# Properties

 Label 693.2.m.g Level 693 Weight 2 Character orbit 693.m Analytic conductor 5.534 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 693.m (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.53363286007$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.159390625.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} ) q^{5} + \beta_{3} q^{7} + ( 2 - 3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{4} q^{2} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} ) q^{5} + \beta_{3} q^{7} + ( 2 - 3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{8} + ( -2 - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{10} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{11} + ( 2 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{13} + \beta_{7} q^{14} + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{16} + ( \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{17} + ( -2 + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{19} + ( -2 + \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{20} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{22} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{6} + 2 \beta_{7} ) q^{23} + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{6} + 3 \beta_{7} ) q^{25} + ( -3 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{26} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{28} + ( 3 \beta_{1} - 3 \beta_{4} - 3 \beta_{7} ) q^{29} + ( -3 - \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} ) 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_{4} + \beta_{6} ) q^{35} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{37} + ( -\beta_{1} - 7 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{38} + ( -3 + 3 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} ) q^{40} + ( 2 + \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{41} + ( -1 + 6 \beta_{2} + 6 \beta_{6} ) q^{43} + ( -5 + 2 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{44} + ( 4 - 3 \beta_{1} - 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{46} + ( -1 + 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{47} -\beta_{2} q^{49} + ( -\beta_{1} + 9 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{50} + ( 6 - 3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 6 \beta_{6} ) q^{52} + ( -1 - 3 \beta_{1} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{53} + ( 1 + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{55} + ( 2 - \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{56} + ( -3 - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{58} + ( -2 + \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + \beta_{4} + \beta_{7} ) q^{59} + ( 4 \beta_{1} + 7 \beta_{2} + \beta_{4} - 5 \beta_{5} - 5 \beta_{7} ) q^{61} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{62} + ( -\beta_{1} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) 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 - \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{68} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{70} + ( -\beta_{1} + \beta_{2} + 5 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} - 5 \beta_{7} ) q^{73} + ( 3 - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{74} + ( -4 + 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{76} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{77} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{79} + ( 1 + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{80} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} + \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{82} + ( 2 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} ) q^{83} + ( -3 + 3 \beta_{2} + 3 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{85} + ( -6 \beta_{1} - 5 \beta_{4} + 6 \beta_{5} ) q^{86} + ( -2 - \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + \beta_{4} + 3 \beta_{5} + 4 \beta_{7} ) q^{88} + ( 1 + 5 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} - 5 \beta_{7} ) q^{89} + ( 2 - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{91} + ( 7 - 2 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} ) q^{92} + ( -6 \beta_{1} - \beta_{2} - 7 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 7 \beta_{6} + 4 \beta_{7} ) q^{94} + ( -5 + 5 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} ) q^{95} + ( 2 - 5 \beta_{1} - 2 \beta_{3} + 5 \beta_{5} + 4 \beta_{6} ) q^{97} + ( \beta_{4} - \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + q^{2} + 3q^{4} - 3q^{5} + 2q^{7} - 3q^{8} + O(q^{10})$$ $$8q + q^{2} + 3q^{4} - 3q^{5} + 2q^{7} - 3q^{8} - 28q^{10} - 5q^{11} + 5q^{13} - q^{14} - 3q^{16} + 11q^{17} - 9q^{19} - 21q^{20} - q^{22} + 16q^{23} + 5q^{25} - 21q^{26} + 7q^{28} + 9q^{29} - 11q^{31} + 20q^{32} - 24q^{34} + 3q^{35} + 6q^{37} - 35q^{38} - 16q^{40} + 22q^{41} + 16q^{43} - 29q^{44} + 29q^{46} - 7q^{47} - 2q^{49} + 34q^{50} + 21q^{52} - 2q^{53} + 26q^{55} + 18q^{56} - 39q^{58} - 25q^{59} + 7q^{61} + 5q^{62} + q^{64} - 24q^{65} - 30q^{67} - 8q^{68} - 2q^{70} + 14q^{71} + 3q^{73} + 9q^{74} - 52q^{76} + 5q^{77} - 9q^{79} + 33q^{80} + 31q^{82} - 23q^{83} - 10q^{85} + 17q^{86} - 7q^{88} + 34q^{89} + 5q^{91} + 34q^{92} - 30q^{94} - 24q^{95} + 30q^{97} - 4q^{98} + O(q^{100})$$

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 $$\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$442$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 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 −0.762262 − 2.34600i
−0.549273 + 1.69049i 0 −0.938015 0.681508i 0.858290 + 2.64154i 0 0.809017 + 0.587785i −1.20872 + 0.878189i 0 −4.93693
64.2 0.240256 0.739431i 0 1.12900 + 0.820265i 0.0687611 + 0.211625i 0 0.809017 + 0.587785i 2.13577 1.55173i 0 0.173002
190.1 −1.18661 + 0.862123i 0 0.0467549 0.143897i 0.377594 + 0.274338i 0 −0.309017 + 0.951057i −0.837913 2.57883i 0 −0.684570
190.2 1.99563 1.44991i 0 1.26226 3.88484i −2.80464 2.03769i 0 −0.309017 + 0.951057i −1.58914 4.89086i 0 −8.55150
379.1 −0.549273 1.69049i 0 −0.938015 + 0.681508i 0.858290 2.64154i 0 0.809017 0.587785i −1.20872 0.878189i 0 −4.93693
379.2 0.240256 + 0.739431i 0 1.12900 0.820265i 0.0687611 0.211625i 0 0.809017 0.587785i 2.13577 + 1.55173i 0 0.173002
631.1 −1.18661 0.862123i 0 0.0467549 + 0.143897i 0.377594 0.274338i 0 −0.309017 0.951057i −0.837913 + 2.57883i 0 −0.684570
631.2 1.99563 + 1.44991i 0 1.26226 + 3.88484i −2.80464 + 2.03769i 0 −0.309017 0.951057i −1.58914 + 4.89086i 0 −8.55150
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 631.2 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 693.2.m.g 8
3.b odd 2 1 77.2.f.a 8
11.c even 5 1 inner 693.2.m.g 8
11.c even 5 1 7623.2.a.ch 4
11.d odd 10 1 7623.2.a.co 4
21.c even 2 1 539.2.f.d 8
21.g even 6 2 539.2.q.b 16
21.h odd 6 2 539.2.q.c 16
33.d even 2 1 847.2.f.q 8
33.f even 10 1 847.2.a.k 4
33.f even 10 1 847.2.f.q 8
33.f even 10 2 847.2.f.s 8
33.h odd 10 1 77.2.f.a 8
33.h odd 10 1 847.2.a.l 4
33.h odd 10 2 847.2.f.p 8
231.r odd 10 1 5929.2.a.bb 4
231.u even 10 1 539.2.f.d 8
231.u even 10 1 5929.2.a.bi 4
231.z odd 30 2 539.2.q.c 16
231.bc even 30 2 539.2.q.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 3.b odd 2 1
77.2.f.a 8 33.h odd 10 1
539.2.f.d 8 21.c even 2 1
539.2.f.d 8 231.u even 10 1
539.2.q.b 16 21.g even 6 2
539.2.q.b 16 231.bc even 30 2
539.2.q.c 16 21.h odd 6 2
539.2.q.c 16 231.z odd 30 2
693.2.m.g 8 1.a even 1 1 trivial
693.2.m.g 8 11.c even 5 1 inner
847.2.a.k 4 33.f even 10 1
847.2.a.l 4 33.h odd 10 1
847.2.f.p 8 33.h odd 10 2
847.2.f.q 8 33.d even 2 1
847.2.f.q 8 33.f even 10 1
847.2.f.s 8 33.f even 10 2
5929.2.a.bb 4 231.r odd 10 1
5929.2.a.bi 4 231.u even 10 1
7623.2.a.ch 4 11.c even 5 1
7623.2.a.co 4 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - T_{2}^{7} + T_{2}^{6} - T_{2}^{5} + 16 T_{2}^{4} + 25 T_{2}^{3} + 35 T_{2}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(693, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T - 3 T^{2} + 5 T^{3} - 19 T^{5} + 21 T^{6} + 20 T^{7} - 51 T^{8} + 40 T^{9} + 84 T^{10} - 152 T^{11} + 160 T^{13} - 192 T^{14} - 128 T^{15} + 256 T^{16}$$
$3$ 
$5$ $$1 + 3 T - 3 T^{2} - 30 T^{3} - 39 T^{4} + 150 T^{5} + 428 T^{6} - 231 T^{7} - 2169 T^{8} - 1155 T^{9} + 10700 T^{10} + 18750 T^{11} - 24375 T^{12} - 93750 T^{13} - 46875 T^{14} + 234375 T^{15} + 390625 T^{16}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$11$ $$1 + 5 T + 26 T^{2} + 75 T^{3} + 251 T^{4} + 825 T^{5} + 3146 T^{6} + 6655 T^{7} + 14641 T^{8}$$
$13$ $$1 - 5 T + 35 T^{2} - 90 T^{3} + 611 T^{4} - 670 T^{5} + 4380 T^{6} + 8865 T^{7} + 38151 T^{8} + 115245 T^{9} + 740220 T^{10} - 1471990 T^{11} + 17450771 T^{12} - 33416370 T^{13} + 168938315 T^{14} - 313742585 T^{15} + 815730721 T^{16}$$
$17$ $$1 - 11 T + 39 T^{2} - 139 T^{3} + 1249 T^{4} - 7484 T^{5} + 34098 T^{6} - 128922 T^{7} + 457683 T^{8} - 2191674 T^{9} + 9854322 T^{10} - 36768892 T^{11} + 104317729 T^{12} - 197360123 T^{13} + 941365191 T^{14} - 4513725403 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 9 T + 43 T^{2} + 171 T^{3} + 1023 T^{4} + 1674 T^{5} - 12940 T^{6} - 86580 T^{7} - 253639 T^{8} - 1645020 T^{9} - 4671340 T^{10} + 11481966 T^{11} + 133318383 T^{12} + 423412929 T^{13} + 2022972883 T^{14} + 8044845651 T^{15} + 16983563041 T^{16}$$
$23$ $$( 1 - 8 T + 83 T^{2} - 402 T^{3} + 2555 T^{4} - 9246 T^{5} + 43907 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$1 - 9 T - 22 T^{2} + 429 T^{3} - 762 T^{4} - 11754 T^{5} + 81940 T^{6} + 191700 T^{7} - 3727369 T^{8} + 5559300 T^{9} + 68911540 T^{10} - 286668306 T^{11} - 538948122 T^{12} + 8799282921 T^{13} - 13086113062 T^{14} - 155248886781 T^{15} + 500246412961 T^{16}$$
$31$ $$1 + 11 T + 34 T^{2} + 183 T^{3} + 3210 T^{4} + 12836 T^{5} - 31430 T^{6} + 25700 T^{7} + 2517721 T^{8} + 796700 T^{9} - 30204230 T^{10} + 382397276 T^{11} + 2964502410 T^{12} + 5239134633 T^{13} + 30175125154 T^{14} + 302638755221 T^{15} + 852891037441 T^{16}$$
$37$ $$1 - 6 T - 63 T^{2} + 520 T^{3} + 1350 T^{4} - 12954 T^{5} - 20209 T^{6} + 92390 T^{7} + 1366879 T^{8} + 3418430 T^{9} - 27666121 T^{10} - 656158962 T^{11} + 2530117350 T^{12} + 36058857640 T^{13} - 161640763767 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 22 T + 160 T^{2} - 414 T^{3} + 2641 T^{4} - 46342 T^{5} + 334288 T^{6} - 1062556 T^{7} + 2274033 T^{8} - 43564796 T^{9} + 561938128 T^{10} - 3193936982 T^{11} + 7462834801 T^{12} - 47964467214 T^{13} + 760016678560 T^{14} - 4284594025382 T^{15} + 7984925229121 T^{16}$$
$43$ $$( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$1 + 7 T - 100 T^{2} - 1414 T^{3} + 286 T^{4} + 103523 T^{5} + 579636 T^{6} - 2543800 T^{7} - 42803781 T^{8} - 119558600 T^{9} + 1280415924 T^{10} + 10748068429 T^{11} + 1395588766 T^{12} - 324293839898 T^{13} - 1077921532900 T^{14} + 3546361843241 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 2 T - 173 T^{2} - 964 T^{3} + 9936 T^{4} + 104742 T^{5} + 122655 T^{6} - 3229712 T^{7} - 31796061 T^{8} - 171174736 T^{9} + 344537895 T^{10} + 15593674734 T^{11} + 78399819216 T^{12} - 403140455252 T^{13} - 3834434475317 T^{14} + 2349422279674 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 25 T + 206 T^{2} - 25 T^{3} - 15220 T^{4} - 157300 T^{5} - 429276 T^{6} + 8492650 T^{7} + 113408229 T^{8} + 501066350 T^{9} - 1494309756 T^{10} - 32306116700 T^{11} - 184426234420 T^{12} - 17873107475 T^{13} + 8689189930046 T^{14} + 62216287120475 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 7 T - 108 T^{2} + 1783 T^{3} + 318 T^{4} - 164342 T^{5} + 1092660 T^{6} + 5279480 T^{7} - 100824709 T^{8} + 322048280 T^{9} + 4065787860 T^{10} - 37302511502 T^{11} + 4402977438 T^{12} + 1505915204683 T^{13} - 5564200430988 T^{14} - 21999199852147 T^{15} + 191707312997281 T^{16}$$
$67$ $$( 1 + 15 T + 335 T^{2} + 3060 T^{3} + 35713 T^{4} + 205020 T^{5} + 1503815 T^{6} + 4511445 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$1 - 14 T - 9 T^{2} + 147 T^{3} + 6927 T^{4} - 52059 T^{5} + 728307 T^{6} - 4047848 T^{7} - 14238189 T^{8} - 287397208 T^{9} + 3671395587 T^{10} - 18632488749 T^{11} + 176026714287 T^{12} + 265221714597 T^{13} - 1152902555289 T^{14} - 127331682217474 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 - 3 T - 2 T^{2} - 345 T^{3} + 220 T^{4} - 100292 T^{5} + 620974 T^{6} + 13580 T^{7} + 31898639 T^{8} + 991340 T^{9} + 3309170446 T^{10} - 39015292964 T^{11} + 6247613020 T^{12} - 715209699585 T^{13} - 302668452578 T^{14} - 33142195557291 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 + 9 T - 75 T^{2} - 768 T^{3} + 10031 T^{4} + 24564 T^{5} - 1553178 T^{6} - 1473567 T^{7} + 120297853 T^{8} - 116411793 T^{9} - 9693383898 T^{10} + 12111009996 T^{11} + 390708262511 T^{12} - 2363179314432 T^{13} - 18231559164075 T^{14} + 172835180875431 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 + 23 T + 98 T^{2} - 1745 T^{3} - 15470 T^{4} + 98632 T^{5} + 1750974 T^{6} - 1504700 T^{7} - 137136651 T^{8} - 124890100 T^{9} + 12062459886 T^{10} + 56396495384 T^{11} - 734180225870 T^{12} - 6873625922035 T^{13} + 32040156590162 T^{14} + 624129172761421 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 17 T + 312 T^{2} - 3419 T^{3} + 38939 T^{4} - 304291 T^{5} + 2471352 T^{6} - 11984473 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 30 T + 361 T^{2} - 2160 T^{3} + 10062 T^{4} - 83370 T^{5} - 577417 T^{6} + 32754150 T^{7} - 458148745 T^{8} + 3177152550 T^{9} - 5432916553 T^{10} - 76089548010 T^{11} + 890781625422 T^{12} - 18548654955120 T^{13} + 300702893779369 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16}$$
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