# Properties

 Label 108.6.e.a Level 108 Weight 6 Character orbit 108.e Analytic conductor 17.321 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 108.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.3214525398$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3^{16}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 - 4 \beta_{1} + \beta_{7} ) q^{5} + ( 6 \beta_{1} + \beta_{2} + \beta_{7} + \beta_{8} ) q^{7} +O(q^{10})$$ $$q + ( 4 - 4 \beta_{1} + \beta_{7} ) q^{5} + ( 6 \beta_{1} + \beta_{2} + \beta_{7} + \beta_{8} ) q^{7} + ( -36 \beta_{1} - \beta_{2} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{11} + ( -34 + 34 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - 8 \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( -229 + 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} ) q^{17} + ( -86 - 17 \beta_{2} - 7 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{19} + ( -78 + 78 \beta_{1} - 15 \beta_{3} - 6 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} - 5 \beta_{9} ) q^{23} + ( -955 \beta_{1} + 33 \beta_{2} + 14 \beta_{3} - 14 \beta_{4} + 33 \beta_{7} - 12 \beta_{8} + 22 \beta_{9} ) q^{25} + ( 1198 \beta_{1} + 2 \beta_{2} - 15 \beta_{3} + 15 \beta_{4} + 2 \beta_{7} + 27 \beta_{8} + 15 \beta_{9} ) q^{29} + ( 562 - 562 \beta_{1} - 7 \beta_{3} + 12 \beta_{5} + 25 \beta_{6} + 6 \beta_{7} - 12 \beta_{8} + 25 \beta_{9} ) q^{31} + ( -3726 - 7 \beta_{2} - 30 \beta_{4} - 21 \beta_{5} - 40 \beta_{6} ) q^{35} + ( -1512 + 23 \beta_{2} - 49 \beta_{4} - 13 \beta_{5} - 23 \beta_{6} ) q^{37} + ( 3693 - 3693 \beta_{1} - 45 \beta_{3} + 51 \beta_{5} - 19 \beta_{6} - 23 \beta_{7} - 51 \beta_{8} - 19 \beta_{9} ) q^{41} + ( 256 \beta_{1} - 162 \beta_{2} + 21 \beta_{3} - 21 \beta_{4} - 162 \beta_{7} + 54 \beta_{8} + 24 \beta_{9} ) q^{43} + ( 5026 \beta_{1} + 33 \beta_{2} + 18 \beta_{3} - 18 \beta_{4} + 33 \beta_{7} - 87 \beta_{8} + 38 \beta_{9} ) q^{47} + ( -817 + 817 \beta_{1} - 35 \beta_{3} - 53 \beta_{5} - \beta_{6} + 62 \beta_{7} + 53 \beta_{8} - \beta_{9} ) q^{49} + ( -11704 - 53 \beta_{2} + 15 \beta_{4} + 27 \beta_{5} - 15 \beta_{6} ) q^{53} + ( 1542 + 264 \beta_{2} + 41 \beta_{4} + 66 \beta_{5} + 49 \beta_{6} ) q^{55} + ( 18076 - 18076 \beta_{1} + 75 \beta_{3} - 138 \beta_{5} + 50 \beta_{6} + 182 \beta_{7} + 138 \beta_{8} + 50 \beta_{9} ) q^{59} + ( 304 \beta_{1} + 120 \beta_{2} - 49 \beta_{3} + 49 \beta_{4} + 120 \beta_{7} - 87 \beta_{8} - 23 \beta_{9} ) q^{61} + ( 29692 \beta_{1} - 140 \beta_{2} - 45 \beta_{3} + 45 \beta_{4} - 140 \beta_{7} + 87 \beta_{8} - 119 \beta_{9} ) q^{65} + ( 2692 - 2692 \beta_{1} + 138 \beta_{3} + 75 \beta_{5} - 57 \beta_{6} + 195 \beta_{7} - 75 \beta_{8} - 57 \beta_{9} ) q^{67} + ( -22780 + 347 \beta_{2} + 81 \beta_{4} + 129 \beta_{5} + 259 \beta_{6} ) q^{71} + ( 1331 - 690 \beta_{2} + 133 \beta_{4} - 141 \beta_{5} - 61 \beta_{6} ) q^{73} + ( 42430 - 42430 \beta_{1} + 132 \beta_{3} + 24 \beta_{5} - 52 \beta_{6} - 403 \beta_{7} - 24 \beta_{8} - 52 \beta_{9} ) q^{77} + ( 6134 \beta_{1} + 122 \beta_{2} - 91 \beta_{3} + 91 \beta_{4} + 122 \beta_{7} - 112 \beta_{8} - 323 \beta_{9} ) q^{79} + ( 45886 \beta_{1} - 21 \beta_{2} + 240 \beta_{3} - 240 \beta_{4} - 21 \beta_{7} + 129 \beta_{8} - 130 \beta_{9} ) q^{83} + ( -9780 + 9780 \beta_{1} - 91 \beta_{3} + 165 \beta_{5} - 197 \beta_{6} - 1065 \beta_{7} - 165 \beta_{8} - 197 \beta_{9} ) q^{85} + ( -59962 - 529 \beta_{2} + 225 \beta_{4} - 375 \beta_{5} + 55 \beta_{6} ) q^{89} + ( 12562 + 84 \beta_{2} + 357 \beta_{4} - 24 \beta_{5} + 309 \beta_{6} ) q^{91} + ( 79024 - 79024 \beta_{1} + 390 \beta_{3} + 438 \beta_{5} + 386 \beta_{6} + 26 \beta_{7} - 438 \beta_{8} + 386 \beta_{9} ) q^{95} + ( 8219 \beta_{1} + 749 \beta_{2} - 281 \beta_{3} + 281 \beta_{4} + 749 \beta_{7} + 605 \beta_{8} - 43 \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 21q^{5} + 29q^{7} + O(q^{10})$$ $$10q + 21q^{5} + 29q^{7} - 177q^{11} - 181q^{13} - 2280q^{17} - 832q^{19} - 399q^{23} - 4778q^{25} + 6033q^{29} + 2759q^{31} - 37146q^{35} - 15172q^{37} + 18435q^{41} + 1469q^{43} + 25155q^{47} - 4056q^{49} - 116844q^{53} + 14778q^{55} + 90537q^{59} + 1403q^{61} + 148407q^{65} + 13907q^{67} - 229368q^{71} + 15200q^{73} + 211983q^{77} + 29993q^{79} + 228951q^{83} - 49662q^{85} - 598332q^{89} + 124930q^{91} + 394764q^{95} + 40541q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 175 x^{8} + 8800 x^{6} + 124623 x^{4} + 498609 x^{2} + 442368$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-99 \nu^{9} - 23021 \nu^{7} - 1847072 \nu^{5} - 56550029 \nu^{3} - 389674035 \nu + 207097728$$$$)/ 414195456$$ $$\beta_{2}$$ $$=$$ $$($$$$-47225 \nu^{8} - 9472484 \nu^{6} - 609253100 \nu^{4} - 13835630451 \nu^{2} - 61649294112$$$$)/ 427139064$$ $$\beta_{3}$$ $$=$$ $$($$$$1089 \nu^{9} + 102528 \nu^{8} + 253231 \nu^{7} + 17565696 \nu^{6} + 20317792 \nu^{5} + 795822336 \nu^{4} + 622050319 \nu^{3} + 6125611392 \nu^{2} - 6896862927 \nu + 1489775232$$$$)/ 414195456$$ $$\beta_{4}$$ $$=$$ $$($$$$267 \nu^{8} + 45744 \nu^{6} + 2072454 \nu^{4} + 15952113 \nu^{2} + 3879623$$$$)/539317$$ $$\beta_{5}$$ $$=$$ $$($$$$-497 \nu^{8} - 86492 \nu^{6} - 4241924 \nu^{4} - 51195963 \nu^{2} - 57765168$$$$)/563508$$ $$\beta_{6}$$ $$=$$ $$($$$$136611 \nu^{8} + 23186844 \nu^{6} + 1089497700 \nu^{4} + 12357836001 \nu^{2} + 24330997448$$$$)/47459896$$ $$\beta_{7}$$ $$=$$ $$($$$$1172161 \nu^{9} + 188900 \nu^{8} + 202512139 \nu^{7} + 37889936 \nu^{6} + 9878166064 \nu^{5} + 2437012400 \nu^{4} + 126820981119 \nu^{3} + 55342521804 \nu^{2} + 425243755557 \nu + 246597176448$$$$)/ 3417112512$$ $$\beta_{8}$$ $$=$$ $$($$$$1720421 \nu^{9} - 1506904 \nu^{8} + 291062195 \nu^{7} - 262243744 \nu^{6} + 13499523392 \nu^{5} - 12861513568 \nu^{4} + 142053572235 \nu^{3} - 155226159816 \nu^{2} + 159206103117 \nu - 175143989376$$$$)/ 3417112512$$ $$\beta_{9}$$ $$=$$ $$($$$$6094287 \nu^{9} - 6557328 \nu^{8} + 1053931697 \nu^{7} - 1112968512 \nu^{6} + 51382170848 \nu^{5} - 52295889600 \nu^{4} + 643758588833 \nu^{3} - 593176128048 \nu^{2} + 1517046597423 \nu - 1167887877504$$$$)/ 4556150016$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - 2 \beta_{3} - 22 \beta_{1} + 11$$$$)/54$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{6} + 27 \beta_{5} + 13 \beta_{4} - 27 \beta_{2} - 3786$$$$)/108$$ $$\nu^{3}$$ $$=$$ $$($$$$-18 \beta_{9} - 36 \beta_{8} + 126 \beta_{7} - 9 \beta_{6} + 18 \beta_{5} - 74 \beta_{4} + 148 \beta_{3} + 63 \beta_{2} + 5894 \beta_{1} - 2947$$$$)/54$$ $$\nu^{4}$$ $$=$$ $$($$$$-182 \beta_{6} - 1188 \beta_{5} - 805 \beta_{4} + 1134 \beta_{2} + 140985$$$$)/54$$ $$\nu^{5}$$ $$=$$ $$($$$$2338 \beta_{9} + 11106 \beta_{8} - 26370 \beta_{7} + 1169 \beta_{6} - 5553 \beta_{5} + 12955 \beta_{4} - 25910 \beta_{3} - 13185 \beta_{2} - 1652084 \beta_{1} + 826042$$$$)/108$$ $$\nu^{6}$$ $$=$$ $$($$$$12040 \beta_{6} + 101061 \beta_{5} + 85859 \beta_{4} - 108270 \beta_{2} - 12057045$$$$)/54$$ $$\nu^{7}$$ $$=$$ $$($$$$-60644 \beta_{9} - 671580 \beta_{8} + 1278036 \beta_{7} - 30322 \beta_{6} + 335790 \beta_{5} - 595221 \beta_{4} + 1190442 \beta_{3} + 639018 \beta_{2} + 93262802 \beta_{1} - 46631401$$$$)/54$$ $$\nu^{8}$$ $$=$$ $$($$$$-1598887 \beta_{6} - 17799345 \beta_{5} - 17481455 \beta_{4} + 21107817 \beta_{2} + 2167343742$$$$)/108$$ $$\nu^{9}$$ $$=$$ $$($$$$2573322 \beta_{9} + 73125792 \beta_{8} - 123164910 \beta_{7} + 1286661 \beta_{6} - 36562896 \beta_{5} + 55890928 \beta_{4} - 111781856 \beta_{3} - 61582455 \beta_{2} - 9781247602 \beta_{1} + 4890623801$$$$)/54$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$

## 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
 3.71922i − 9.84603i 7.64342i − 2.13639i − 1.11227i − 3.71922i 9.84603i − 7.64342i 2.13639i 1.11227i
0 0 0 −40.7270 + 70.5412i 0 89.6312 + 155.246i 0 0 0
37.2 0 0 0 −13.1603 + 22.7942i 0 −31.6287 54.7826i 0 0 0
37.3 0 0 0 −4.88422 + 8.45972i 0 −68.3340 118.358i 0 0 0
37.4 0 0 0 14.0718 24.3731i 0 75.7039 + 131.123i 0 0 0
37.5 0 0 0 55.1996 95.6086i 0 −50.8724 88.1135i 0 0 0
73.1 0 0 0 −40.7270 70.5412i 0 89.6312 155.246i 0 0 0
73.2 0 0 0 −13.1603 22.7942i 0 −31.6287 + 54.7826i 0 0 0
73.3 0 0 0 −4.88422 8.45972i 0 −68.3340 + 118.358i 0 0 0
73.4 0 0 0 14.0718 + 24.3731i 0 75.7039 131.123i 0 0 0
73.5 0 0 0 55.1996 + 95.6086i 0 −50.8724 + 88.1135i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.e.a 10
3.b odd 2 1 36.6.e.a 10
4.b odd 2 1 432.6.i.d 10
9.c even 3 1 inner 108.6.e.a 10
9.c even 3 1 324.6.a.d 5
9.d odd 6 1 36.6.e.a 10
9.d odd 6 1 324.6.a.e 5
12.b even 2 1 144.6.i.d 10
36.f odd 6 1 432.6.i.d 10
36.h even 6 1 144.6.i.d 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.e.a 10 3.b odd 2 1
36.6.e.a 10 9.d odd 6 1
108.6.e.a 10 1.a even 1 1 trivial
108.6.e.a 10 9.c even 3 1 inner
144.6.i.d 10 12.b even 2 1
144.6.i.d 10 36.h even 6 1
324.6.a.d 5 9.c even 3 1
324.6.a.e 5 9.d odd 6 1
432.6.i.d 10 4.b odd 2 1
432.6.i.d 10 36.f odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(108, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 21 T - 5203 T^{2} + 519930 T^{3} + 14035794 T^{4} - 2854822770 T^{5} + 76722872007 T^{6} + 9761967315441 T^{7} - 599011867854189 T^{8} - 11924309583255600 T^{9} + 2533145723872694124 T^{10} - 37263467447673750000 T^{11} -$$$$58\!\cdots\!25$$$$T^{12} +$$$$29\!\cdots\!25$$$$T^{13} +$$$$73\!\cdots\!75$$$$T^{14} -$$$$85\!\cdots\!50$$$$T^{15} +$$$$13\!\cdots\!50$$$$T^{16} +$$$$15\!\cdots\!50$$$$T^{17} -$$$$47\!\cdots\!75$$$$T^{18} -$$$$59\!\cdots\!25$$$$T^{19} +$$$$88\!\cdots\!25$$$$T^{20}$$
$7$ $$1 - 29 T - 39569 T^{2} + 3762444 T^{3} + 440397336 T^{4} - 77352503496 T^{5} - 2769093584103 T^{6} - 560172784984473 T^{7} + 238615372451780007 T^{8} + 16031898530170676332 T^{9} -$$$$69\!\cdots\!60$$$$T^{10} +$$$$26\!\cdots\!24$$$$T^{11} +$$$$67\!\cdots\!43$$$$T^{12} -$$$$26\!\cdots\!39$$$$T^{13} -$$$$22\!\cdots\!03$$$$T^{14} -$$$$10\!\cdots\!72$$$$T^{15} +$$$$99\!\cdots\!64$$$$T^{16} +$$$$14\!\cdots\!92$$$$T^{17} -$$$$25\!\cdots\!69$$$$T^{18} -$$$$31\!\cdots\!03$$$$T^{19} +$$$$17\!\cdots\!49$$$$T^{20}$$
$11$ $$1 + 177 T - 396232 T^{2} + 71434269 T^{3} + 104816625882 T^{4} - 33726096455301 T^{5} - 6913987980717606 T^{6} + 8552599160812456257 T^{7} -$$$$10\!\cdots\!51$$$$T^{8} -$$$$50\!\cdots\!82$$$$T^{9} +$$$$47\!\cdots\!16$$$$T^{10} -$$$$81\!\cdots\!82$$$$T^{11} -$$$$27\!\cdots\!51$$$$T^{12} +$$$$35\!\cdots\!07$$$$T^{13} -$$$$46\!\cdots\!06$$$$T^{14} -$$$$36\!\cdots\!51$$$$T^{15} +$$$$18\!\cdots\!82$$$$T^{16} +$$$$20\!\cdots\!19$$$$T^{17} -$$$$17\!\cdots\!32$$$$T^{18} +$$$$12\!\cdots\!27$$$$T^{19} +$$$$11\!\cdots\!01$$$$T^{20}$$
$13$ $$1 + 181 T - 1012331 T^{2} + 14482182 T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 192479505557683773 T^{6} - 802846347498861897 T^{7} +$$$$98\!\cdots\!51$$$$T^{8} +$$$$77\!\cdots\!72$$$$T^{9} -$$$$41\!\cdots\!24$$$$T^{10} +$$$$28\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!99$$$$T^{12} -$$$$41\!\cdots\!29$$$$T^{13} -$$$$36\!\cdots\!73$$$$T^{14} -$$$$59\!\cdots\!66$$$$T^{15} +$$$$11\!\cdots\!26$$$$T^{16} +$$$$14\!\cdots\!74$$$$T^{17} -$$$$36\!\cdots\!31$$$$T^{18} +$$$$24\!\cdots\!33$$$$T^{19} +$$$$49\!\cdots\!49$$$$T^{20}$$
$17$ $$( 1 + 1140 T + 4980550 T^{2} + 3443850354 T^{3} + 10068870522169 T^{4} + 5069379208548852 T^{5} + 14296356292995309833 T^{6} +$$$$69\!\cdots\!46$$$$T^{7} +$$$$14\!\cdots\!50$$$$T^{8} +$$$$46\!\cdots\!40$$$$T^{9} +$$$$57\!\cdots\!57$$$$T^{10} )^{2}$$
$19$ $$( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} +$$$$38\!\cdots\!44$$$$T^{7} +$$$$76\!\cdots\!42$$$$T^{8} +$$$$15\!\cdots\!16$$$$T^{9} +$$$$93\!\cdots\!99$$$$T^{10} )^{2}$$
$23$ $$1 + 399 T - 16077241 T^{2} + 38108825820 T^{3} + 155650662506976 T^{4} - 562944417983120520 T^{5} - 48522958516353490863 T^{6} +$$$$48\!\cdots\!51$$$$T^{7} -$$$$71\!\cdots\!41$$$$T^{8} -$$$$11\!\cdots\!52$$$$T^{9} +$$$$81\!\cdots\!80$$$$T^{10} -$$$$74\!\cdots\!36$$$$T^{11} -$$$$29\!\cdots\!09$$$$T^{12} +$$$$12\!\cdots\!57$$$$T^{13} -$$$$83\!\cdots\!63$$$$T^{14} -$$$$62\!\cdots\!60$$$$T^{15} +$$$$11\!\cdots\!24$$$$T^{16} +$$$$17\!\cdots\!40$$$$T^{17} -$$$$47\!\cdots\!41$$$$T^{18} +$$$$75\!\cdots\!57$$$$T^{19} +$$$$12\!\cdots\!49$$$$T^{20}$$
$29$ $$1 - 6033 T + 3652157 T^{2} - 31641196734 T^{3} + 283528398607854 T^{4} - 668469168127712358 T^{5} +$$$$64\!\cdots\!39$$$$T^{6} -$$$$82\!\cdots\!55$$$$T^{7} -$$$$90\!\cdots\!17$$$$T^{8} -$$$$14\!\cdots\!60$$$$T^{9} +$$$$58\!\cdots\!16$$$$T^{10} -$$$$29\!\cdots\!40$$$$T^{11} -$$$$38\!\cdots\!17$$$$T^{12} -$$$$71\!\cdots\!95$$$$T^{13} +$$$$11\!\cdots\!39$$$$T^{14} -$$$$24\!\cdots\!42$$$$T^{15} +$$$$21\!\cdots\!54$$$$T^{16} -$$$$48\!\cdots\!66$$$$T^{17} +$$$$11\!\cdots\!57$$$$T^{18} -$$$$38\!\cdots\!17$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$
$31$ $$1 - 2759 T - 54902477 T^{2} - 189444651072 T^{3} + 2052158291804100 T^{4} + 13274031992302596720 T^{5} -$$$$32\!\cdots\!47$$$$T^{6} -$$$$42\!\cdots\!39$$$$T^{7} -$$$$13\!\cdots\!49$$$$T^{8} +$$$$36\!\cdots\!48$$$$T^{9} +$$$$63\!\cdots\!24$$$$T^{10} +$$$$10\!\cdots\!48$$$$T^{11} -$$$$11\!\cdots\!49$$$$T^{12} -$$$$99\!\cdots\!89$$$$T^{13} -$$$$21\!\cdots\!47$$$$T^{14} +$$$$25\!\cdots\!20$$$$T^{15} +$$$$11\!\cdots\!00$$$$T^{16} -$$$$29\!\cdots\!72$$$$T^{17} -$$$$24\!\cdots\!77$$$$T^{18} -$$$$35\!\cdots\!09$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$37$ $$( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} +$$$$13\!\cdots\!82$$$$T^{6} +$$$$38\!\cdots\!04$$$$T^{7} +$$$$67\!\cdots\!49$$$$T^{8} +$$$$17\!\cdots\!86$$$$T^{9} +$$$$16\!\cdots\!57$$$$T^{10} )^{2}$$
$41$ $$1 - 18435 T - 117679042 T^{2} + 4344492069675 T^{3} - 505249106564622 T^{4} -$$$$52\!\cdots\!97$$$$T^{5} +$$$$16\!\cdots\!40$$$$T^{6} +$$$$39\!\cdots\!43$$$$T^{7} -$$$$31\!\cdots\!03$$$$T^{8} -$$$$10\!\cdots\!42$$$$T^{9} +$$$$29\!\cdots\!40$$$$T^{10} -$$$$11\!\cdots\!42$$$$T^{11} -$$$$41\!\cdots\!03$$$$T^{12} +$$$$62\!\cdots\!43$$$$T^{13} +$$$$30\!\cdots\!40$$$$T^{14} -$$$$10\!\cdots\!97$$$$T^{15} -$$$$12\!\cdots\!22$$$$T^{16} +$$$$12\!\cdots\!75$$$$T^{17} -$$$$38\!\cdots\!42$$$$T^{18} -$$$$69\!\cdots\!35$$$$T^{19} +$$$$43\!\cdots\!01$$$$T^{20}$$
$43$ $$1 - 1469 T - 271863536 T^{2} + 4016430594327 T^{3} + 12129147672135834 T^{4} -$$$$75\!\cdots\!27$$$$T^{5} +$$$$55\!\cdots\!62$$$$T^{6} -$$$$12\!\cdots\!57$$$$T^{7} -$$$$29\!\cdots\!39$$$$T^{8} +$$$$61\!\cdots\!62$$$$T^{9} -$$$$11\!\cdots\!84$$$$T^{10} +$$$$90\!\cdots\!66$$$$T^{11} -$$$$64\!\cdots\!11$$$$T^{12} -$$$$38\!\cdots\!99$$$$T^{13} +$$$$26\!\cdots\!62$$$$T^{14} -$$$$52\!\cdots\!61$$$$T^{15} +$$$$12\!\cdots\!66$$$$T^{16} +$$$$59\!\cdots\!89$$$$T^{17} -$$$$59\!\cdots\!36$$$$T^{18} -$$$$47\!\cdots\!67$$$$T^{19} +$$$$47\!\cdots\!49$$$$T^{20}$$
$47$ $$1 - 25155 T - 401246233 T^{2} + 14349179861244 T^{3} + 97557609874842960 T^{4} -$$$$41\!\cdots\!12$$$$T^{5} -$$$$25\!\cdots\!27$$$$T^{6} +$$$$55\!\cdots\!85$$$$T^{7} +$$$$12\!\cdots\!11$$$$T^{8} -$$$$42\!\cdots\!56$$$$T^{9} -$$$$37\!\cdots\!60$$$$T^{10} -$$$$96\!\cdots\!92$$$$T^{11} +$$$$67\!\cdots\!39$$$$T^{12} +$$$$67\!\cdots\!55$$$$T^{13} -$$$$71\!\cdots\!27$$$$T^{14} -$$$$26\!\cdots\!84$$$$T^{15} +$$$$14\!\cdots\!40$$$$T^{16} +$$$$47\!\cdots\!92$$$$T^{17} -$$$$30\!\cdots\!33$$$$T^{18} -$$$$44\!\cdots\!85$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20}$$
$53$ $$( 1 + 58422 T + 3354568213 T^{2} + 110313236959296 T^{3} + 3390725554692289246 T^{4} +$$$$71\!\cdots\!28$$$$T^{5} +$$$$14\!\cdots\!78$$$$T^{6} +$$$$19\!\cdots\!04$$$$T^{7} +$$$$24\!\cdots\!41$$$$T^{8} +$$$$17\!\cdots\!22$$$$T^{9} +$$$$12\!\cdots\!93$$$$T^{10} )^{2}$$
$59$ $$1 - 90537 T + 2831117840 T^{2} - 13805150996349 T^{3} - 966660594685472478 T^{4} +$$$$10\!\cdots\!09$$$$T^{5} +$$$$69\!\cdots\!78$$$$T^{6} -$$$$39\!\cdots\!93$$$$T^{7} +$$$$84\!\cdots\!01$$$$T^{8} +$$$$17\!\cdots\!74$$$$T^{9} -$$$$12\!\cdots\!20$$$$T^{10} +$$$$12\!\cdots\!26$$$$T^{11} +$$$$42\!\cdots\!01$$$$T^{12} -$$$$14\!\cdots\!07$$$$T^{13} +$$$$18\!\cdots\!78$$$$T^{14} +$$$$19\!\cdots\!91$$$$T^{15} -$$$$12\!\cdots\!78$$$$T^{16} -$$$$13\!\cdots\!51$$$$T^{17} +$$$$19\!\cdots\!40$$$$T^{18} -$$$$44\!\cdots\!63$$$$T^{19} +$$$$34\!\cdots\!01$$$$T^{20}$$
$61$ $$1 - 1403 T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} +$$$$54\!\cdots\!90$$$$T^{5} -$$$$10\!\cdots\!33$$$$T^{6} -$$$$70\!\cdots\!89$$$$T^{7} +$$$$11\!\cdots\!67$$$$T^{8} +$$$$32\!\cdots\!04$$$$T^{9} -$$$$10\!\cdots\!12$$$$T^{10} +$$$$27\!\cdots\!04$$$$T^{11} +$$$$80\!\cdots\!67$$$$T^{12} -$$$$42\!\cdots\!89$$$$T^{13} -$$$$51\!\cdots\!33$$$$T^{14} +$$$$23\!\cdots\!90$$$$T^{15} +$$$$25\!\cdots\!98$$$$T^{16} -$$$$13\!\cdots\!46$$$$T^{17} -$$$$91\!\cdots\!83$$$$T^{18} -$$$$30\!\cdots\!03$$$$T^{19} +$$$$18\!\cdots\!01$$$$T^{20}$$
$67$ $$1 - 13907 T - 3876685544 T^{2} - 77425491657903 T^{3} + 10014688417385231130 T^{4} +$$$$30\!\cdots\!39$$$$T^{5} -$$$$79\!\cdots\!54$$$$T^{6} -$$$$69\!\cdots\!51$$$$T^{7} -$$$$33\!\cdots\!67$$$$T^{8} +$$$$37\!\cdots\!46$$$$T^{9} +$$$$22\!\cdots\!76$$$$T^{10} +$$$$51\!\cdots\!22$$$$T^{11} -$$$$60\!\cdots\!83$$$$T^{12} -$$$$16\!\cdots\!93$$$$T^{13} -$$$$26\!\cdots\!54$$$$T^{14} +$$$$13\!\cdots\!73$$$$T^{15} +$$$$60\!\cdots\!70$$$$T^{16} -$$$$63\!\cdots\!29$$$$T^{17} -$$$$42\!\cdots\!44$$$$T^{18} -$$$$20\!\cdots\!49$$$$T^{19} +$$$$20\!\cdots\!49$$$$T^{20}$$
$71$ $$( 1 + 114684 T + 7758380659 T^{2} + 426246123888336 T^{3} + 19260501229393543450 T^{4} +$$$$77\!\cdots\!40$$$$T^{5} +$$$$34\!\cdots\!50$$$$T^{6} +$$$$13\!\cdots\!36$$$$T^{7} +$$$$45\!\cdots\!09$$$$T^{8} +$$$$12\!\cdots\!84$$$$T^{9} +$$$$19\!\cdots\!51$$$$T^{10} )^{2}$$
$73$ $$( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} -$$$$80\!\cdots\!84$$$$T^{5} +$$$$25\!\cdots\!33$$$$T^{6} -$$$$13\!\cdots\!86$$$$T^{7} +$$$$32\!\cdots\!22$$$$T^{8} -$$$$14\!\cdots\!00$$$$T^{9} +$$$$38\!\cdots\!93$$$$T^{10} )^{2}$$
$79$ $$1 - 29993 T - 5352351629 T^{2} - 358913063028768 T^{3} + 26234825811851125236 T^{4} +$$$$21\!\cdots\!52$$$$T^{5} +$$$$27\!\cdots\!85$$$$T^{6} -$$$$84\!\cdots\!45$$$$T^{7} -$$$$35\!\cdots\!45$$$$T^{8} +$$$$81\!\cdots\!80$$$$T^{9} +$$$$16\!\cdots\!00$$$$T^{10} +$$$$25\!\cdots\!20$$$$T^{11} -$$$$33\!\cdots\!45$$$$T^{12} -$$$$24\!\cdots\!55$$$$T^{13} +$$$$24\!\cdots\!85$$$$T^{14} +$$$$60\!\cdots\!48$$$$T^{15} +$$$$22\!\cdots\!36$$$$T^{16} -$$$$93\!\cdots\!32$$$$T^{17} -$$$$43\!\cdots\!29$$$$T^{18} -$$$$74\!\cdots\!07$$$$T^{19} +$$$$76\!\cdots\!01$$$$T^{20}$$
$83$ $$1 - 228951 T + 21403431983 T^{2} - 1202282302650156 T^{3} + 62567029919071222368 T^{4} -$$$$36\!\cdots\!68$$$$T^{5} +$$$$11\!\cdots\!01$$$$T^{6} +$$$$11\!\cdots\!41$$$$T^{7} -$$$$18\!\cdots\!73$$$$T^{8} +$$$$14\!\cdots\!84$$$$T^{9} -$$$$88\!\cdots\!72$$$$T^{10} +$$$$55\!\cdots\!12$$$$T^{11} -$$$$28\!\cdots\!77$$$$T^{12} +$$$$67\!\cdots\!87$$$$T^{13} +$$$$28\!\cdots\!01$$$$T^{14} -$$$$34\!\cdots\!24$$$$T^{15} +$$$$23\!\cdots\!32$$$$T^{16} -$$$$17\!\cdots\!92$$$$T^{17} +$$$$12\!\cdots\!83$$$$T^{18} -$$$$52\!\cdots\!93$$$$T^{19} +$$$$89\!\cdots\!49$$$$T^{20}$$
$89$ $$( 1 + 299166 T + 52616244181 T^{2} + 6660261403977288 T^{3} +$$$$67\!\cdots\!10$$$$T^{4} +$$$$55\!\cdots\!64$$$$T^{5} +$$$$37\!\cdots\!90$$$$T^{6} +$$$$20\!\cdots\!88$$$$T^{7} +$$$$91\!\cdots\!69$$$$T^{8} +$$$$29\!\cdots\!66$$$$T^{9} +$$$$54\!\cdots\!49$$$$T^{10} )^{2}$$
$97$ $$1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} -$$$$30\!\cdots\!95$$$$T^{5} +$$$$10\!\cdots\!20$$$$T^{6} +$$$$21\!\cdots\!29$$$$T^{7} -$$$$20\!\cdots\!15$$$$T^{8} -$$$$65\!\cdots\!06$$$$T^{9} +$$$$19\!\cdots\!00$$$$T^{10} -$$$$56\!\cdots\!42$$$$T^{11} -$$$$15\!\cdots\!35$$$$T^{12} +$$$$13\!\cdots\!97$$$$T^{13} +$$$$56\!\cdots\!20$$$$T^{14} -$$$$14\!\cdots\!15$$$$T^{15} +$$$$39\!\cdots\!90$$$$T^{16} +$$$$77\!\cdots\!49$$$$T^{17} -$$$$52\!\cdots\!38$$$$T^{18} -$$$$10\!\cdots\!37$$$$T^{19} +$$$$21\!\cdots\!49$$$$T^{20}$$