2} + 76 \nu - 64$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} - 7 \nu^{6} + 3 \nu^{5} + 11 \nu^{4} - 15 \nu^{3} - 11 \nu^{2} + 40 \nu - 28$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} - 9 \nu^{6} + 5 \nu^{5} + 13 \nu^{4} - 21 \nu^{3} - 13 \nu^{2} + 54 \nu - 40$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{7} - 24 \nu^{6} + 13 \nu^{5} + 38 \nu^{4} - 51 \nu^{3} - 32 \nu^{2} + 132 \nu - 104$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - 3 \beta_{1} + 3$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 3 \beta_{2}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_{1} + 2$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} + 4 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 5 \beta_{1} + 8$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{7} - 5 \beta_{6} + 9 \beta_{5} - 12 \beta_{3} - 3 \beta_{2} + 3$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-4 \beta_{7} - 14 \beta_{6} + 24 \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - 7 \beta_{2} + \beta_{1} + 4$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1
 1.20036 + 0.747754i 1.40994 − 0.109843i −1.27597 + 0.609843i 0.665665 − 1.24775i 1.20036 − 0.747754i 1.40994 + 0.109843i −1.27597 − 0.609843i 0.665665 + 1.24775i
−1.24775 2.16117i 1.41342 + 2.44811i −2.11378 + 3.66117i −1.00000 3.52720 6.10929i −0.952606 + 1.64996i 5.55889 −2.49551 + 4.32235i 1.24775 + 2.16117i
146.2 −0.609843 1.05628i −1.16612 2.01978i 0.256182 0.443720i −1.00000 −1.42231 + 2.46350i −1.80010 + 3.11786i −3.06430 −1.21969 + 2.11256i 0.609843 + 1.05628i
146.3 0.109843 + 0.190254i 0.800098 + 1.38581i 0.975869 1.69025i −1.00000 −0.175771 + 0.304444i 0.166123 0.287734i 0.868145 0.219687 0.380509i −0.109843 0.190254i
146.4 0.747754 + 1.29515i −0.0473938 0.0820885i −0.118272 + 0.204852i −1.00000 0.0708778 0.122764i −2.41342 + 4.18016i 2.63726 1.49551 2.59030i −0.747754 1.29515i
191.1 −1.24775 + 2.16117i 1.41342 2.44811i −2.11378 3.66117i −1.00000 3.52720 + 6.10929i −0.952606 1.64996i 5.55889 −2.49551 4.32235i 1.24775 2.16117i
191.2 −0.609843 + 1.05628i −1.16612 + 2.01978i 0.256182 + 0.443720i −1.00000 −1.42231 2.46350i −1.80010 3.11786i −3.06430 −1.21969 2.11256i 0.609843 1.05628i
191.3 0.109843 0.190254i 0.800098 1.38581i 0.975869 + 1.69025i −1.00000 −0.175771 0.304444i 0.166123 + 0.287734i 0.868145 0.219687 + 0.380509i −0.109843 + 0.190254i
191.4 0.747754 1.29515i −0.0473938 + 0.0820885i −0.118272 0.204852i −1.00000 0.0708778 + 0.122764i −2.41342 4.18016i 2.63726 1.49551 + 2.59030i −0.747754 + 1.29515i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.e.m 8
13.b even 2 1 845.2.e.n 8
13.c even 3 1 845.2.a.m 4
13.c even 3 1 inner 845.2.e.m 8
13.d odd 4 1 65.2.m.a 8
13.d odd 4 1 845.2.m.g 8
13.e even 6 1 845.2.a.l 4
13.e even 6 1 845.2.e.n 8
13.f odd 12 1 65.2.m.a 8
13.f odd 12 2 845.2.c.g 8
13.f odd 12 1 845.2.m.g 8
39.f even 4 1 585.2.bu.c 8
39.h odd 6 1 7605.2.a.cj 4
39.i odd 6 1 7605.2.a.cf 4
39.k even 12 1 585.2.bu.c 8
52.f even 4 1 1040.2.da.b 8
52.l even 12 1 1040.2.da.b 8
65.f even 4 1 325.2.m.c 8
65.g odd 4 1 325.2.n.d 8
65.k even 4 1 325.2.m.b 8
65.l even 6 1 4225.2.a.bl 4
65.n even 6 1 4225.2.a.bi 4
65.o even 12 1 325.2.m.b 8
65.s odd 12 1 325.2.n.d 8
65.t even 12 1 325.2.m.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 13.d odd 4 1
65.2.m.a 8 13.f odd 12 1
325.2.m.b 8 65.k even 4 1
325.2.m.b 8 65.o even 12 1
325.2.m.c 8 65.f even 4 1
325.2.m.c 8 65.t even 12 1
325.2.n.d 8 65.g odd 4 1
325.2.n.d 8 65.s odd 12 1
585.2.bu.c 8 39.f even 4 1
585.2.bu.c 8 39.k even 12 1
845.2.a.l 4 13.e even 6 1
845.2.a.m 4 13.c even 3 1
845.2.c.g 8 13.f odd 12 2
845.2.e.m 8 1.a even 1 1 trivial
845.2.e.m 8 13.c even 3 1 inner
845.2.e.n 8 13.b even 2 1
845.2.e.n 8 13.e even 6 1
845.2.m.g 8 13.d odd 4 1
845.2.m.g 8 13.f odd 12 1
1040.2.da.b 8 52.f even 4 1
1040.2.da.b 8 52.l even 12 1
4225.2.a.bi 4 65.n even 6 1
4225.2.a.bl 4 65.l even 6 1
7605.2.a.cf 4 39.i odd 6 1
7605.2.a.cj 4 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$:

 $$T_{2}^{8} + \cdots$$ $$T_{7}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 19 T^{2} + 8 T^{3} + 16 T^{4} + 2 T^{5} + 7 T^{6} + 2 T^{7} + T^{8}$$
$3$ $$1 + 10 T + 106 T^{2} - 56 T^{3} + 55 T^{4} - 8 T^{5} + 10 T^{6} - 2 T^{7} + T^{8}$$
$5$ $$( 1 + T )^{8}$$
$7$ $$121 - 242 T + 814 T^{2} + 880 T^{3} + 691 T^{4} + 256 T^{5} + 70 T^{6} + 10 T^{7} + T^{8}$$
$11$ $$1089 + 990 T^{2} + 867 T^{4} + 30 T^{6} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$169 + 130 T + 334 T^{2} - 128 T^{3} + 331 T^{4} + 16 T^{5} + 22 T^{6} - 2 T^{7} + T^{8}$$
$19$ $$( 169 + 104 T + 51 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$23$ $$89401 - 43654 T + 23110 T^{2} - 5104 T^{3} + 1795 T^{4} - 352 T^{5} + 94 T^{6} - 10 T^{7} + T^{8}$$
$29$ $$1 - 40 T + 1618 T^{2} + 704 T^{3} + 643 T^{4} - 64 T^{5} + 82 T^{6} + 8 T^{7} + T^{8}$$
$31$ $$( -8 - 4 T + T^{2} )^{4}$$
$37$ $$1 - 38 T + 1498 T^{2} + 2056 T^{3} + 2839 T^{4} + 184 T^{5} + 58 T^{6} - 2 T^{7} + T^{8}$$
$41$ $$( 1 + 4 T + 15 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$43$ $$169 + 130 T + 334 T^{2} - 128 T^{3} + 331 T^{4} + 16 T^{5} + 22 T^{6} - 2 T^{7} + T^{8}$$
$47$ $$( -1328 + 736 T - 72 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$53$ $$( -48 + 36 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$59$ $$9 - 36 T + 234 T^{2} + 432 T^{3} + 759 T^{4} + 336 T^{5} + 114 T^{6} + 12 T^{7} + T^{8}$$
$61$ $$1590121 + 1215604 T + 603958 T^{2} + 178096 T^{3} + 38311 T^{4} + 5296 T^{5} + 526 T^{6} + 28 T^{7} + T^{8}$$
$67$ $$7667361 + 4369482 T + 1576314 T^{2} + 354600 T^{3} + 58791 T^{4} + 6744 T^{5} + 570 T^{6} + 30 T^{7} + T^{8}$$
$71$ $$109767529 - 4484156 T + 2383354 T^{2} + 6064 T^{3} + 35335 T^{4} + 16 T^{5} + 226 T^{6} + 4 T^{7} + T^{8}$$
$73$ $$( -1712 - 832 T - 84 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$79$ $$( 4432 - 640 T - 132 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$83$ $$( -192 - 288 T - 24 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$89$ $$78375609 + 19016244 T + 6685506 T^{2} - 290160 T^{3} + 71679 T^{4} - 1488 T^{5} + 378 T^{6} - 12 T^{7} + T^{8}$$
$97$ $$196249 - 165682 T + 100006 T^{2} - 31888 T^{3} + 7795 T^{4} - 928 T^{5} + 94 T^{6} + 2 T^{7} + T^{8}$$