# Properties

 Label 198.4.f.d Level $198$ Weight $4$ Character orbit 198.f Analytic conductor $11.682$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$198 = 2 \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 198.f (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.6823781811$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 71 x^{6} - 141 x^{5} + 2911 x^{4} + 2710 x^{3} + 75340 x^{2} + 169400 x + 5856400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{3} q^{2} + 4 \beta_{2} q^{4} + ( 4 + 8 \beta_{2} + 8 \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{6} ) q^{5} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} -8 \beta_{4} q^{8} +O(q^{10})$$ $$q + 2 \beta_{3} q^{2} + 4 \beta_{2} q^{4} + ( 4 + 8 \beta_{2} + 8 \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{6} ) q^{5} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} -8 \beta_{4} q^{8} + ( -10 + 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{4} - 2 \beta_{7} ) q^{10} + ( 18 + 8 \beta_{2} - 6 \beta_{3} + 16 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{11} + ( 7 + 3 \beta_{1} + 7 \beta_{2} + 16 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{13} + ( -4 + 6 \beta_{1} + 2 \beta_{3} + 4 \beta_{5} - 6 \beta_{6} ) q^{14} + ( -16 - 16 \beta_{2} - 16 \beta_{3} + 16 \beta_{4} ) q^{16} + ( -23 + 2 \beta_{2} + 2 \beta_{3} + 27 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{17} + ( -57 + 2 \beta_{1} - 55 \beta_{3} + 28 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} ) q^{19} + ( -12 - 4 \beta_{1} - 12 \beta_{2} - 32 \beta_{3} - 4 \beta_{5} + 4 \beta_{7} ) q^{20} + ( 36 - 6 \beta_{1} + 20 \beta_{2} + 62 \beta_{3} - 48 \beta_{4} + 4 \beta_{6} + 6 \beta_{7} ) q^{22} + ( -42 - 6 \beta_{1} + 76 \beta_{2} - 76 \beta_{4} - 6 \beta_{6} + 6 \beta_{7} ) q^{23} + ( -10 + 9 \beta_{1} - \beta_{3} + 7 \beta_{4} + 2 \beta_{5} - 9 \beta_{6} ) q^{25} + ( 12 - 6 \beta_{1} + 32 \beta_{2} + 14 \beta_{3} - 14 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} ) q^{26} + ( 12 + 4 \beta_{2} + 4 \beta_{3} - 12 \beta_{5} + 12 \beta_{6} + 4 \beta_{7} ) q^{28} + ( -9 + 10 \beta_{1} + 111 \beta_{2} - 49 \beta_{3} + 49 \beta_{4} + 10 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} ) q^{29} + ( 95 - 25 \beta_{1} + 95 \beta_{2} + 90 \beta_{3} - 21 \beta_{5} + 21 \beta_{7} ) q^{31} + 32 q^{32} + ( 60 - 8 \beta_{1} + 58 \beta_{2} - 58 \beta_{4} + 6 \beta_{6} + 8 \beta_{7} ) q^{34} + ( 76 + 3 \beta_{1} + 76 \beta_{2} - 135 \beta_{3} - 16 \beta_{5} + 16 \beta_{7} ) q^{35} + ( -27 + 32 \beta_{1} + 53 \beta_{2} - 43 \beta_{3} + 43 \beta_{4} + 32 \beta_{5} - 27 \beta_{6} - 27 \beta_{7} ) q^{37} + ( 60 - 54 \beta_{2} - 54 \beta_{3} - 56 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 12 \beta_{7} ) q^{38} + ( 8 \beta_{1} - 64 \beta_{2} - 24 \beta_{3} + 24 \beta_{4} + 8 \beta_{5} ) q^{40} + ( -81 + 24 \beta_{1} - 57 \beta_{3} + 18 \beta_{4} + 17 \beta_{5} - 24 \beta_{6} ) q^{41} + ( 276 - 7 \beta_{1} + 174 \beta_{2} - 174 \beta_{4} + 14 \beta_{6} + 7 \beta_{7} ) q^{43} + ( -96 + 4 \beta_{1} + 28 \beta_{2} - 32 \beta_{3} + 56 \beta_{4} + 12 \beta_{5} - 12 \beta_{7} ) q^{44} + ( -152 + 24 \beta_{1} - 152 \beta_{2} - 224 \beta_{3} + 12 \beta_{5} - 12 \beta_{7} ) q^{46} + ( 178 - 9 \beta_{1} + 169 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} + 9 \beta_{6} ) q^{47} + ( -237 - 175 \beta_{2} - 175 \beta_{3} + 232 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{49} + ( 32 + 12 \beta_{2} + 12 \beta_{3} - 14 \beta_{4} - 18 \beta_{5} + 18 \beta_{6} + 14 \beta_{7} ) q^{50} + ( -16 - 12 \beta_{1} - 28 \beta_{3} - 36 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} ) q^{52} + ( 89 - 47 \beta_{1} + 89 \beta_{2} + 64 \beta_{3} - 39 \beta_{5} + 39 \beta_{7} ) q^{53} + ( 64 - 50 \beta_{1} + 206 \beta_{2} + 82 \beta_{3} + 71 \beta_{4} - \beta_{5} + 5 \beta_{6} + 14 \beta_{7} ) q^{55} + ( 8 - 24 \beta_{1} + 8 \beta_{2} - 8 \beta_{4} + 8 \beta_{6} + 24 \beta_{7} ) q^{56} + ( 100 - 2 \beta_{1} + 98 \beta_{3} - 320 \beta_{4} - 20 \beta_{5} + 2 \beta_{6} ) q^{58} + ( -26 + 40 \beta_{1} - 229 \beta_{2} + 83 \beta_{3} - 83 \beta_{4} + 40 \beta_{5} - 26 \beta_{6} - 26 \beta_{7} ) q^{59} + ( -218 - 200 \beta_{2} - 200 \beta_{3} + 245 \beta_{4} - 27 \beta_{5} + 27 \beta_{6} + 28 \beta_{7} ) q^{61} + ( -8 + 50 \beta_{1} + 180 \beta_{2} + 190 \beta_{3} - 190 \beta_{4} + 50 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} ) q^{62} + 64 \beta_{3} q^{64} + ( 11 - 27 \beta_{1} - 79 \beta_{2} + 79 \beta_{4} + 53 \beta_{6} + 27 \beta_{7} ) q^{65} + ( -19 + 59 \beta_{1} - 102 \beta_{2} + 102 \beta_{4} - 51 \beta_{6} - 59 \beta_{7} ) q^{67} + ( -116 + 4 \beta_{1} - 116 \beta_{2} - 8 \beta_{3} + 16 \beta_{5} - 16 \beta_{7} ) q^{68} + ( 38 - 6 \beta_{1} - 270 \beta_{2} + 152 \beta_{3} - 152 \beta_{4} - 6 \beta_{5} + 38 \beta_{6} + 38 \beta_{7} ) q^{70} + ( -294 - 366 \beta_{2} - 366 \beta_{3} + 243 \beta_{4} + 51 \beta_{5} - 51 \beta_{6} - 32 \beta_{7} ) q^{71} + ( -45 - 17 \beta_{1} + 373 \beta_{2} + 143 \beta_{3} - 143 \beta_{4} - 17 \beta_{5} - 45 \beta_{6} - 45 \beta_{7} ) q^{73} + ( 96 - 10 \beta_{1} + 86 \beta_{3} - 192 \beta_{4} - 64 \beta_{5} + 10 \beta_{6} ) q^{74} + ( -136 - 8 \beta_{1} - 220 \beta_{2} + 220 \beta_{4} - 24 \beta_{6} + 8 \beta_{7} ) q^{76} + ( 93 - 8 \beta_{1} + 65 \beta_{2} + 581 \beta_{3} + 31 \beta_{4} - 68 \beta_{5} + 11 \beta_{6} - 53 \beta_{7} ) q^{77} + ( 47 + 19 \beta_{1} + 47 \beta_{2} + 170 \beta_{3} + 55 \beta_{5} - 55 \beta_{7} ) q^{79} + ( 64 - 16 \beta_{1} + 48 \beta_{3} + 80 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} ) q^{80} + ( 84 - 78 \beta_{2} - 78 \beta_{3} - 36 \beta_{4} - 48 \beta_{5} + 48 \beta_{6} + 14 \beta_{7} ) q^{82} + ( -537 - 306 \beta_{2} - 306 \beta_{3} + 549 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} - 92 \beta_{7} ) q^{83} + ( -110 - 55 \beta_{1} - 165 \beta_{3} + 270 \beta_{4} - 44 \beta_{5} + 55 \beta_{6} ) q^{85} + ( -348 - 14 \beta_{1} - 348 \beta_{2} + 176 \beta_{3} + 14 \beta_{5} - 14 \beta_{7} ) q^{86} + ( 96 - 8 \beta_{1} + 48 \beta_{2} - 80 \beta_{3} - 168 \beta_{4} - 8 \beta_{5} - 16 \beta_{6} - 16 \beta_{7} ) q^{88} + ( 434 + 61 \beta_{1} + 374 \beta_{2} - 374 \beta_{4} - 26 \beta_{6} - 61 \beta_{7} ) q^{89} + ( 660 + 28 \beta_{1} + 688 \beta_{3} + 241 \beta_{4} - 3 \beta_{5} - 28 \beta_{6} ) q^{91} + ( 24 - 48 \beta_{1} - 448 \beta_{2} - 304 \beta_{3} + 304 \beta_{4} - 48 \beta_{5} + 24 \beta_{6} + 24 \beta_{7} ) q^{92} + ( -2 + 354 \beta_{2} + 354 \beta_{3} - 16 \beta_{4} + 18 \beta_{5} - 18 \beta_{6} - 42 \beta_{7} ) q^{94} + ( 15 - 59 \beta_{1} - 425 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 59 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} ) q^{95} + ( -819 - 10 \beta_{1} - 819 \beta_{2} + 112 \beta_{3} + 10 \beta_{5} - 10 \beta_{7} ) q^{97} + ( 468 + 10 \beta_{1} + 114 \beta_{2} - 114 \beta_{4} + 4 \beta_{6} - 10 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} - 8q^{4} - 5q^{5} - q^{7} - 16q^{8} + O(q^{10})$$ $$8q - 4q^{2} - 8q^{4} - 5q^{5} - q^{7} - 16q^{8} - 100q^{10} + 155q^{11} + 7q^{13} - 2q^{14} - 32q^{16} - 161q^{17} - 272q^{19} - 20q^{20} - 628q^{23} - 17q^{25} - 96q^{26} + 16q^{28} - 33q^{29} + 323q^{31} + 256q^{32} + 208q^{34} + 697q^{35} + 49q^{37} + 576q^{38} + 240q^{40} - 361q^{41} + 1442q^{43} - 620q^{44} - 416q^{46} + 1069q^{47} - 709q^{49} + 76q^{50} - 192q^{52} + 281q^{53} - 7q^{55} - 48q^{56} - 66q^{58} + 128q^{59} - 617q^{61} - 1044q^{62} - 128q^{64} + 138q^{65} + 578q^{67} - 644q^{68} + 34q^{70} - 115q^{71} - 1487q^{73} + 98q^{74} - 128q^{76} - 553q^{77} + 71q^{79} + 480q^{80} + 658q^{82} - 1942q^{83} - 329q^{85} - 2426q^{86} + 560q^{88} + 2202q^{89} + 4523q^{91} + 2088q^{92} - 1332q^{94} + 793q^{95} - 5128q^{97} + 3292q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 71 x^{6} - 141 x^{5} + 2911 x^{4} + 2710 x^{3} + 75340 x^{2} + 169400 x + 5856400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$22554025143 \nu^{7} + 151013926876 \nu^{6} - 1356924294556 \nu^{5} + 28230146638036 \nu^{4} - 111071208987556 \nu^{3} + 8011043329394989 \nu^{2} - 4059658693406480 \nu + 48229141584676800$$$$)/ 342693986325717620$$ $$\beta_{3}$$ $$=$$ $$($$$$14112299722 \nu^{7} + 955320989289 \nu^{6} + 1256420838616 \nu^{5} - 5250112809736 \nu^{4} + 12754848436776 \nu^{3} - 1313471371492594 \nu^{2} + 8498101996878245 \nu - 66484077419717320$$$$)/ 171346993162858810$$ $$\beta_{4}$$ $$=$$ $$($$$$-123788106187 \nu^{7} + 5732748190805 \nu^{6} - 11662655347575 \nu^{5} + 374269556401525 \nu^{4} - 772584075715295 \nu^{3} + 13504262211521188 \nu^{2} + 13624366122837560 \nu + 426480737760227920$$$$)/ 685387972651435240$$ $$\beta_{5}$$ $$=$$ $$($$$$15778904729 \nu^{7} - 268932734519 \nu^{6} + 2855478562109 \nu^{5} - 16065997834439 \nu^{4} + 722720174659769 \nu^{3} - 523534449789100 \nu^{2} + 4037135429586600 \nu - 12007762986133200$$$$)/ 31153998756883420$$ $$\beta_{6}$$ $$=$$ $$($$$$-15728307097 \nu^{7} - 23011705883 \nu^{6} - 3489045673392 \nu^{5} - 619537249643 \nu^{4} - 38031215360567 \nu^{3} - 53736302216310 \nu^{2} + 582451195267480 \nu - 22785233088495405$$$$)/ 7788499689220855$$ $$\beta_{7}$$ $$=$$ $$($$$$254952731119 \nu^{7} - 130622718559 \nu^{6} + 16218883337689 \nu^{5} - 18738040845679 \nu^{4} + 629078544513089 \nu^{3} + 1043207365589370 \nu^{2} + 20338656497650260 \nu + 32952393866979400$$$$)/ 31153998756883420$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} + \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + 54 \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{6} + 47 \beta_{5} + 46 \beta_{4} + 8 \beta_{3} - 8 \beta_{1} + 16$$ $$\nu^{4}$$ $$=$$ $$109 \beta_{7} + 16 \beta_{6} - 16 \beta_{5} + 2226 \beta_{4} - 3034 \beta_{3} - 3034 \beta_{2} - 2210$$ $$\nu^{5}$$ $$=$$ $$-824 \beta_{7} - 3143 \beta_{6} - 1608 \beta_{4} + 1608 \beta_{2} + 824 \beta_{1} - 7549$$ $$\nu^{6}$$ $$=$$ $$-2432 \beta_{7} + 2432 \beta_{5} + 176314 \beta_{3} + 63048 \beta_{2} - 6725 \beta_{1} + 63048$$ $$\nu^{7}$$ $$=$$ $$185471 \beta_{7} + 185471 \beta_{6} - 119991 \beta_{5} + 185128 \beta_{4} - 185128 \beta_{3} - 513934 \beta_{2} - 119991 \beta_{1} + 185471$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/198\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$155$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 5.60402 + 4.07156i −4.79501 − 3.48378i 5.60402 − 4.07156i −4.79501 + 3.48378i −2.53202 + 7.79275i 2.22300 − 6.84169i −2.53202 − 7.79275i 2.22300 + 6.84169i
−1.61803 1.17557i 0 1.23607 + 3.80423i −6.52241 + 4.73881i 0 −8.05890 24.8027i 2.47214 7.60845i 0 16.1243
37.2 −1.61803 1.17557i 0 1.23607 + 3.80423i 10.3036 7.48598i 0 7.24988 + 22.3128i 2.47214 7.60845i 0 −25.4718
91.1 −1.61803 + 1.17557i 0 1.23607 3.80423i −6.52241 4.73881i 0 −8.05890 + 24.8027i 2.47214 + 7.60845i 0 16.1243
91.2 −1.61803 + 1.17557i 0 1.23607 3.80423i 10.3036 + 7.48598i 0 7.24988 22.3128i 2.47214 + 7.60845i 0 −25.4718
163.1 0.618034 1.90211i 0 −3.23607 2.35114i −4.60996 14.1880i 0 −17.6106 12.7948i −6.47214 + 4.70228i 0 −29.8363
163.2 0.618034 1.90211i 0 −3.23607 2.35114i −1.67119 5.14341i 0 17.9196 + 13.0193i −6.47214 + 4.70228i 0 −10.8162
181.1 0.618034 + 1.90211i 0 −3.23607 + 2.35114i −4.60996 + 14.1880i 0 −17.6106 + 12.7948i −6.47214 4.70228i 0 −29.8363
181.2 0.618034 + 1.90211i 0 −3.23607 + 2.35114i −1.67119 + 5.14341i 0 17.9196 13.0193i −6.47214 4.70228i 0 −10.8162
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.2 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 198.4.f.d 8
3.b odd 2 1 22.4.c.b 8
11.c even 5 1 inner 198.4.f.d 8
11.c even 5 1 2178.4.a.by 4
11.d odd 10 1 2178.4.a.bt 4
12.b even 2 1 176.4.m.b 8
33.d even 2 1 242.4.c.q 8
33.f even 10 1 242.4.a.o 4
33.f even 10 2 242.4.c.n 8
33.f even 10 1 242.4.c.q 8
33.h odd 10 1 22.4.c.b 8
33.h odd 10 1 242.4.a.n 4
33.h odd 10 2 242.4.c.r 8
132.n odd 10 1 1936.4.a.bm 4
132.o even 10 1 176.4.m.b 8
132.o even 10 1 1936.4.a.bn 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.b 8 3.b odd 2 1
22.4.c.b 8 33.h odd 10 1
176.4.m.b 8 12.b even 2 1
176.4.m.b 8 132.o even 10 1
198.4.f.d 8 1.a even 1 1 trivial
198.4.f.d 8 11.c even 5 1 inner
242.4.a.n 4 33.h odd 10 1
242.4.a.o 4 33.f even 10 1
242.4.c.n 8 33.f even 10 2
242.4.c.q 8 33.d even 2 1
242.4.c.q 8 33.f even 10 1
242.4.c.r 8 33.h odd 10 2
1936.4.a.bm 4 132.n odd 10 1
1936.4.a.bn 4 132.o even 10 1
2178.4.a.bt 4 11.d odd 10 1
2178.4.a.by 4 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(198, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 4 T^{2} + 8 T^{3} + 16 T^{4} )^{2}$$
$3$ 1
$5$ $$1 + 5 T - 104 T^{2} + 1630 T^{3} + 10006 T^{4} - 283885 T^{5} + 2835571 T^{6} + 31638350 T^{7} - 387953844 T^{8} + 3954793750 T^{9} + 44305796875 T^{10} - 554462890625 T^{11} + 2442871093750 T^{12} + 49743652343750 T^{13} - 396728515625000 T^{14} + 2384185791015625 T^{15} + 59604644775390625 T^{16}$$
$7$ $$1 + T + 12 T^{2} - 1104 T^{3} - 87888 T^{4} - 922401 T^{5} + 28062801 T^{6} + 211635072 T^{7} + 13215739812 T^{8} + 72590829696 T^{9} + 3301560474849 T^{10} - 37222207450407 T^{11} - 1216483049521488 T^{12} - 5241307906977072 T^{13} + 19540963174925388 T^{14} + 558545864083284007 T^{15} +$$$$19\!\cdots\!01$$$$T^{16}$$
$11$ $$1 - 155 T + 13111 T^{2} - 755095 T^{3} + 31999176 T^{4} - 1005031445 T^{5} + 23226936271 T^{6} - 365481892105 T^{7} + 3138428376721 T^{8}$$
$13$ $$1 - 7 T + 1930 T^{2} + 79384 T^{3} + 10224596 T^{4} + 326819897 T^{5} + 10009625561 T^{6} + 1235524128730 T^{7} + 47709276128944 T^{8} + 2714446510819810 T^{9} + 48314550744464849 T^{10} + 3465761392820424581 T^{11} +$$$$23\!\cdots\!76$$$$T^{12} +$$$$40\!\cdots\!88$$$$T^{13} +$$$$21\!\cdots\!70$$$$T^{14} -$$$$17\!\cdots\!91$$$$T^{15} +$$$$54\!\cdots\!61$$$$T^{16}$$
$17$ $$1 + 161 T + 5020 T^{2} - 211188 T^{3} + 21130156 T^{4} + 1323457999 T^{5} - 265862283281 T^{6} - 13237803594360 T^{7} + 277440196391544 T^{8} - 65037329059090680 T^{9} - 6417269207192683889 T^{10} +$$$$15\!\cdots\!03$$$$T^{11} +$$$$12\!\cdots\!16$$$$T^{12} -$$$$60\!\cdots\!84$$$$T^{13} +$$$$70\!\cdots\!80$$$$T^{14} +$$$$11\!\cdots\!37$$$$T^{15} +$$$$33\!\cdots\!21$$$$T^{16}$$
$19$ $$1 + 272 T + 33201 T^{2} + 3039584 T^{3} + 292957015 T^{4} + 21884364992 T^{5} + 909384182595 T^{6} + 32108296725040 T^{7} + 2715838419478696 T^{8} + 220230807237049360 T^{9} + 42782780037646641195 T^{10} +$$$$70\!\cdots\!68$$$$T^{11} +$$$$64\!\cdots\!15$$$$T^{12} +$$$$46\!\cdots\!16$$$$T^{13} +$$$$34\!\cdots\!41$$$$T^{14} +$$$$19\!\cdots\!68$$$$T^{15} +$$$$48\!\cdots\!21$$$$T^{16}$$
$23$ $$( 1 + 314 T + 61816 T^{2} + 8042722 T^{3} + 950275950 T^{4} + 97855798574 T^{5} + 9150986514424 T^{6} + 565561935699382 T^{7} + 21914624432020321 T^{8} )^{2}$$
$29$ $$1 + 33 T + 14176 T^{2} - 2091144 T^{3} + 419909220 T^{4} - 81926257017 T^{5} + 7198320043835 T^{6} - 2719607629962300 T^{7} + 63828590693142176 T^{8} - 66328510487150534700 T^{9} +$$$$42\!\cdots\!35$$$$T^{10} -$$$$11\!\cdots\!73$$$$T^{11} +$$$$14\!\cdots\!20$$$$T^{12} -$$$$18\!\cdots\!56$$$$T^{13} +$$$$29\!\cdots\!36$$$$T^{14} +$$$$16\!\cdots\!57$$$$T^{15} +$$$$12\!\cdots\!81$$$$T^{16}$$
$31$ $$1 - 323 T - 4098 T^{2} + 5799192 T^{3} + 305184598 T^{4} - 93424183073 T^{5} + 36115996851965 T^{6} - 3925905573337100 T^{7} - 564631402757797204 T^{8} -$$$$11\!\cdots\!00$$$$T^{9} +$$$$32\!\cdots\!65$$$$T^{10} -$$$$24\!\cdots\!83$$$$T^{11} +$$$$24\!\cdots\!78$$$$T^{12} +$$$$13\!\cdots\!92$$$$T^{13} -$$$$28\!\cdots\!18$$$$T^{14} -$$$$67\!\cdots\!13$$$$T^{15} +$$$$62\!\cdots\!21$$$$T^{16}$$
$37$ $$1 - 49 T - 7868 T^{2} + 4260746 T^{3} - 566286138 T^{4} + 1042572020189 T^{5} + 91349107260671 T^{6} - 14416097444880138 T^{7} + 5705343181934634812 T^{8} -$$$$73\!\cdots\!14$$$$T^{9} +$$$$23\!\cdots\!39$$$$T^{10} +$$$$13\!\cdots\!53$$$$T^{11} -$$$$37\!\cdots\!78$$$$T^{12} +$$$$14\!\cdots\!78$$$$T^{13} -$$$$13\!\cdots\!72$$$$T^{14} -$$$$41\!\cdots\!13$$$$T^{15} +$$$$43\!\cdots\!61$$$$T^{16}$$
$41$ $$1 + 361 T - 19140 T^{2} - 12862580 T^{3} + 3853225260 T^{4} + 605378813703 T^{5} - 283363657590617 T^{6} - 99783171614693080 T^{7} - 22290482644766813160 T^{8} -$$$$68\!\cdots\!80$$$$T^{9} -$$$$13\!\cdots\!97$$$$T^{10} +$$$$19\!\cdots\!83$$$$T^{11} +$$$$86\!\cdots\!60$$$$T^{12} -$$$$20\!\cdots\!80$$$$T^{13} -$$$$20\!\cdots\!40$$$$T^{14} +$$$$26\!\cdots\!01$$$$T^{15} +$$$$50\!\cdots\!61$$$$T^{16}$$
$43$ $$( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 12280589284047 T^{5} + 2655712080056733 T^{6} - 362369273206463803 T^{7} + 39959630797262576401 T^{8} )^{2}$$
$47$ $$1 - 1069 T + 301070 T^{2} + 43368042 T^{3} - 24642745634 T^{4} - 6223420821981 T^{5} + 4635059307315839 T^{6} - 746453682627128500 T^{7} + 29698413240018564184 T^{8} -$$$$77\!\cdots\!00$$$$T^{9} +$$$$49\!\cdots\!31$$$$T^{10} -$$$$69\!\cdots\!27$$$$T^{11} -$$$$28\!\cdots\!94$$$$T^{12} +$$$$52\!\cdots\!06$$$$T^{13} +$$$$37\!\cdots\!30$$$$T^{14} -$$$$13\!\cdots\!43$$$$T^{15} +$$$$13\!\cdots\!81$$$$T^{16}$$
$53$ $$1 - 281 T - 343802 T^{2} + 60864958 T^{3} + 33213391926 T^{4} - 4932564267181 T^{5} + 8874412003121445 T^{6} - 107096647673733916 T^{7} -$$$$25\!\cdots\!56$$$$T^{8} -$$$$15\!\cdots\!32$$$$T^{9} +$$$$19\!\cdots\!05$$$$T^{10} -$$$$16\!\cdots\!73$$$$T^{11} +$$$$16\!\cdots\!66$$$$T^{12} +$$$$44\!\cdots\!06$$$$T^{13} -$$$$37\!\cdots\!78$$$$T^{14} -$$$$45\!\cdots\!93$$$$T^{15} +$$$$24\!\cdots\!81$$$$T^{16}$$
$59$ $$1 - 128 T - 87119 T^{2} + 50674984 T^{3} + 39669673335 T^{4} - 496570131448 T^{5} - 4049042036547885 T^{6} + 22790010341236160 T^{7} +$$$$27\!\cdots\!36$$$$T^{8} +$$$$46\!\cdots\!40$$$$T^{9} -$$$$17\!\cdots\!85$$$$T^{10} -$$$$43\!\cdots\!72$$$$T^{11} +$$$$70\!\cdots\!35$$$$T^{12} +$$$$18\!\cdots\!16$$$$T^{13} -$$$$65\!\cdots\!99$$$$T^{14} -$$$$19\!\cdots\!52$$$$T^{15} +$$$$31\!\cdots\!61$$$$T^{16}$$
$61$ $$1 + 617 T - 60798 T^{2} - 112995618 T^{3} + 22253617738 T^{4} + 7295748837377 T^{5} - 13214872371636415 T^{6} + 3429112403997460700 T^{7} +$$$$68\!\cdots\!36$$$$T^{8} +$$$$77\!\cdots\!00$$$$T^{9} -$$$$68\!\cdots\!15$$$$T^{10} +$$$$85\!\cdots\!57$$$$T^{11} +$$$$59\!\cdots\!98$$$$T^{12} -$$$$68\!\cdots\!18$$$$T^{13} -$$$$83\!\cdots\!38$$$$T^{14} +$$$$19\!\cdots\!37$$$$T^{15} +$$$$70\!\cdots\!41$$$$T^{16}$$
$67$ $$( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 73178257244967 T^{5} + 62120937078828215 T^{6} - 7862688440529239683 T^{7} +$$$$81\!\cdots\!61$$$$T^{8} )^{2}$$
$71$ $$1 + 115 T - 138428 T^{2} - 42960000 T^{3} + 77925723838 T^{4} + 71653925274035 T^{5} + 12191505393571699 T^{6} - 3285417296443548450 T^{7} +$$$$11\!\cdots\!80$$$$T^{8} -$$$$11\!\cdots\!50$$$$T^{9} +$$$$15\!\cdots\!79$$$$T^{10} +$$$$32\!\cdots\!85$$$$T^{11} +$$$$12\!\cdots\!58$$$$T^{12} -$$$$25\!\cdots\!00$$$$T^{13} -$$$$29\!\cdots\!08$$$$T^{14} +$$$$86\!\cdots\!65$$$$T^{15} +$$$$26\!\cdots\!81$$$$T^{16}$$
$73$ $$1 + 1487 T + 142728 T^{2} - 1045087272 T^{3} - 569519703768 T^{4} + 366060407928453 T^{5} + 393291607153115139 T^{6} - 68019158899540158384 T^{7} -$$$$19\!\cdots\!28$$$$T^{8} -$$$$26\!\cdots\!28$$$$T^{9} +$$$$59\!\cdots\!71$$$$T^{10} +$$$$21\!\cdots\!89$$$$T^{11} -$$$$13\!\cdots\!28$$$$T^{12} -$$$$93\!\cdots\!04$$$$T^{13} +$$$$49\!\cdots\!32$$$$T^{14} +$$$$20\!\cdots\!51$$$$T^{15} +$$$$52\!\cdots\!41$$$$T^{16}$$
$79$ $$1 - 71 T - 500222 T^{2} - 73013624 T^{3} + 350790961998 T^{4} + 97550067497959 T^{5} - 161731343227399015 T^{6} + 8534351363848547400 T^{7} +$$$$59\!\cdots\!76$$$$T^{8} +$$$$42\!\cdots\!00$$$$T^{9} -$$$$39\!\cdots\!15$$$$T^{10} +$$$$11\!\cdots\!21$$$$T^{11} +$$$$20\!\cdots\!18$$$$T^{12} -$$$$21\!\cdots\!76$$$$T^{13} -$$$$71\!\cdots\!42$$$$T^{14} -$$$$50\!\cdots\!09$$$$T^{15} +$$$$34\!\cdots\!81$$$$T^{16}$$
$83$ $$1 + 1942 T + 875395 T^{2} - 857035884 T^{3} - 790010610449 T^{4} + 618864428088728 T^{5} + 845503237658086801 T^{6} -$$$$17\!\cdots\!30$$$$T^{7} -$$$$59\!\cdots\!96$$$$T^{8} -$$$$10\!\cdots\!10$$$$T^{9} +$$$$27\!\cdots\!69$$$$T^{10} +$$$$11\!\cdots\!84$$$$T^{11} -$$$$84\!\cdots\!89$$$$T^{12} -$$$$52\!\cdots\!88$$$$T^{13} +$$$$30\!\cdots\!55$$$$T^{14} +$$$$38\!\cdots\!86$$$$T^{15} +$$$$11\!\cdots\!21$$$$T^{16}$$
$89$ $$( 1 - 1101 T + 2406895 T^{2} - 1914547747 T^{3} + 2334666485008 T^{4} - 1349696810654843 T^{5} + 1196181784307576095 T^{6} -$$$$38\!\cdots\!09$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8} )^{2}$$
$97$ $$1 + 5128 T + 10343177 T^{2} + 9500759764 T^{3} + 1479496697751 T^{4} - 4959875190278012 T^{5} - 1475957592224572125 T^{6} +$$$$96\!\cdots\!12$$$$T^{7} +$$$$15\!\cdots\!64$$$$T^{8} +$$$$88\!\cdots\!76$$$$T^{9} -$$$$12\!\cdots\!25$$$$T^{10} -$$$$37\!\cdots\!04$$$$T^{11} +$$$$10\!\cdots\!91$$$$T^{12} +$$$$60\!\cdots\!52$$$$T^{13} +$$$$59\!\cdots\!53$$$$T^{14} +$$$$27\!\cdots\!16$$$$T^{15} +$$$$48\!\cdots\!81$$$$T^{16}$$