# Properties

 Label 847.2.f.p Level 847 Weight 2 Character orbit 847.f Analytic conductor 6.763 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 847.f (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.159390625.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6 x^{6} - 11 x^{5} + 21 x^{4} - 5 x^{3} + 10 x^{2} + 25 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$555 \nu^{7} - 2159 \nu^{6} + 7489 \nu^{5} - 18164 \nu^{4} + 40069 \nu^{3} - 84434 \nu^{2} + 43855 \nu + 375$$$$)/94655$$ $$\beta_{3}$$ $$=$$ $$($$$$-970 \nu^{7} - 1002 \nu^{6} - 6608 \nu^{5} + 9063 \nu^{4} - 14943 \nu^{3} + 27673 \nu^{2} - 68120 \nu + 35160$$$$)/94655$$ $$\beta_{4}$$ $$=$$ $$($$$$-1604 \nu^{7} + 4159 \nu^{6} - 12059 \nu^{5} + 28414 \nu^{4} - 81659 \nu^{3} + 38305 \nu^{2} - 13500 \nu - 13875$$$$)/94655$$ $$\beta_{5}$$ $$=$$ $$($$$$-2052 \nu^{7} + 2252 \nu^{6} - 19912 \nu^{5} + 21007 \nu^{4} - 82042 \nu^{3} + 35785 \nu^{2} - 19395 \nu - 90925$$$$)/94655$$ $$\beta_{6}$$ $$=$$ $$($$$$-2667 \nu^{7} + 6691 \nu^{6} - 17466 \nu^{5} + 50856 \nu^{4} - 82441 \nu^{3} + 72554 \nu^{2} - 4035 \nu - 12035$$$$)/94655$$ $$\beta_{7}$$ $$=$$ $$($$$$4024 \nu^{7} - 1464 \nu^{6} + 21519 \nu^{5} - 26434 \nu^{4} + 59219 \nu^{3} + 22635 \nu^{2} + 54640 \nu + 66675$$$$)/94655$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3 \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} + \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{7} + 7 \beta_{6} + 2 \beta_{5} + 13 \beta_{3} + 13 \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{7} - 11 \beta_{6} - 20 \beta_{5} + 20 \beta_{4} - 11 \beta_{2} + 8 \beta_{1} - 12$$ $$\nu^{6}$$ $$=$$ $$-19 \beta_{7} - 19 \beta_{4} - 68 \beta_{3} - 36 \beta_{2} - 24 \beta_{1} + 36$$ $$\nu^{7}$$ $$=$$ $$111 \beta_{7} + 81 \beta_{6} + 111 \beta_{5} - 55 \beta_{4} + 81 \beta_{3} + 148 \beta_{2} - 56 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 0.453245 + 1.39494i −0.762262 − 2.34600i 1.43801 + 1.04478i −0.628998 − 0.456994i 0.453245 − 1.39494i −0.762262 + 2.34600i 1.43801 − 1.04478i −0.628998 + 0.456994i
−0.453245 1.39494i 1.30902 + 0.951057i −0.122406 + 0.0889332i 0.144228 0.443888i 0.733366 2.25707i 0.809017 0.587785i −2.19369 1.59381i −0.118034 0.363271i −0.684570
148.2 0.762262 + 2.34600i 1.30902 + 0.951057i −3.30464 + 2.40097i −1.07128 + 3.29706i −1.23337 + 3.79591i 0.809017 0.587785i −4.16042 3.02272i −0.118034 0.363271i −8.55150
323.1 −1.43801 1.04478i 0.190983 0.587785i 0.358290 + 1.10270i 2.24703 1.63256i −0.888742 + 0.645709i −0.309017 0.951057i −0.461691 + 1.42094i 2.11803 + 1.53884i −4.93693
323.2 0.628998 + 0.456994i 0.190983 0.587785i −0.431239 1.32722i 0.180019 0.130791i 0.388742 0.282438i −0.309017 0.951057i 0.815793 2.51075i 2.11803 + 1.53884i 0.173002
372.1 −0.453245 + 1.39494i 1.30902 0.951057i −0.122406 0.0889332i 0.144228 + 0.443888i 0.733366 + 2.25707i 0.809017 + 0.587785i −2.19369 + 1.59381i −0.118034 + 0.363271i −0.684570
372.2 0.762262 2.34600i 1.30902 0.951057i −3.30464 2.40097i −1.07128 3.29706i −1.23337 3.79591i 0.809017 + 0.587785i −4.16042 + 3.02272i −0.118034 + 0.363271i −8.55150
729.1 −1.43801 + 1.04478i 0.190983 + 0.587785i 0.358290 1.10270i 2.24703 + 1.63256i −0.888742 0.645709i −0.309017 + 0.951057i −0.461691 1.42094i 2.11803 1.53884i −4.93693
729.2 0.628998 0.456994i 0.190983 + 0.587785i −0.431239 + 1.32722i 0.180019 + 0.130791i 0.388742 + 0.282438i −0.309017 + 0.951057i 0.815793 + 2.51075i 2.11803 1.53884i 0.173002
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.p 8
11.b odd 2 1 847.2.f.s 8
11.c even 5 2 77.2.f.a 8
11.c even 5 1 847.2.a.l 4
11.c even 5 1 inner 847.2.f.p 8
11.d odd 10 1 847.2.a.k 4
11.d odd 10 2 847.2.f.q 8
11.d odd 10 1 847.2.f.s 8
33.f even 10 1 7623.2.a.co 4
33.h odd 10 2 693.2.m.g 8
33.h odd 10 1 7623.2.a.ch 4
77.j odd 10 2 539.2.f.d 8
77.j odd 10 1 5929.2.a.bi 4
77.l even 10 1 5929.2.a.bb 4
77.m even 15 4 539.2.q.c 16
77.p odd 30 4 539.2.q.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 11.c even 5 2
539.2.f.d 8 77.j odd 10 2
539.2.q.b 16 77.p odd 30 4
539.2.q.c 16 77.m even 15 4
693.2.m.g 8 33.h odd 10 2
847.2.a.k 4 11.d odd 10 1
847.2.a.l 4 11.c even 5 1
847.2.f.p 8 1.a even 1 1 trivial
847.2.f.p 8 11.c even 5 1 inner
847.2.f.q 8 11.d odd 10 2
847.2.f.s 8 11.b odd 2 1
847.2.f.s 8 11.d odd 10 1
5929.2.a.bb 4 77.l even 10 1
5929.2.a.bi 4 77.j odd 10 1
7623.2.a.ch 4 33.h odd 10 1
7623.2.a.co 4 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{8} + \cdots$$ $$T_{3}^{4} - 3 T_{3}^{3} + 4 T_{3}^{2} - 2 T_{3} + 1$$ $$T_{13}^{8} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 2 T^{2} + 5 T^{3} + 5 T^{4} + 19 T^{5} + 26 T^{6} + 25 T^{7} + 59 T^{8} + 50 T^{9} + 104 T^{10} + 152 T^{11} + 80 T^{12} + 160 T^{13} + 128 T^{14} + 128 T^{15} + 256 T^{16}$$
$3$ $$( 1 - 3 T + T^{2} + T^{3} + 4 T^{4} + 3 T^{5} + 9 T^{6} - 81 T^{7} + 81 T^{8} )^{2}$$
$5$ $$1 - 3 T + 2 T^{2} + 6 T^{4} - 75 T^{5} + 208 T^{6} - 234 T^{7} + 431 T^{8} - 1170 T^{9} + 5200 T^{10} - 9375 T^{11} + 3750 T^{12} + 31250 T^{14} - 234375 T^{15} + 390625 T^{16}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$11$ 
$13$ $$1 + 5 T - 30 T^{2} - 170 T^{3} + 286 T^{4} + 2815 T^{5} + 5030 T^{6} - 15820 T^{7} - 130849 T^{8} - 205660 T^{9} + 850070 T^{10} + 6184555 T^{11} + 8168446 T^{12} - 63119810 T^{13} - 144804270 T^{14} + 313742585 T^{15} + 815730721 T^{16}$$
$17$ $$1 - 14 T + 79 T^{2} - 281 T^{3} + 1089 T^{4} - 2031 T^{5} - 15507 T^{6} + 106442 T^{7} - 393507 T^{8} + 1809514 T^{9} - 4481523 T^{10} - 9978303 T^{11} + 90954369 T^{12} - 398979817 T^{13} + 1906867951 T^{14} - 5744741422 T^{15} + 6975757441 T^{16}$$
$19$ $$1 - 6 T - 27 T^{2} + 111 T^{3} + 483 T^{4} - 2061 T^{5} + 20515 T^{6} - 13260 T^{7} - 552129 T^{8} - 251940 T^{9} + 7405915 T^{10} - 14136399 T^{11} + 62945043 T^{12} + 274846989 T^{13} - 1270238787 T^{14} - 5363230434 T^{15} + 16983563041 T^{16}$$
$23$ $$( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 9246 T^{5} + 43907 T^{6} + 97336 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$1 - 6 T + 23 T^{2} + 6 T^{3} + 138 T^{4} + 5484 T^{5} - 19805 T^{6} + 121230 T^{7} + 57131 T^{8} + 3515670 T^{9} - 16656005 T^{10} + 133749276 T^{11} + 97604778 T^{12} + 123066894 T^{13} + 13680936383 T^{14} - 103499257854 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 14 T + 29 T^{2} + 248 T^{3} + 530 T^{4} - 15954 T^{5} + 65935 T^{6} - 166230 T^{7} + 667951 T^{8} - 5153130 T^{9} + 63363535 T^{10} - 475285614 T^{11} + 489466130 T^{12} + 7100029448 T^{13} + 25737606749 T^{14} - 385176597554 T^{15} + 852891037441 T^{16}$$
$37$ $$1 - T - 48 T^{2} + 175 T^{3} + 2760 T^{4} - 1334 T^{5} - 138534 T^{6} - 40270 T^{7} + 6227939 T^{8} - 1489990 T^{9} - 189653046 T^{10} - 67571102 T^{11} + 5172684360 T^{12} + 12135192475 T^{13} - 123154867632 T^{14} - 94931877133 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 18 T + 145 T^{2} - 1046 T^{3} + 9461 T^{4} - 59718 T^{5} + 237483 T^{6} - 1463634 T^{7} + 11961308 T^{8} - 60008994 T^{9} + 399208923 T^{10} - 4115824278 T^{11} + 26734524821 T^{12} - 121185586246 T^{13} + 688765114945 T^{14} - 3505576929858 T^{15} + 7984925229121 T^{16}$$
$43$ $$( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$1 - 7 T + 85 T^{2} - 1176 T^{3} + 9351 T^{4} - 81508 T^{5} + 731706 T^{6} - 5276705 T^{7} + 35329489 T^{8} - 248005135 T^{9} + 1616338554 T^{10} - 8462405084 T^{11} + 45629897031 T^{12} - 269709728232 T^{13} + 916233302965 T^{14} - 3546361843241 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 7 T + 132 T^{2} - 1421 T^{3} + 16876 T^{4} - 131512 T^{5} + 1370850 T^{6} - 9985608 T^{7} + 80785479 T^{8} - 529237224 T^{9} + 3850717650 T^{10} - 19579112024 T^{11} + 133159757356 T^{12} - 594255795553 T^{13} + 2925695669028 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 9 T^{2} + 800 T^{3} + 2280 T^{4} - 15200 T^{5} + 247289 T^{6} + 987600 T^{7} - 4469641 T^{8} + 58268400 T^{9} + 860813009 T^{10} - 3121760800 T^{11} + 27627583080 T^{12} + 571939439200 T^{13} - 379624802769 T^{14} + 146830437604321 T^{16}$$
$61$ $$1 - 12 T + 147 T^{2} - 2312 T^{3} + 22368 T^{4} - 224532 T^{5} + 2244305 T^{6} - 18537880 T^{7} + 147792991 T^{8} - 1130810680 T^{9} + 8351058905 T^{10} - 50964497892 T^{11} + 309703771488 T^{12} - 1952706647912 T^{13} + 7573495031067 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16}$$
$67$ $$( 1 + 15 T + 335 T^{2} + 3060 T^{3} + 35713 T^{4} + 205020 T^{5} + 1503815 T^{6} + 4511445 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$1 - 21 T + 151 T^{2} - 497 T^{3} + 5457 T^{4} - 14896 T^{5} - 888488 T^{6} + 11215568 T^{7} - 84203719 T^{8} + 796305328 T^{9} - 4478868008 T^{10} - 5331442256 T^{11} + 138671543217 T^{12} - 896701987447 T^{13} + 19343142872071 T^{14} - 190997523326211 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 - 8 T - 77 T^{2} + 990 T^{3} - 2170 T^{4} - 132362 T^{5} + 1139369 T^{6} + 4655830 T^{7} - 91523921 T^{8} + 339875590 T^{9} + 6071697401 T^{10} - 51491068154 T^{11} - 61624182970 T^{12} + 2052340877070 T^{13} - 11652735424253 T^{14} - 88379188152776 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - T - 160 T^{2} - 548 T^{3} + 13806 T^{4} + 31769 T^{5} - 282588 T^{6} - 2159462 T^{7} + 16750183 T^{8} - 170597498 T^{9} - 1763631708 T^{10} + 15663355991 T^{11} + 537744818286 T^{12} - 1686226906652 T^{13} - 38893992883360 T^{14} - 19203908986159 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 + 22 T + 313 T^{2} + 2340 T^{3} + 7050 T^{4} - 116402 T^{5} - 1950441 T^{6} - 18988930 T^{7} - 158760641 T^{8} - 1576081190 T^{9} - 13436588049 T^{10} - 66557150374 T^{11} + 334581163050 T^{12} + 9217355104620 T^{13} + 102332336864497 T^{14} + 596993121771794 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 17 T + 312 T^{2} + 3419 T^{3} + 38939 T^{4} + 304291 T^{5} + 2471352 T^{6} + 11984473 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 15 T + 36 T^{2} + 375 T^{3} + 2412 T^{4} - 52890 T^{5} + 1250458 T^{6} + 18589350 T^{7} + 83790255 T^{8} + 1803166950 T^{9} + 11765559322 T^{10} - 48271274970 T^{11} + 213532625772 T^{12} + 3220252596375 T^{13} + 29986992177444 T^{14} + 1211974267171695 T^{15} + 7837433594376961 T^{16}$$