# Properties

 Label 201.4.j.a Level 201 Weight 4 Character orbit 201.j Analytic conductor 11.859 Analytic rank 0 Dimension 20 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.j (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8593839112$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{22})$$ Coefficient field: $$\Q(\zeta_{33})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{10}\cdot 3^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{33}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 \zeta_{33}^{8} - 3 \zeta_{33}^{19} ) q^{3} + ( 8 - 8 \zeta_{33}^{2} + 8 \zeta_{33}^{3} - 8 \zeta_{33}^{5} + 8 \zeta_{33}^{6} - 8 \zeta_{33}^{8} + 8 \zeta_{33}^{9} + 8 \zeta_{33}^{12} - 8 \zeta_{33}^{13} + 8 \zeta_{33}^{15} - 8 \zeta_{33}^{16} + 8 \zeta_{33}^{18} - 8 \zeta_{33}^{19} ) q^{4} + ( -18 \zeta_{33}^{4} - \zeta_{33}^{6} + \zeta_{33}^{15} + 18 \zeta_{33}^{17} ) q^{7} + ( 27 \zeta_{33}^{5} + 27 \zeta_{33}^{16} ) q^{9} +O(q^{10})$$ $$q + ( 3 \zeta_{33}^{8} - 3 \zeta_{33}^{19} ) q^{3} + ( 8 - 8 \zeta_{33}^{2} + 8 \zeta_{33}^{3} - 8 \zeta_{33}^{5} + 8 \zeta_{33}^{6} - 8 \zeta_{33}^{8} + 8 \zeta_{33}^{9} + 8 \zeta_{33}^{12} - 8 \zeta_{33}^{13} + 8 \zeta_{33}^{15} - 8 \zeta_{33}^{16} + 8 \zeta_{33}^{18} - 8 \zeta_{33}^{19} ) q^{4} + ( -18 \zeta_{33}^{4} - \zeta_{33}^{6} + \zeta_{33}^{15} + 18 \zeta_{33}^{17} ) q^{7} + ( 27 \zeta_{33}^{5} + 27 \zeta_{33}^{16} ) q^{9} + ( 48 \zeta_{33}^{7} + 24 \zeta_{33}^{18} ) q^{12} + ( 36 \zeta_{33} + 17 \zeta_{33}^{2} + 53 \zeta_{33}^{12} + 53 \zeta_{33}^{13} ) q^{13} + 64 \zeta_{33}^{9} q^{16} + ( -90 + 90 \zeta_{33} - 163 \zeta_{33}^{3} + 90 \zeta_{33}^{4} - 90 \zeta_{33}^{6} + 90 \zeta_{33}^{7} - 73 \zeta_{33}^{9} + 90 \zeta_{33}^{10} - 90 \zeta_{33}^{11} + 90 \zeta_{33}^{13} - 180 \zeta_{33}^{14} + 90 \zeta_{33}^{16} - 90 \zeta_{33}^{17} + 90 \zeta_{33}^{19} ) q^{19} + ( -60 \zeta_{33} - 111 \zeta_{33}^{3} - 111 \zeta_{33}^{12} - 60 \zeta_{33}^{14} ) q^{21} -125 \zeta_{33}^{18} q^{25} + ( -81 \zeta_{33}^{2} + 81 \zeta_{33}^{13} ) q^{27} + ( -152 \zeta_{33}^{3} - 152 \zeta_{33}^{5} - 144 \zeta_{33}^{14} - 8 \zeta_{33}^{16} ) q^{28} + ( 109 \zeta_{33}^{2} + 109 \zeta_{33}^{6} + 199 \zeta_{33}^{13} - 90 \zeta_{33}^{17} ) q^{31} + 216 \zeta_{33}^{15} q^{36} + ( -252 \zeta_{33}^{4} + 252 \zeta_{33}^{7} - 181 \zeta_{33}^{15} + 71 \zeta_{33}^{18} ) q^{37} + ( -477 + 210 \zeta_{33} + 267 \zeta_{33}^{2} - 477 \zeta_{33}^{3} + 210 \zeta_{33}^{4} + 267 \zeta_{33}^{5} - 477 \zeta_{33}^{6} + 210 \zeta_{33}^{7} + 267 \zeta_{33}^{8} - 210 \zeta_{33}^{9} + 420 \zeta_{33}^{10} - 210 \zeta_{33}^{11} - 267 \zeta_{33}^{12} + 477 \zeta_{33}^{13} - 210 \zeta_{33}^{14} - 267 \zeta_{33}^{15} + 477 \zeta_{33}^{16} - 210 \zeta_{33}^{17} - 267 \zeta_{33}^{18} + 477 \zeta_{33}^{19} ) q^{39} + ( 197 \zeta_{33}^{5} - 323 \zeta_{33}^{8} + 323 \zeta_{33}^{16} - 197 \zeta_{33}^{19} ) q^{43} + ( 192 \zeta_{33}^{6} + 384 \zeta_{33}^{17} ) q^{48} + ( 343 + 360 \zeta_{33} - 343 \zeta_{33}^{2} + 343 \zeta_{33}^{3} - 343 \zeta_{33}^{5} + 343 \zeta_{33}^{6} - 20 \zeta_{33}^{8} + 343 \zeta_{33}^{9} + 380 \zeta_{33}^{12} - 343 \zeta_{33}^{13} + 343 \zeta_{33}^{15} - 343 \zeta_{33}^{16} + 343 \zeta_{33}^{18} - 380 \zeta_{33}^{19} ) q^{49} + ( -136 - 288 \zeta_{33} + 288 \zeta_{33}^{11} + 136 \zeta_{33}^{12} ) q^{52} + ( 321 - 489 \zeta_{33}^{6} - 168 \zeta_{33}^{11} - 168 \zeta_{33}^{17} ) q^{57} + ( 361 - 361 \zeta_{33}^{2} + 361 \zeta_{33}^{3} - 182 \zeta_{33}^{5} + 361 \zeta_{33}^{6} - 361 \zeta_{33}^{8} + 361 \zeta_{33}^{9} - 540 \zeta_{33}^{10} + 361 \zeta_{33}^{12} - 361 \zeta_{33}^{13} + 361 \zeta_{33}^{15} - 722 \zeta_{33}^{16} + 361 \zeta_{33}^{18} - 361 \zeta_{33}^{19} ) q^{61} + ( 513 - 486 \zeta_{33} + 486 \zeta_{33}^{3} - 486 \zeta_{33}^{4} + 486 \zeta_{33}^{6} - 486 \zeta_{33}^{7} - 27 \zeta_{33}^{9} - 486 \zeta_{33}^{10} - 486 \zeta_{33}^{13} + 486 \zeta_{33}^{14} - 486 \zeta_{33}^{16} + 486 \zeta_{33}^{17} - 486 \zeta_{33}^{19} ) q^{63} + ( 512 \zeta_{33}^{8} + 512 \zeta_{33}^{19} ) q^{64} + ( -378 \zeta_{33}^{7} + 251 \zeta_{33}^{18} ) q^{67} + ( -487 + 216 \zeta_{33}^{4} + 216 \zeta_{33}^{11} - 487 \zeta_{33}^{15} ) q^{73} + ( 750 \zeta_{33}^{4} + 375 \zeta_{33}^{15} ) q^{75} + ( 136 \zeta_{33}^{2} - 584 \zeta_{33}^{8} - 584 \zeta_{33}^{13} + 136 \zeta_{33}^{19} ) q^{76} + ( -127 + 757 \zeta_{33}^{3} + 630 \zeta_{33}^{11} + 630 \zeta_{33}^{14} ) q^{79} + ( -729 + 729 \zeta_{33}^{2} - 729 \zeta_{33}^{3} + 729 \zeta_{33}^{5} - 729 \zeta_{33}^{6} + 729 \zeta_{33}^{8} - 729 \zeta_{33}^{9} - 729 \zeta_{33}^{12} + 729 \zeta_{33}^{13} - 729 \zeta_{33}^{15} + 729 \zeta_{33}^{16} - 729 \zeta_{33}^{18} + 729 \zeta_{33}^{19} ) q^{81} + ( 408 - 408 \zeta_{33}^{2} - 480 \zeta_{33}^{11} - 888 \zeta_{33}^{13} ) q^{84} + ( -701 \zeta_{33}^{5} - 359 \zeta_{33}^{6} - 990 \zeta_{33}^{7} - 971 \zeta_{33}^{8} - 971 \zeta_{33}^{16} - 990 \zeta_{33}^{17} - 359 \zeta_{33}^{18} - 701 \zeta_{33}^{19} ) q^{91} + ( -867 + 867 \zeta_{33}^{2} + 867 \zeta_{33}^{5} - 867 \zeta_{33}^{6} + 867 \zeta_{33}^{8} - 867 \zeta_{33}^{9} + 924 \zeta_{33}^{10} - 867 \zeta_{33}^{12} + 867 \zeta_{33}^{13} + 924 \zeta_{33}^{14} - 867 \zeta_{33}^{15} + 867 \zeta_{33}^{16} - 867 \zeta_{33}^{18} + 867 \zeta_{33}^{19} ) q^{93} + ( 1061 \zeta_{33}^{3} + 1061 \zeta_{33}^{8} + 792 \zeta_{33}^{14} + 269 \zeta_{33}^{19} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 16q^{4} + 54q^{9} + O(q^{10})$$ $$20q + 16q^{4} + 54q^{9} - 128q^{16} + 112q^{19} + 324q^{21} + 250q^{25} - 432q^{36} + 220q^{37} - 648q^{39} + 1258q^{49} - 6160q^{52} + 8910q^{57} + 5940q^{63} + 1024q^{64} - 880q^{67} - 10710q^{73} - 896q^{76} - 9724q^{79} - 1458q^{81} + 11664q^{84} - 3888q^{91} - 1620q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$$.

 $$n$$ $$68$$ $$136$$ $$\chi(n)$$ $$-1$$ $$1 + \zeta_{33} + \zeta_{33}^{2} + \zeta_{33}^{3} + \zeta_{33}^{4} - \zeta_{33}^{5} + \zeta_{33}^{6} - \zeta_{33}^{7} + \zeta_{33}^{8} - \zeta_{33}^{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.928368 − 0.371662i −0.786053 − 0.618159i 0.235759 − 0.971812i 0.723734 + 0.690079i −0.327068 + 0.945001i 0.981929 − 0.189251i −0.327068 − 0.945001i 0.981929 + 0.189251i 0.580057 − 0.814576i −0.995472 − 0.0950560i 0.580057 + 0.814576i −0.995472 + 0.0950560i −0.888835 − 0.458227i 0.0475819 + 0.998867i 0.928368 + 0.371662i −0.786053 + 0.618159i 0.235759 + 0.971812i 0.723734 − 0.690079i −0.888835 + 0.458227i 0.0475819 − 0.998867i
0 −4.72659 + 2.15856i 1.13852 + 7.91857i 0 0 18.3144 + 15.8695i 0 17.6812 20.4052i 0
5.2 0 4.72659 2.15856i 1.13852 + 7.91857i 0 0 11.6079 + 10.0583i 0 17.6812 20.4052i 0
8.1 0 −3.92699 + 3.40276i 7.67594 + 2.25386i 0 0 −24.0323 3.45532i 0 3.84250 26.7252i 0
8.2 0 3.92699 3.40276i 7.67594 + 2.25386i 0 0 35.1868 + 5.05910i 0 3.84250 26.7252i 0
53.1 0 −2.80925 + 4.37128i 5.23889 6.04600i 0 0 4.82246 2.20234i 0 −11.2162 24.5601i 0
53.2 0 2.80925 4.37128i 5.23889 6.04600i 0 0 −32.3206 + 14.7603i 0 −11.2162 24.5601i 0
110.1 0 −2.80925 4.37128i 5.23889 + 6.04600i 0 0 4.82246 + 2.20234i 0 −11.2162 + 24.5601i 0
110.2 0 2.80925 + 4.37128i 5.23889 + 6.04600i 0 0 −32.3206 14.7603i 0 −11.2162 + 24.5601i 0
119.1 0 −1.46393 4.98567i −3.32332 + 7.27706i 0 0 −2.83365 4.40925i 0 −22.7138 + 14.5973i 0
119.2 0 1.46393 + 4.98567i −3.32332 + 7.27706i 0 0 −16.8377 26.2000i 0 −22.7138 + 14.5973i 0
125.1 0 −1.46393 + 4.98567i −3.32332 7.27706i 0 0 −2.83365 + 4.40925i 0 −22.7138 14.5973i 0
125.2 0 1.46393 4.98567i −3.32332 7.27706i 0 0 −16.8377 + 26.2000i 0 −22.7138 14.5973i 0
137.1 0 −5.14326 0.739490i −6.73003 4.32513i 0 0 10.4355 35.5401i 0 25.9063 + 7.60678i 0
137.2 0 5.14326 + 0.739490i −6.73003 4.32513i 0 0 −4.34287 + 14.7905i 0 25.9063 + 7.60678i 0
161.1 0 −4.72659 2.15856i 1.13852 7.91857i 0 0 18.3144 15.8695i 0 17.6812 + 20.4052i 0
161.2 0 4.72659 + 2.15856i 1.13852 7.91857i 0 0 11.6079 10.0583i 0 17.6812 + 20.4052i 0
176.1 0 −3.92699 3.40276i 7.67594 2.25386i 0 0 −24.0323 + 3.45532i 0 3.84250 + 26.7252i 0
176.2 0 3.92699 + 3.40276i 7.67594 2.25386i 0 0 35.1868 5.05910i 0 3.84250 + 26.7252i 0
179.1 0 −5.14326 + 0.739490i −6.73003 + 4.32513i 0 0 10.4355 + 35.5401i 0 25.9063 7.60678i 0
179.2 0 5.14326 0.739490i −6.73003 + 4.32513i 0 0 −4.34287 14.7905i 0 25.9063 7.60678i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
67.f odd 22 1 inner
201.j even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.j.a 20
3.b odd 2 1 CM 201.4.j.a 20
67.f odd 22 1 inner 201.4.j.a 20
201.j even 22 1 inner 201.4.j.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.j.a 20 1.a even 1 1 trivial
201.4.j.a 20 3.b odd 2 1 CM
201.4.j.a 20 67.f odd 22 1 inner
201.4.j.a 20 201.j even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 8 T^{2} + 64 T^{4} - 512 T^{6} + 4096 T^{8} - 32768 T^{10} + 262144 T^{12} - 2097152 T^{14} + 16777216 T^{16} - 134217728 T^{18} + 1073741824 T^{20} )^{2}$$
$3$ $$1 - 27 T^{2} + 729 T^{4} - 19683 T^{6} + 531441 T^{8} - 14348907 T^{10} + 387420489 T^{12} - 10460353203 T^{14} + 282429536481 T^{16} - 7625597484987 T^{18} + 205891132094649 T^{20}$$
$5$ $$( 1 - 125 T^{2} + 15625 T^{4} - 1953125 T^{6} + 244140625 T^{8} - 30517578125 T^{10} + 3814697265625 T^{12} - 476837158203125 T^{14} + 59604644775390625 T^{16} - 7450580596923828125 T^{18} +$$$$93\!\cdots\!25$$$$T^{20} )^{2}$$
$7$ $$( 1 - 20 T + 57 T^{2} + 5720 T^{3} - 133951 T^{4} + 717060 T^{5} + 31603993 T^{6} - 878031440 T^{7} + 6720459201 T^{8} + 166755599900 T^{9} - 5640229503943 T^{10} + 57197170765700 T^{11} + 790655304538449 T^{12} - 35431735663404080 T^{13} + 437439943811393593 T^{14} + 3404286456319727580 T^{15} -$$$$21\!\cdots\!99$$$$T^{16} +$$$$31\!\cdots\!40$$$$T^{17} +$$$$10\!\cdots\!57$$$$T^{18} -$$$$13\!\cdots\!60$$$$T^{19} +$$$$22\!\cdots\!49$$$$T^{20} )( 1 + 20 T + 57 T^{2} - 5720 T^{3} - 133951 T^{4} - 717060 T^{5} + 31603993 T^{6} + 878031440 T^{7} + 6720459201 T^{8} - 166755599900 T^{9} - 5640229503943 T^{10} - 57197170765700 T^{11} + 790655304538449 T^{12} + 35431735663404080 T^{13} + 437439943811393593 T^{14} - 3404286456319727580 T^{15} -$$$$21\!\cdots\!99$$$$T^{16} -$$$$31\!\cdots\!40$$$$T^{17} +$$$$10\!\cdots\!57$$$$T^{18} +$$$$13\!\cdots\!60$$$$T^{19} +$$$$22\!\cdots\!49$$$$T^{20} )$$
$11$ $$( 1 - 121 T + 6655 T^{2} - 161051 T^{3} - 1771561 T^{4} + 214358881 T^{5} - 2357947691 T^{6} - 285311670611 T^{7} + 15692141883605 T^{8} - 379749833583241 T^{9} + 4177248169415651 T^{10} )^{2}( 1 + 121 T + 6655 T^{2} + 161051 T^{3} - 1771561 T^{4} - 214358881 T^{5} - 2357947691 T^{6} + 285311670611 T^{7} + 15692141883605 T^{8} + 379749833583241 T^{9} + 4177248169415651 T^{10} )^{2}$$
$13$ $$( 1 - 70 T + 2703 T^{2} - 35420 T^{3} - 3459091 T^{4} + 319954110 T^{5} - 14797164773 T^{6} + 332862354440 T^{7} + 9209006195481 T^{8} - 1375929026388350 T^{9} + 76082845235712743 T^{10} - 3022916070975204950 T^{11} + 44450113985403450129 T^{12} +$$$$35\!\cdots\!20$$$$T^{13} -$$$$34\!\cdots\!13$$$$T^{14} +$$$$16\!\cdots\!70$$$$T^{15} -$$$$38\!\cdots\!39$$$$T^{16} -$$$$87\!\cdots\!60$$$$T^{17} +$$$$14\!\cdots\!83$$$$T^{18} -$$$$83\!\cdots\!90$$$$T^{19} +$$$$26\!\cdots\!49$$$$T^{20} )( 1 + 70 T + 2703 T^{2} + 35420 T^{3} - 3459091 T^{4} - 319954110 T^{5} - 14797164773 T^{6} - 332862354440 T^{7} + 9209006195481 T^{8} + 1375929026388350 T^{9} + 76082845235712743 T^{10} + 3022916070975204950 T^{11} + 44450113985403450129 T^{12} -$$$$35\!\cdots\!20$$$$T^{13} -$$$$34\!\cdots\!13$$$$T^{14} -$$$$16\!\cdots\!70$$$$T^{15} -$$$$38\!\cdots\!39$$$$T^{16} +$$$$87\!\cdots\!60$$$$T^{17} +$$$$14\!\cdots\!83$$$$T^{18} +$$$$83\!\cdots\!90$$$$T^{19} +$$$$26\!\cdots\!49$$$$T^{20} )$$
$17$ $$( 1 + 4913 T^{2} + 24137569 T^{4} + 118587876497 T^{6} + 582622237229761 T^{8} + 2862423051509815793 T^{10} +$$$$14\!\cdots\!09$$$$T^{12} +$$$$69\!\cdots\!17$$$$T^{14} +$$$$33\!\cdots\!21$$$$T^{16} +$$$$16\!\cdots\!73$$$$T^{18} +$$$$81\!\cdots\!49$$$$T^{20} )^{2}$$
$19$ $$( 1 - 56 T - 3723 T^{2} + 592592 T^{3} - 7649095 T^{4} - 3636239208 T^{5} + 256094538253 T^{6} + 10599670585504 T^{7} - 2350133990665551 T^{8} + 58904362931298920 T^{9} + 12820924717822274789 T^{10} +$$$$40\!\cdots\!80$$$$T^{11} -$$$$11\!\cdots\!31$$$$T^{12} +$$$$34\!\cdots\!16$$$$T^{13} +$$$$56\!\cdots\!33$$$$T^{14} -$$$$55\!\cdots\!92$$$$T^{15} -$$$$79\!\cdots\!95$$$$T^{16} +$$$$42\!\cdots\!48$$$$T^{17} -$$$$18\!\cdots\!83$$$$T^{18} -$$$$18\!\cdots\!84$$$$T^{19} +$$$$23\!\cdots\!01$$$$T^{20} )^{2}$$
$23$ $$( 1 + 12167 T^{2} + 148035889 T^{4} + 1801152661463 T^{6} + 21914624432020321 T^{8} +$$$$26\!\cdots\!07$$$$T^{10} +$$$$32\!\cdots\!69$$$$T^{12} +$$$$39\!\cdots\!23$$$$T^{14} +$$$$48\!\cdots\!41$$$$T^{16} +$$$$58\!\cdots\!47$$$$T^{18} +$$$$71\!\cdots\!49$$$$T^{20} )^{2}$$
$29$ $$( 1 - 24389 T^{2} )^{20}$$
$31$ $$( 1 - 308 T + 65073 T^{2} - 10866856 T^{3} + 1408401905 T^{4} - 110053279644 T^{5} - 8061291021503 T^{6} + 5761474888497328 T^{7} - 1534380344835581151 T^{8} +$$$$30\!\cdots\!60$$$$T^{9} -$$$$46\!\cdots\!39$$$$T^{10} +$$$$89\!\cdots\!60$$$$T^{11} -$$$$13\!\cdots\!31$$$$T^{12} +$$$$15\!\cdots\!88$$$$T^{13} -$$$$63\!\cdots\!83$$$$T^{14} -$$$$25\!\cdots\!44$$$$T^{15} +$$$$98\!\cdots\!05$$$$T^{16} -$$$$22\!\cdots\!36$$$$T^{17} +$$$$40\!\cdots\!33$$$$T^{18} -$$$$56\!\cdots\!88$$$$T^{19} +$$$$55\!\cdots\!01$$$$T^{20} )( 1 + 308 T + 65073 T^{2} + 10866856 T^{3} + 1408401905 T^{4} + 110053279644 T^{5} - 8061291021503 T^{6} - 5761474888497328 T^{7} - 1534380344835581151 T^{8} -$$$$30\!\cdots\!60$$$$T^{9} -$$$$46\!\cdots\!39$$$$T^{10} -$$$$89\!\cdots\!60$$$$T^{11} -$$$$13\!\cdots\!31$$$$T^{12} -$$$$15\!\cdots\!88$$$$T^{13} -$$$$63\!\cdots\!83$$$$T^{14} +$$$$25\!\cdots\!44$$$$T^{15} +$$$$98\!\cdots\!05$$$$T^{16} +$$$$22\!\cdots\!36$$$$T^{17} +$$$$40\!\cdots\!33$$$$T^{18} +$$$$56\!\cdots\!88$$$$T^{19} +$$$$55\!\cdots\!01$$$$T^{20} )$$
$37$ $$( 1 - 110 T - 38553 T^{2} + 9812660 T^{3} + 873432509 T^{4} - 593118242970 T^{5} + 21001029848323 T^{6} + 27733105077843880 T^{7} - 4114406723469931719 T^{8} -$$$$95\!\cdots\!50$$$$T^{9} +$$$$31\!\cdots\!07$$$$T^{10} -$$$$48\!\cdots\!50$$$$T^{11} -$$$$10\!\cdots\!71$$$$T^{12} +$$$$36\!\cdots\!60$$$$T^{13} +$$$$13\!\cdots\!63$$$$T^{14} -$$$$19\!\cdots\!10$$$$T^{15} +$$$$14\!\cdots\!61$$$$T^{16} +$$$$83\!\cdots\!20$$$$T^{17} -$$$$16\!\cdots\!33$$$$T^{18} -$$$$24\!\cdots\!30$$$$T^{19} +$$$$11\!\cdots\!49$$$$T^{20} )^{2}$$
$41$ $$( 1 - 68921 T^{2} + 4750104241 T^{4} - 327381934393961 T^{6} + 22563490300366186081 T^{8} -$$$$15\!\cdots\!01$$$$T^{10} +$$$$10\!\cdots\!21$$$$T^{12} -$$$$73\!\cdots\!41$$$$T^{14} +$$$$50\!\cdots\!61$$$$T^{16} -$$$$35\!\cdots\!81$$$$T^{18} +$$$$24\!\cdots\!01$$$$T^{20} )^{2}$$
$43$ $$( 1 - 520 T + 190893 T^{2} - 57920720 T^{3} + 14941444649 T^{4} - 3164448532440 T^{5} + 457563797160757 T^{6} + 13662634945113440 T^{7} - 43484094992319295599 T^{8} +$$$$21\!\cdots\!00$$$$T^{9} -$$$$77\!\cdots\!07$$$$T^{10} +$$$$17\!\cdots\!00$$$$T^{11} -$$$$27\!\cdots\!51$$$$T^{12} +$$$$68\!\cdots\!20$$$$T^{13} +$$$$18\!\cdots\!57$$$$T^{14} -$$$$10\!\cdots\!80$$$$T^{15} +$$$$37\!\cdots\!01$$$$T^{16} -$$$$11\!\cdots\!60$$$$T^{17} +$$$$30\!\cdots\!93$$$$T^{18} -$$$$66\!\cdots\!40$$$$T^{19} +$$$$10\!\cdots\!49$$$$T^{20} )( 1 + 520 T + 190893 T^{2} + 57920720 T^{3} + 14941444649 T^{4} + 3164448532440 T^{5} + 457563797160757 T^{6} - 13662634945113440 T^{7} - 43484094992319295599 T^{8} -$$$$21\!\cdots\!00$$$$T^{9} -$$$$77\!\cdots\!07$$$$T^{10} -$$$$17\!\cdots\!00$$$$T^{11} -$$$$27\!\cdots\!51$$$$T^{12} -$$$$68\!\cdots\!20$$$$T^{13} +$$$$18\!\cdots\!57$$$$T^{14} +$$$$10\!\cdots\!80$$$$T^{15} +$$$$37\!\cdots\!01$$$$T^{16} +$$$$11\!\cdots\!60$$$$T^{17} +$$$$30\!\cdots\!93$$$$T^{18} +$$$$66\!\cdots\!40$$$$T^{19} +$$$$10\!\cdots\!49$$$$T^{20} )$$
$47$ $$( 1 + 103823 T^{2} + 10779215329 T^{4} + 1119130473102767 T^{6} +$$$$11\!\cdots\!41$$$$T^{8} +$$$$12\!\cdots\!43$$$$T^{10} +$$$$12\!\cdots\!89$$$$T^{12} +$$$$13\!\cdots\!47$$$$T^{14} +$$$$13\!\cdots\!81$$$$T^{16} +$$$$14\!\cdots\!63$$$$T^{18} +$$$$14\!\cdots\!49$$$$T^{20} )^{2}$$
$53$ $$( 1 - 148877 T^{2} + 22164361129 T^{4} - 3299763591802133 T^{6} +$$$$49\!\cdots\!41$$$$T^{8} -$$$$73\!\cdots\!57$$$$T^{10} +$$$$10\!\cdots\!89$$$$T^{12} -$$$$16\!\cdots\!53$$$$T^{14} +$$$$24\!\cdots\!81$$$$T^{16} -$$$$35\!\cdots\!37$$$$T^{18} +$$$$53\!\cdots\!49$$$$T^{20} )^{2}$$
$59$ $$( 1 + 205379 T^{2} + 42180533641 T^{4} + 8662995818654939 T^{6} +$$$$17\!\cdots\!81$$$$T^{8} +$$$$36\!\cdots\!99$$$$T^{10} +$$$$75\!\cdots\!21$$$$T^{12} +$$$$15\!\cdots\!59$$$$T^{14} +$$$$31\!\cdots\!61$$$$T^{16} +$$$$65\!\cdots\!19$$$$T^{18} +$$$$13\!\cdots\!01$$$$T^{20} )^{2}$$
$61$ $$( 1 - 182 T - 193857 T^{2} + 76592516 T^{3} + 30062017805 T^{4} - 22856333114706 T^{5} - 2663654236520213 T^{6} + 5672738417755761352 T^{7} -$$$$42\!\cdots\!11$$$$T^{8} -$$$$12\!\cdots\!10$$$$T^{9} +$$$$31\!\cdots\!11$$$$T^{10} -$$$$27\!\cdots\!10$$$$T^{11} -$$$$22\!\cdots\!71$$$$T^{12} +$$$$66\!\cdots\!32$$$$T^{13} -$$$$70\!\cdots\!73$$$$T^{14} -$$$$13\!\cdots\!06$$$$T^{15} +$$$$41\!\cdots\!05$$$$T^{16} +$$$$23\!\cdots\!76$$$$T^{17} -$$$$13\!\cdots\!37$$$$T^{18} -$$$$29\!\cdots\!22$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20} )( 1 + 182 T - 193857 T^{2} - 76592516 T^{3} + 30062017805 T^{4} + 22856333114706 T^{5} - 2663654236520213 T^{6} - 5672738417755761352 T^{7} -$$$$42\!\cdots\!11$$$$T^{8} +$$$$12\!\cdots\!10$$$$T^{9} +$$$$31\!\cdots\!11$$$$T^{10} +$$$$27\!\cdots\!10$$$$T^{11} -$$$$22\!\cdots\!71$$$$T^{12} -$$$$66\!\cdots\!32$$$$T^{13} -$$$$70\!\cdots\!73$$$$T^{14} +$$$$13\!\cdots\!06$$$$T^{15} +$$$$41\!\cdots\!05$$$$T^{16} -$$$$23\!\cdots\!76$$$$T^{17} -$$$$13\!\cdots\!37$$$$T^{18} +$$$$29\!\cdots\!22$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20} )$$
$67$ $$1 + 880 T + 473637 T^{2} + 152129120 T^{3} - 8578859431 T^{4} - 53304206817840 T^{5} - 44327498500653347 T^{6} - 22976265525420933440 T^{7} -$$$$68\!\cdots\!39$$$$T^{8} +$$$$84\!\cdots\!00$$$$T^{9} +$$$$28\!\cdots\!57$$$$T^{10} +$$$$25\!\cdots\!00$$$$T^{11} -$$$$62\!\cdots\!91$$$$T^{12} -$$$$62\!\cdots\!80$$$$T^{13} -$$$$36\!\cdots\!67$$$$T^{14} -$$$$13\!\cdots\!20$$$$T^{15} -$$$$63\!\cdots\!79$$$$T^{16} +$$$$33\!\cdots\!40$$$$T^{17} +$$$$31\!\cdots\!77$$$$T^{18} +$$$$17\!\cdots\!40$$$$T^{19} +$$$$60\!\cdots\!49$$$$T^{20}$$
$71$ $$( 1 + 357911 T^{2} + 128100283921 T^{4} + 45848500718449031 T^{6} +$$$$16\!\cdots\!41$$$$T^{8} +$$$$58\!\cdots\!51$$$$T^{10} +$$$$21\!\cdots\!61$$$$T^{12} +$$$$75\!\cdots\!71$$$$T^{14} +$$$$26\!\cdots\!81$$$$T^{16} +$$$$96\!\cdots\!91$$$$T^{18} +$$$$34\!\cdots\!01$$$$T^{20} )^{2}$$
$73$ $$( 1 + 1190 T + 389017 T^{2} )^{10}( 1 - 1190 T + 1027083 T^{2} - 759298540 T^{3} + 504012515189 T^{4} - 304394852939730 T^{5} + 166160438376999487 T^{6} - 79316149162574444120 T^{7} +$$$$29\!\cdots\!21$$$$T^{8} -$$$$45\!\cdots\!50$$$$T^{9} -$$$$61\!\cdots\!57$$$$T^{10} -$$$$17\!\cdots\!50$$$$T^{11} +$$$$45\!\cdots\!69$$$$T^{12} -$$$$46\!\cdots\!60$$$$T^{13} +$$$$38\!\cdots\!27$$$$T^{14} -$$$$27\!\cdots\!10$$$$T^{15} +$$$$17\!\cdots\!41$$$$T^{16} -$$$$10\!\cdots\!20$$$$T^{17} +$$$$53\!\cdots\!03$$$$T^{18} -$$$$24\!\cdots\!30$$$$T^{19} +$$$$79\!\cdots\!49$$$$T^{20} )$$
$79$ $$( 1 + 884 T + 493039 T^{2} )^{10}( 1 + 884 T + 288417 T^{2} - 180885848 T^{3} - 302103918895 T^{4} - 177876086691108 T^{5} - 8293446566867567 T^{6} + 80368441140986267984 T^{7} +$$$$75\!\cdots\!69$$$$T^{8} +$$$$26\!\cdots\!20$$$$T^{9} -$$$$13\!\cdots\!11$$$$T^{10} +$$$$13\!\cdots\!80$$$$T^{11} +$$$$18\!\cdots\!49$$$$T^{12} +$$$$96\!\cdots\!96$$$$T^{13} -$$$$49\!\cdots\!47$$$$T^{14} -$$$$51\!\cdots\!92$$$$T^{15} -$$$$43\!\cdots\!95$$$$T^{16} -$$$$12\!\cdots\!92$$$$T^{17} +$$$$10\!\cdots\!77$$$$T^{18} +$$$$15\!\cdots\!56$$$$T^{19} +$$$$84\!\cdots\!01$$$$T^{20} )$$
$83$ $$( 1 + 571787 T^{2} + 326940373369 T^{4} + 186940255267540403 T^{6} +$$$$10\!\cdots\!61$$$$T^{8} +$$$$61\!\cdots\!07$$$$T^{10} +$$$$34\!\cdots\!09$$$$T^{12} +$$$$19\!\cdots\!83$$$$T^{14} +$$$$11\!\cdots\!21$$$$T^{16} +$$$$65\!\cdots\!27$$$$T^{18} +$$$$37\!\cdots\!49$$$$T^{20} )^{2}$$
$89$ $$( 1 + 704969 T^{2} + 496981290961 T^{4} + 350356403707485209 T^{6} +$$$$24\!\cdots\!21$$$$T^{8} +$$$$17\!\cdots\!49$$$$T^{10} +$$$$12\!\cdots\!81$$$$T^{12} +$$$$86\!\cdots\!89$$$$T^{14} +$$$$61\!\cdots\!41$$$$T^{16} +$$$$43\!\cdots\!29$$$$T^{18} +$$$$30\!\cdots\!01$$$$T^{20} )^{2}$$
$97$ $$( 1 - 1330 T + 856227 T^{2} + 75073180 T^{3} - 881302594171 T^{4} + 1103615185837290 T^{5} - 663467114633766617 T^{6} -$$$$12\!\cdots\!60$$$$T^{7} +$$$$77\!\cdots\!41$$$$T^{8} -$$$$91\!\cdots\!50$$$$T^{9} +$$$$50\!\cdots\!07$$$$T^{10} -$$$$83\!\cdots\!50$$$$T^{11} +$$$$64\!\cdots\!89$$$$T^{12} -$$$$94\!\cdots\!20$$$$T^{13} -$$$$46\!\cdots\!97$$$$T^{14} +$$$$69\!\cdots\!70$$$$T^{15} -$$$$50\!\cdots\!19$$$$T^{16} +$$$$39\!\cdots\!60$$$$T^{17} +$$$$41\!\cdots\!87$$$$T^{18} -$$$$58\!\cdots\!90$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20} )( 1 + 1330 T + 856227 T^{2} - 75073180 T^{3} - 881302594171 T^{4} - 1103615185837290 T^{5} - 663467114633766617 T^{6} +$$$$12\!\cdots\!60$$$$T^{7} +$$$$77\!\cdots\!41$$$$T^{8} +$$$$91\!\cdots\!50$$$$T^{9} +$$$$50\!\cdots\!07$$$$T^{10} +$$$$83\!\cdots\!50$$$$T^{11} +$$$$64\!\cdots\!89$$$$T^{12} +$$$$94\!\cdots\!20$$$$T^{13} -$$$$46\!\cdots\!97$$$$T^{14} -$$$$69\!\cdots\!70$$$$T^{15} -$$$$50\!\cdots\!19$$$$T^{16} -$$$$39\!\cdots\!60$$$$T^{17} +$$$$41\!\cdots\!87$$$$T^{18} +$$$$58\!\cdots\!90$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20} )$$