# Properties

 Label 930.2.n.a Level $930$ Weight $2$ Character orbit 930.n Analytic conductor $7.426$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.13140625.1 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{7} q^{3} + \beta_{4} q^{4} + q^{5} - q^{6} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{7} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{8} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{7} q^{3} + \beta_{4} q^{4} + q^{5} - q^{6} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{7} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{8} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{9} -\beta_{3} q^{10} + ( 2 - \beta_{2} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{11} + \beta_{3} q^{12} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{13} + ( 2 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{14} -\beta_{7} q^{15} + \beta_{7} q^{16} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{17} + \beta_{7} q^{18} + ( 3 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{19} + \beta_{4} q^{20} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{21} + ( -5 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{22} + ( 2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{23} -\beta_{4} q^{24} + q^{25} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{7} ) q^{26} -\beta_{4} q^{27} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{28} + ( 1 + \beta_{1} + 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{29} - q^{30} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{31} + q^{32} + ( 2 - 5 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{33} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{34} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{35} + q^{36} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{7} ) q^{37} + ( -2 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{38} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{39} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{40} + ( 4 + \beta_{1} - \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} ) q^{41} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{42} + ( -2 + \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{43} + ( -\beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{44} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{45} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{6} - 2 \beta_{7} ) q^{46} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - 5 \beta_{7} ) q^{47} + ( 1 - \beta_{3} + \beta_{4} + \beta_{7} ) q^{48} + ( -5 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 5 \beta_{6} - 4 \beta_{7} ) q^{49} -\beta_{3} q^{50} + ( -2 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{51} + ( 2 - 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{52} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{53} + ( 1 - \beta_{3} + \beta_{4} + \beta_{7} ) q^{54} + ( 2 - \beta_{2} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{55} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{56} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{57} + ( -1 - 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{58} + ( 7 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 7 \beta_{6} + \beta_{7} ) q^{59} + \beta_{3} q^{60} + ( -3 \beta_{3} + 3 \beta_{7} ) q^{61} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{62} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{63} -\beta_{3} q^{64} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{65} + ( -2 + \beta_{2} - 2 \beta_{4} - \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{66} + ( -2 - 5 \beta_{1} - 5 \beta_{2} + 7 \beta_{3} - 3 \beta_{5} - 7 \beta_{7} ) q^{67} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{68} + ( -2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{69} + ( 2 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{70} + ( 7 - 6 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} + 9 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} + 9 \beta_{7} ) q^{71} -\beta_{3} q^{72} + ( -6 - \beta_{4} + 10 \beta_{6} - 6 \beta_{7} ) q^{73} + ( 1 - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{74} -\beta_{7} q^{75} + ( 3 + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} ) q^{76} + ( 7 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 7 \beta_{6} - \beta_{7} ) q^{77} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{78} + ( -12 + \beta_{1} - 3 \beta_{2} + 12 \beta_{3} - 10 \beta_{4} - \beta_{5} + 3 \beta_{6} - 10 \beta_{7} ) q^{79} + \beta_{7} q^{80} -\beta_{3} q^{81} + ( -4 + \beta_{2} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{82} + ( 3 - 10 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} ) q^{83} + ( -2 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{84} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{85} + ( 2 - 2 \beta_{2} + 6 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{86} + ( 3 - 4 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{87} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - 5 \beta_{5} - 2 \beta_{7} ) q^{88} + ( 2 - \beta_{2} - 8 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{89} + \beta_{7} q^{90} + ( -2 + 5 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{91} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{5} ) q^{92} + ( -5 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{93} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 5 \beta_{5} - \beta_{7} ) q^{94} + ( 3 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{95} -\beta_{7} q^{96} + ( -6 - 4 \beta_{2} - 9 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{97} + ( 5 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} - 7 \beta_{5} + 4 \beta_{7} ) q^{98} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - 5 \beta_{5} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} + 2q^{3} - 2q^{4} + 8q^{5} - 8q^{6} + 9q^{7} - 2q^{8} - 2q^{9} + O(q^{10})$$ $$8q - 2q^{2} + 2q^{3} - 2q^{4} + 8q^{5} - 8q^{6} + 9q^{7} - 2q^{8} - 2q^{9} - 2q^{10} - q^{11} + 2q^{12} + q^{13} + 9q^{14} + 2q^{15} - 2q^{16} + 13q^{17} - 2q^{18} - 2q^{19} - 2q^{20} + q^{21} - q^{22} + 8q^{23} + 2q^{24} + 8q^{25} + 16q^{26} + 2q^{27} - q^{28} - q^{29} - 8q^{30} + 3q^{31} + 8q^{32} + q^{33} - 12q^{34} + 9q^{35} + 8q^{36} + 8q^{37} + 8q^{38} + 9q^{39} - 2q^{40} + 29q^{41} - 9q^{42} - 25q^{43} - q^{44} - 2q^{45} + 3q^{46} + 13q^{47} + 2q^{48} + 25q^{49} - 2q^{50} - 13q^{51} + q^{52} - 19q^{53} + 2q^{54} - q^{55} - 16q^{56} + 12q^{57} + 4q^{58} + 23q^{59} + 2q^{60} - 12q^{61} - 27q^{62} - 16q^{63} - 2q^{64} + q^{65} + q^{66} - 24q^{67} - 2q^{68} - 8q^{69} + 9q^{70} + 3q^{71} - 2q^{72} - 14q^{73} + 8q^{74} + 2q^{75} + 8q^{76} - 24q^{77} - q^{78} - 34q^{79} - 2q^{80} - 2q^{81} - 21q^{82} + 8q^{83} - 9q^{84} + 13q^{85} + 6q^{87} + 4q^{88} + 25q^{89} - 2q^{90} + 3q^{91} - 22q^{92} - 18q^{93} - 42q^{94} - 2q^{95} + 2q^{96} - 28q^{97} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{3} - \beta_{4} - \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
481.1
 1.69513 − 1.23158i −0.386111 + 0.280526i −0.227943 + 0.701538i 0.418926 − 1.28932i −0.227943 − 0.701538i 0.418926 + 1.28932i 1.69513 + 1.23158i −0.386111 − 0.280526i
−0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i 1.00000 −1.00000 −0.518200 + 1.59485i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.2 −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i 1.00000 −1.00000 −0.0268854 + 0.0827449i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
721.1 0.309017 0.951057i −0.309017 0.951057i −0.809017 0.587785i 1.00000 −1.00000 1.15245 + 0.837304i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.2 0.309017 0.951057i −0.309017 0.951057i −0.809017 0.587785i 1.00000 −1.00000 3.89263 + 2.82816i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
841.1 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i 1.00000 −1.00000 1.15245 0.837304i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.2 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i 1.00000 −1.00000 3.89263 2.82816i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
901.1 −0.809017 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i 1.00000 −1.00000 −0.518200 1.59485i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.2 −0.809017 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i 1.00000 −1.00000 −0.0268854 0.0827449i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.a 8
31.d even 5 1 inner 930.2.n.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.a 8 1.a even 1 1 trivial
930.2.n.a 8 31.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$( -1 + T )^{8}$$
$7$ $$1 + 6 T + 125 T^{2} - 141 T^{3} + 94 T^{4} - 51 T^{5} + 35 T^{6} - 9 T^{7} + T^{8}$$
$11$ $$410881 - 5769 T + 29090 T^{2} - 2131 T^{3} + 949 T^{4} + 19 T^{5} + 10 T^{6} + T^{7} + T^{8}$$
$13$ $$121 - 44 T + 177 T^{2} - 247 T^{3} + 180 T^{4} + 137 T^{5} + 37 T^{6} - T^{7} + T^{8}$$
$17$ $$121 - 737 T + 1875 T^{2} - 1417 T^{3} + 894 T^{4} - 313 T^{5} + 85 T^{6} - 13 T^{7} + T^{8}$$
$19$ $$7921 + 5518 T + 5705 T^{2} + 1158 T^{3} + 179 T^{4} - 18 T^{5} + 5 T^{6} + 2 T^{7} + T^{8}$$
$23$ $$1681 + 3034 T + 2919 T^{2} + 1538 T^{3} + 659 T^{4} + 86 T^{5} + 21 T^{6} - 8 T^{7} + T^{8}$$
$29$ $$323761 - 41537 T + 57786 T^{2} + 5229 T^{3} + 1869 T^{4} - 53 T^{5} - 6 T^{6} + T^{7} + T^{8}$$
$31$ $$923521 - 89373 T - 1922 T^{2} - 5301 T^{3} + 1275 T^{4} - 171 T^{5} - 2 T^{6} - 3 T^{7} + T^{8}$$
$37$ $$( -139 + 124 T - 23 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$41$ $$346921 - 305102 T + 172461 T^{2} - 63301 T^{3} + 16754 T^{4} - 3103 T^{5} + 399 T^{6} - 29 T^{7} + T^{8}$$
$43$ $$225625 - 59375 T + 157875 T^{2} + 48125 T^{3} + 9350 T^{4} + 1625 T^{5} + 285 T^{6} + 25 T^{7} + T^{8}$$
$47$ $$346921 - 325717 T + 228195 T^{2} - 86697 T^{3} + 18894 T^{4} - 2053 T^{5} + 165 T^{6} - 13 T^{7} + T^{8}$$
$53$ $$361 + 589 T + 1335 T^{2} + 2841 T^{3} + 4284 T^{4} + 1111 T^{5} + 195 T^{6} + 19 T^{7} + T^{8}$$
$59$ $$6974881 + 4708903 T + 1276170 T^{2} - 5107 T^{3} + 21789 T^{4} - 2933 T^{5} + 370 T^{6} - 23 T^{7} + T^{8}$$
$61$ $$( -9 + 3 T + T^{2} )^{4}$$
$67$ $$( -8209 - 2613 T - 172 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$71$ $$244328161 + 3173093 T + 3175935 T^{2} + 249893 T^{3} + 24354 T^{4} - 1303 T^{5} + 125 T^{6} - 3 T^{7} + T^{8}$$
$73$ $$144264121 - 29715214 T + 2732397 T^{2} + 37878 T^{3} + 70455 T^{4} + 4262 T^{5} + 297 T^{6} + 14 T^{7} + T^{8}$$
$79$ $$17147881 + 8000412 T + 3779911 T^{2} + 713796 T^{3} + 81509 T^{4} + 6768 T^{5} + 589 T^{6} + 34 T^{7} + T^{8}$$
$83$ $$1907161 - 2411226 T + 1356334 T^{2} - 380532 T^{3} + 53684 T^{4} - 2964 T^{5} + 361 T^{6} - 8 T^{7} + T^{8}$$
$89$ $$25391521 - 4006005 T + 971198 T^{2} - 185855 T^{3} + 27649 T^{4} - 2665 T^{5} + 322 T^{6} - 25 T^{7} + T^{8}$$
$97$ $$214300321 - 9412877 T + 192133 T^{2} + 54571 T^{3} + 53580 T^{4} + 6641 T^{5} + 603 T^{6} + 28 T^{7} + T^{8}$$