# Properties

 Label 625.2.e.a Level $625$ Weight $2$ Character orbit 625.e Analytic conductor $4.991$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.e (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.58140625.2 Defining polynomial: $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800$$$$)/1355$$ $$\beta_{3}$$ $$=$$ $$($$$$420 \nu^{7} - 776 \nu^{6} + 698 \nu^{5} - 1924 \nu^{4} + 2297 \nu^{3} + 5129 \nu^{2} + 1055 \nu - 10265$$$$)/1355$$ $$\beta_{4}$$ $$=$$ $$($$$$728 \nu^{7} - 1327 \nu^{6} + 1246 \nu^{5} - 3353 \nu^{4} + 3584 \nu^{3} + 8547 \nu^{2} + 2190 \nu - 15715$$$$)/1355$$ $$\beta_{5}$$ $$=$$ $$($$$$-857 \nu^{7} + 1666 \nu^{6} - 1743 \nu^{5} + 4424 \nu^{4} - 4907 \nu^{3} - 9470 \nu^{2} - 2485 \nu + 18200$$$$)/1355$$ $$\beta_{6}$$ $$=$$ $$($$$$891 \nu^{7} - 1623 \nu^{6} + 1624 \nu^{5} - 4492 \nu^{4} + 4991 \nu^{3} + 9520 \nu^{2} + 3235 \nu - 18960$$$$)/1355$$ $$\beta_{7}$$ $$=$$ $$($$$$955 \nu^{7} - 1829 \nu^{6} + 1942 \nu^{5} - 4891 \nu^{4} + 5723 \nu^{3} + 9646 \nu^{2} + 2415 \nu - 20550$$$$)/1355$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 6 \beta_{6} - 7 \beta_{3} + 11 \beta_{2} + 4 \beta_{1} - 13$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{6} + 7 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} + 21 \beta_{2} - 2 \beta_{1} - 21$$ $$\nu^{7}$$ $$=$$ $$23 \beta_{7} + 38 \beta_{5} + 23 \beta_{4} - 23 \beta_{3} + 12 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 1.66637 − 0.917186i −0.357358 + 1.86824i −0.983224 + 0.644389i 1.17421 − 0.0566033i −0.983224 − 0.644389i 1.17421 + 0.0566033i 1.66637 + 0.917186i −0.357358 − 1.86824i
−1.07822 0.350334i −1.52988 + 2.10569i −0.578217 0.420099i 0 2.38723 1.73443i 0.407162i 1.80902 + 2.48990i −1.16637 3.58973i 0
124.2 2.19625 + 0.713605i −0.279141 + 0.384204i 2.69625 + 1.95894i 0 −0.887234 + 0.644613i 3.03582i 1.80902 + 2.48990i 0.857358 + 2.63868i 0
249.1 −1.22570 + 1.68703i −2.09089 + 0.679371i −0.725700 2.23347i 0 1.41668 4.36010i 0.992398i 0.690983 + 0.224514i 1.48322 1.07763i 0
249.2 0.107666 0.148189i 1.39991 0.454857i 0.607666 + 1.87020i 0 0.0833172 0.256424i 3.26086i 0.690983 + 0.224514i −0.674207 + 0.489840i 0
374.1 −1.22570 1.68703i −2.09089 0.679371i −0.725700 + 2.23347i 0 1.41668 + 4.36010i 0.992398i 0.690983 0.224514i 1.48322 + 1.07763i 0
374.2 0.107666 + 0.148189i 1.39991 + 0.454857i 0.607666 1.87020i 0 0.0833172 + 0.256424i 3.26086i 0.690983 0.224514i −0.674207 0.489840i 0
499.1 −1.07822 + 0.350334i −1.52988 2.10569i −0.578217 + 0.420099i 0 2.38723 + 1.73443i 0.407162i 1.80902 2.48990i −1.16637 + 3.58973i 0
499.2 2.19625 0.713605i −0.279141 0.384204i 2.69625 1.95894i 0 −0.887234 0.644613i 3.03582i 1.80902 2.48990i 0.857358 2.63868i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 499.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.a 8
5.b even 2 1 625.2.e.i 8
5.c odd 4 2 625.2.d.o 16
25.d even 5 1 25.2.e.a 8
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.b.c 8
25.d even 5 1 625.2.e.i 8
25.e even 10 1 25.2.e.a 8
25.e even 10 1 125.2.e.b 8
25.e even 10 1 625.2.b.c 8
25.e even 10 1 inner 625.2.e.a 8
25.f odd 20 4 125.2.d.b 16
25.f odd 20 2 625.2.a.f 8
25.f odd 20 2 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.j odd 10 1 225.2.m.a 8
75.l even 20 2 5625.2.a.x 8
100.h odd 10 1 400.2.y.c 8
100.j odd 10 1 400.2.y.c 8
100.l even 20 2 10000.2.a.bj 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 25.d even 5 1
25.2.e.a 8 25.e even 10 1
125.2.d.b 16 25.f odd 20 4
125.2.e.b 8 25.d even 5 1
125.2.e.b 8 25.e even 10 1
225.2.m.a 8 75.h odd 10 1
225.2.m.a 8 75.j odd 10 1
400.2.y.c 8 100.h odd 10 1
400.2.y.c 8 100.j odd 10 1
625.2.a.f 8 25.f odd 20 2
625.2.b.c 8 25.d even 5 1
625.2.b.c 8 25.e even 10 1
625.2.d.o 16 5.c odd 4 2
625.2.d.o 16 25.f odd 20 2
625.2.e.a 8 1.a even 1 1 trivial
625.2.e.a 8 25.e even 10 1 inner
625.2.e.i 8 5.b even 2 1
625.2.e.i 8 25.d even 5 1
5625.2.a.x 8 75.l even 20 2
10000.2.a.bj 8 100.l even 20 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{8} - 4 T_{2}^{6} - 10 T_{2}^{5} + 11 T_{2}^{4} + 40 T_{2}^{3} + 21 T_{2}^{2} - 5 T_{2} + 1$$ $$T_{3}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T + 21 T^{2} + 40 T^{3} + 11 T^{4} - 10 T^{5} - 4 T^{6} + T^{8}$$
$3$ $$16 + 40 T + 64 T^{2} - 20 T^{3} - 39 T^{4} - 5 T^{5} + 9 T^{6} + 5 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$16 + 116 T^{2} + 121 T^{4} + 21 T^{6} + T^{8}$$
$11$ $$( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$13$ $$1 - 5 T + 14 T^{2} - 5 T^{3} - 19 T^{4} + 5 T^{5} + 14 T^{6} + 5 T^{7} + T^{8}$$
$17$ $$1936 - 2200 T + 1416 T^{2} - 580 T^{3} + 341 T^{4} - 230 T^{5} + 86 T^{6} - 15 T^{7} + T^{8}$$
$19$ $$400 - 800 T + 800 T^{2} - 450 T^{3} + 335 T^{4} - 135 T^{5} + 50 T^{6} - 10 T^{7} + T^{8}$$
$23$ $$256 - 960 T + 1744 T^{2} - 1860 T^{3} + 1201 T^{4} - 465 T^{5} + 109 T^{6} - 15 T^{7} + T^{8}$$
$29$ $$483025 - 344025 T + 132750 T^{2} - 33775 T^{3} + 6885 T^{4} - 1105 T^{5} + 150 T^{6} - 15 T^{7} + T^{8}$$
$31$ $$1936 - 9064 T + 105252 T^{2} - 16322 T^{3} + 2255 T^{4} + 67 T^{5} - 3 T^{6} - T^{7} + T^{8}$$
$37$ $$116281 + 20460 T + 116 T^{2} + 8530 T^{3} + 3631 T^{4} + 380 T^{5} + T^{6} + 5 T^{7} + T^{8}$$
$41$ $$13456 - 15544 T + 8332 T^{2} - 1222 T^{3} + 3805 T^{4} + 382 T^{5} + 62 T^{6} + 9 T^{7} + T^{8}$$
$43$ $$246016 + 56784 T^{2} + 4421 T^{4} + 129 T^{6} + T^{8}$$
$47$ $$65536 - 61440 T + 38656 T^{2} + 1200 T^{3} - 939 T^{4} - 450 T^{5} + 26 T^{6} + 15 T^{7} + T^{8}$$
$53$ $$8755681 - 3314080 T + 530284 T^{2} - 92710 T^{3} + 25131 T^{4} - 4840 T^{5} + 549 T^{6} - 35 T^{7} + T^{8}$$
$59$ $$4080400 - 2605800 T + 851400 T^{2} - 172200 T^{3} + 24985 T^{4} - 2610 T^{5} + 220 T^{6} - 15 T^{7} + T^{8}$$
$61$ $$116281 + 140151 T + 38437 T^{2} - 51672 T^{3} + 17305 T^{4} - 318 T^{5} + 252 T^{6} - 6 T^{7} + T^{8}$$
$67$ $$246016 - 79360 T - 24384 T^{2} - 3520 T^{3} + 4976 T^{4} + 880 T^{5} - 4 T^{6} + T^{8}$$
$71$ $$24245776 + 9818456 T + 2542392 T^{2} + 427928 T^{3} + 54105 T^{4} + 5172 T^{5} + 462 T^{6} + 29 T^{7} + T^{8}$$
$73$ $$1 + 5 T - T^{2} - 30 T^{3} + T^{4} + 30 T^{5} + 74 T^{6} - 10 T^{7} + T^{8}$$
$79$ $$33408400 - 4913000 T + 433500 T^{2} + 36550 T^{3} + 7935 T^{4} + 375 T^{5} + 270 T^{6} + 10 T^{7} + T^{8}$$
$83$ $$99856 + 9480 T + 7024 T^{2} - 74940 T^{3} + 36361 T^{4} - 1185 T^{5} - 11 T^{6} - 15 T^{7} + T^{8}$$
$89$ $$1392400 - 1014800 T + 629000 T^{2} - 206550 T^{3} + 48335 T^{4} - 7485 T^{5} + 740 T^{6} - 40 T^{7} + T^{8}$$
$97$ $$301334881 + 110403240 T + 15337196 T^{2} + 795650 T^{3} - 28154 T^{4} - 6200 T^{5} - 119 T^{6} + 10 T^{7} + T^{8}$$