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

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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:

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

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