Properties

Label 201.2.p.a
Level 201
Weight 2
Character orbit 201.p
Analytic conductor 1.605
Analytic rank 0
Dimension 20
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 201.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{66}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{33} + \zeta_{33}^{12} ) q^{3} -2 \zeta_{33}^{4} q^{4} + ( 3 - 3 \zeta_{33} + 3 \zeta_{33}^{3} - 3 \zeta_{33}^{4} + 6 \zeta_{33}^{6} - 3 \zeta_{33}^{7} + 2 \zeta_{33}^{9} - 3 \zeta_{33}^{10} + 3 \zeta_{33}^{11} - 3 \zeta_{33}^{13} + 3 \zeta_{33}^{14} - 3 \zeta_{33}^{16} + 4 \zeta_{33}^{17} - 3 \zeta_{33}^{19} ) q^{7} + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{33} + \zeta_{33}^{12} ) q^{3} -2 \zeta_{33}^{4} q^{4} + ( 3 - 3 \zeta_{33} + 3 \zeta_{33}^{3} - 3 \zeta_{33}^{4} + 6 \zeta_{33}^{6} - 3 \zeta_{33}^{7} + 2 \zeta_{33}^{9} - 3 \zeta_{33}^{10} + 3 \zeta_{33}^{11} - 3 \zeta_{33}^{13} + 3 \zeta_{33}^{14} - 3 \zeta_{33}^{16} + 4 \zeta_{33}^{17} - 3 \zeta_{33}^{19} ) q^{7} + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{9} + ( -4 \zeta_{33}^{5} - 2 \zeta_{33}^{16} ) q^{12} + ( \zeta_{33}^{3} - 4 \zeta_{33}^{7} - 3 \zeta_{33}^{14} - 3 \zeta_{33}^{18} ) q^{13} + 4 \zeta_{33}^{8} q^{16} + ( -5 + 5 \zeta_{33}^{2} - 5 \zeta_{33}^{3} + 5 \zeta_{33}^{5} - 5 \zeta_{33}^{6} + 2 \zeta_{33}^{8} - 5 \zeta_{33}^{9} + 3 \zeta_{33}^{10} - 5 \zeta_{33}^{12} + 5 \zeta_{33}^{13} - 5 \zeta_{33}^{15} + 5 \zeta_{33}^{16} - 5 \zeta_{33}^{18} + 7 \zeta_{33}^{19} ) q^{19} + ( 4 - 4 \zeta_{33}^{2} + 4 \zeta_{33}^{3} - 4 \zeta_{33}^{5} + 4 \zeta_{33}^{6} + 5 \zeta_{33}^{7} - 4 \zeta_{33}^{8} + 4 \zeta_{33}^{9} + \zeta_{33}^{10} + 4 \zeta_{33}^{12} - 4 \zeta_{33}^{13} + 4 \zeta_{33}^{15} - 4 \zeta_{33}^{16} + 8 \zeta_{33}^{18} - 4 \zeta_{33}^{19} ) q^{21} + ( 5 \zeta_{33}^{5} + 5 \zeta_{33}^{16} ) q^{25} + ( 3 \zeta_{33}^{3} + 6 \zeta_{33}^{14} ) q^{27} + ( 2 - 8 \zeta_{33}^{2} + 2 \zeta_{33}^{3} - 2 \zeta_{33}^{5} + 2 \zeta_{33}^{6} - 2 \zeta_{33}^{8} + 2 \zeta_{33}^{9} - 6 \zeta_{33}^{10} + 2 \zeta_{33}^{12} - 6 \zeta_{33}^{13} + 2 \zeta_{33}^{15} - 2 \zeta_{33}^{16} + 2 \zeta_{33}^{18} - 2 \zeta_{33}^{19} ) q^{28} + ( -6 + 6 \zeta_{33} - 12 \zeta_{33}^{3} + 6 \zeta_{33}^{4} - 6 \zeta_{33}^{6} + 6 \zeta_{33}^{7} - 5 \zeta_{33}^{9} + 6 \zeta_{33}^{10} - 6 \zeta_{33}^{11} + 6 \zeta_{33}^{13} - 7 \zeta_{33}^{14} + 6 \zeta_{33}^{16} - 6 \zeta_{33}^{17} + 6 \zeta_{33}^{19} ) q^{31} + ( -6 \zeta_{33}^{6} - 6 \zeta_{33}^{17} ) q^{36} + ( 3 \zeta_{33}^{5} - 4 \zeta_{33}^{6} - 4 \zeta_{33}^{16} + 3 \zeta_{33}^{17} ) q^{37} + ( 5 \zeta_{33}^{4} - 5 \zeta_{33}^{8} - 2 \zeta_{33}^{15} - 7 \zeta_{33}^{19} ) q^{39} + ( -7 \zeta_{33} + \zeta_{33}^{2} - 6 \zeta_{33}^{12} - 6 \zeta_{33}^{13} ) q^{43} + ( -4 + 4 \zeta_{33} - 4 \zeta_{33}^{3} + 4 \zeta_{33}^{4} - 4 \zeta_{33}^{6} + 4 \zeta_{33}^{7} + 4 \zeta_{33}^{9} + 4 \zeta_{33}^{10} - 4 \zeta_{33}^{11} + 4 \zeta_{33}^{13} - 4 \zeta_{33}^{14} + 4 \zeta_{33}^{16} - 4 \zeta_{33}^{17} + 4 \zeta_{33}^{19} ) q^{48} + ( -5 \zeta_{33} + 7 \zeta_{33}^{4} + 3 \zeta_{33}^{7} + 3 \zeta_{33}^{12} + 7 \zeta_{33}^{15} - 5 \zeta_{33}^{18} ) q^{49} + ( -6 - 2 \zeta_{33}^{7} + 2 \zeta_{33}^{11} + 6 \zeta_{33}^{18} ) q^{52} + ( -7 - \zeta_{33} + \zeta_{33}^{3} - \zeta_{33}^{4} + \zeta_{33}^{6} - \zeta_{33}^{7} - 7 \zeta_{33}^{9} - \zeta_{33}^{10} - 6 \zeta_{33}^{11} - \zeta_{33}^{13} + \zeta_{33}^{14} - \zeta_{33}^{16} + \zeta_{33}^{17} - \zeta_{33}^{19} ) q^{57} + ( -4 \zeta_{33}^{2} - 4 \zeta_{33}^{4} + 5 \zeta_{33}^{13} - 9 \zeta_{33}^{15} ) q^{61} + ( 3 + 6 \zeta_{33}^{8} + 9 \zeta_{33}^{11} + 9 \zeta_{33}^{19} ) q^{63} -8 \zeta_{33}^{12} q^{64} + ( -2 \zeta_{33}^{5} + 7 \zeta_{33}^{16} ) q^{67} + ( -1 + \zeta_{33}^{6} - 9 \zeta_{33}^{11} - 8 \zeta_{33}^{17} ) q^{73} + ( 5 \zeta_{33}^{6} + 10 \zeta_{33}^{17} ) q^{75} + ( 4 \zeta_{33} + 10 \zeta_{33}^{3} + 10 \zeta_{33}^{12} + 4 \zeta_{33}^{14} ) q^{76} + ( 7 + 3 \zeta_{33}^{2} - 3 \zeta_{33}^{3} + 3 \zeta_{33}^{5} - 3 \zeta_{33}^{6} + 3 \zeta_{33}^{8} - 3 \zeta_{33}^{9} + 10 \zeta_{33}^{10} + 7 \zeta_{33}^{11} - 3 \zeta_{33}^{12} + 3 \zeta_{33}^{13} - 3 \zeta_{33}^{15} + 3 \zeta_{33}^{16} - 3 \zeta_{33}^{18} + 3 \zeta_{33}^{19} ) q^{79} + 9 \zeta_{33}^{15} q^{81} + ( 8 - 8 \zeta_{33}^{3} - 2 \zeta_{33}^{11} - 10 \zeta_{33}^{14} ) q^{84} + ( 5 + 4 \zeta_{33} + 10 \zeta_{33}^{2} + 5 \zeta_{33}^{3} - 5 \zeta_{33}^{4} - 6 \zeta_{33}^{5} + 5 \zeta_{33}^{6} - 5 \zeta_{33}^{7} + 11 \zeta_{33}^{9} - 5 \zeta_{33}^{10} + 5 \zeta_{33}^{11} - \zeta_{33}^{12} - 4 \zeta_{33}^{13} + 5 \zeta_{33}^{14} - 16 \zeta_{33}^{16} + 5 \zeta_{33}^{17} - 5 \zeta_{33}^{19} ) q^{91} + ( -7 + 7 \zeta_{33}^{2} - 7 \zeta_{33}^{3} - 11 \zeta_{33}^{4} + 7 \zeta_{33}^{5} - 7 \zeta_{33}^{6} + 7 \zeta_{33}^{8} - 7 \zeta_{33}^{9} - 4 \zeta_{33}^{10} - 7 \zeta_{33}^{12} + 7 \zeta_{33}^{13} - 14 \zeta_{33}^{15} + 7 \zeta_{33}^{16} - 7 \zeta_{33}^{18} + 7 \zeta_{33}^{19} ) q^{93} + ( 11 + 11 \zeta_{33} - 11 \zeta_{33}^{2} + 11 \zeta_{33}^{3} - 11 \zeta_{33}^{5} + 11 \zeta_{33}^{6} - 11 \zeta_{33}^{8} + 11 \zeta_{33}^{9} - 8 \zeta_{33}^{10} + 19 \zeta_{33}^{12} - 11 \zeta_{33}^{13} + 11 \zeta_{33}^{15} - 11 \zeta_{33}^{16} + 11 \zeta_{33}^{18} - 11 \zeta_{33}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 2q^{4} - 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q - 2q^{4} - 6q^{7} + 6q^{9} - 6q^{12} - 3q^{13} + 4q^{16} - 8q^{19} + 6q^{21} + 10q^{25} - 12q^{28} + 15q^{31} + 6q^{36} + 10q^{37} - 3q^{39} - 12q^{48} - 5q^{49} - 154q^{52} - 75q^{57} + 15q^{61} - 15q^{63} + 16q^{64} + 5q^{67} + 60q^{73} - 32q^{76} + 134q^{79} - 18q^{81} + 186q^{84} - 12q^{91} - 15q^{93} + 9q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(-\zeta_{33}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.723734 0.690079i
0.580057 0.814576i
0.928368 0.371662i
−0.786053 + 0.618159i
−0.888835 0.458227i
−0.786053 0.618159i
0.0475819 + 0.998867i
0.235759 + 0.971812i
−0.327068 0.945001i
0.981929 0.189251i
−0.327068 + 0.945001i
0.723734 + 0.690079i
−0.995472 + 0.0950560i
0.580057 + 0.814576i
0.981929 + 0.189251i
−0.888835 + 0.458227i
0.235759 0.971812i
−0.995472 0.0950560i
0.928368 + 0.371662i
0.0475819 0.998867i
0 0.487975 1.66189i 1.99094 + 0.190112i 0 0 2.32668 + 4.51312i 0 −2.52376 1.62192i 0
11.1 0 1.57553 0.719520i 1.57211 1.23632i 0 0 −0.656000 + 3.40365i 0 1.96458 2.26725i 0
20.1 0 1.71442 + 0.246497i −0.0951638 + 1.99773i 0 0 −0.730437 + 0.177202i 0 2.87848 + 0.845198i 0
32.1 0 −1.71442 + 0.246497i 1.77767 + 0.916453i 0 0 −3.50336 + 3.67422i 0 2.87848 0.845198i 0
41.1 0 −0.936417 1.45709i 0.654136 1.89000i 0 0 −0.0717192 0.751078i 0 −1.24625 + 2.72890i 0
44.1 0 −1.71442 0.246497i 1.77767 0.916453i 0 0 −3.50336 3.67422i 0 2.87848 + 0.845198i 0
50.1 0 0.936417 + 1.45709i −1.96386 + 0.378502i 0 0 −4.31033 + 3.06937i 0 −1.24625 + 2.72890i 0
74.1 0 −0.487975 + 1.66189i −1.16011 + 1.62915i 0 0 −2.19700 0.104656i 0 −2.52376 1.62192i 0
80.1 0 −1.30900 1.13425i −0.471518 + 1.94362i 0 0 −0.816487 + 2.03949i 0 0.426945 + 2.96946i 0
95.1 0 1.30900 1.13425i −1.44747 + 1.38016i 0 0 2.75125 3.49850i 0 0.426945 2.96946i 0
98.1 0 −1.30900 + 1.13425i −0.471518 1.94362i 0 0 −0.816487 2.03949i 0 0.426945 2.96946i 0
101.1 0 0.487975 + 1.66189i 1.99094 0.190112i 0 0 2.32668 4.51312i 0 −2.52376 + 1.62192i 0
113.1 0 −1.57553 0.719520i −1.85674 + 0.743325i 0 0 4.20741 + 1.45620i 0 1.96458 + 2.26725i 0
128.1 0 1.57553 + 0.719520i 1.57211 + 1.23632i 0 0 −0.656000 3.40365i 0 1.96458 + 2.26725i 0
146.1 0 1.30900 + 1.13425i −1.44747 1.38016i 0 0 2.75125 + 3.49850i 0 0.426945 + 2.96946i 0
152.1 0 −0.936417 + 1.45709i 0.654136 + 1.89000i 0 0 −0.0717192 + 0.751078i 0 −1.24625 2.72890i 0
182.1 0 −0.487975 1.66189i −1.16011 1.62915i 0 0 −2.19700 + 0.104656i 0 −2.52376 + 1.62192i 0
185.1 0 −1.57553 + 0.719520i −1.85674 0.743325i 0 0 4.20741 1.45620i 0 1.96458 2.26725i 0
191.1 0 1.71442 0.246497i −0.0951638 1.99773i 0 0 −0.730437 0.177202i 0 2.87848 0.845198i 0
197.1 0 0.936417 1.45709i −1.96386 0.378502i 0 0 −4.31033 3.06937i 0 −1.24625 2.72890i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.1
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.h odd 66 1 inner
201.p even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.p.a 20
3.b odd 2 1 CM 201.2.p.a 20
67.h odd 66 1 inner 201.2.p.a 20
201.p even 66 1 inner 201.2.p.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.p.a 20 1.a even 1 1 trivial
201.2.p.a 20 3.b odd 2 1 CM
201.2.p.a 20 67.h odd 66 1 inner
201.2.p.a 20 201.p even 66 1 inner

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} - 8 T^{6} - 16 T^{8} + 64 T^{12} + 128 T^{14} - 512 T^{18} - 1024 T^{20} - 2048 T^{22} + 8192 T^{26} + 16384 T^{28} - 65536 T^{32} - 131072 T^{34} + 524288 T^{38} + 1048576 T^{40} \)
$3$ \( 1 - 3 T^{2} + 9 T^{4} - 27 T^{6} + 81 T^{8} - 243 T^{10} + 729 T^{12} - 2187 T^{14} + 6561 T^{16} - 19683 T^{18} + 59049 T^{20} \)
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} - 3125 T^{10} + 15625 T^{12} - 78125 T^{14} + 390625 T^{16} - 1953125 T^{18} + 9765625 T^{20} )^{2} \)
$7$ \( ( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} + 120 T^{5} - 83 T^{6} - 923 T^{7} - 342 T^{8} + 6119 T^{9} + 8513 T^{10} + 42833 T^{11} - 16758 T^{12} - 316589 T^{13} - 199283 T^{14} + 2016840 T^{15} + 3411821 T^{16} - 10706059 T^{17} - 34588806 T^{18} + 40353607 T^{19} + 282475249 T^{20} )( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 360 T^{5} + 757 T^{6} + 1265 T^{7} + 1026 T^{8} - 3725 T^{9} - 25807 T^{10} - 26075 T^{11} + 50274 T^{12} + 433895 T^{13} + 1817557 T^{14} + 6050520 T^{15} + 17529701 T^{16} + 45294865 T^{17} + 103766418 T^{18} + 201768035 T^{19} + 282475249 T^{20} ) \)
$11$ \( ( 1 - 11 T + 66 T^{2} - 363 T^{3} + 1815 T^{4} - 7986 T^{5} + 33275 T^{6} - 131769 T^{7} + 497794 T^{8} - 1771561 T^{9} + 5958887 T^{10} - 19487171 T^{11} + 60233074 T^{12} - 175384539 T^{13} + 487179275 T^{14} - 1286153286 T^{15} + 3215383215 T^{16} - 7073843073 T^{17} + 14147686146 T^{18} - 25937424601 T^{19} + 25937424601 T^{20} )( 1 + 11 T + 66 T^{2} + 363 T^{3} + 1815 T^{4} + 7986 T^{5} + 33275 T^{6} + 131769 T^{7} + 497794 T^{8} + 1771561 T^{9} + 5958887 T^{10} + 19487171 T^{11} + 60233074 T^{12} + 175384539 T^{13} + 487179275 T^{14} + 1286153286 T^{15} + 3215383215 T^{16} + 7073843073 T^{17} + 14147686146 T^{18} + 25937424601 T^{19} + 25937424601 T^{20} ) \)
$13$ \( ( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 453154 T^{11} - 4111263 T^{12} + 14113528 T^{13} + 25219363 T^{14} - 233914590 T^{15} + 139977461 T^{16} + 2760934748 T^{17} - 7341576489 T^{18} - 21208998746 T^{19} + 137858491849 T^{20} )( 1 + 5 T + 12 T^{2} - 5 T^{3} - 181 T^{4} - 840 T^{5} - 1847 T^{6} + 1685 T^{7} + 32436 T^{8} + 140275 T^{9} + 279707 T^{10} + 1823575 T^{11} + 5481684 T^{12} + 3701945 T^{13} - 52752167 T^{14} - 311886120 T^{15} - 873652429 T^{16} - 313742585 T^{17} + 9788768652 T^{18} + 53022496865 T^{19} + 137858491849 T^{20} ) \)
$17$ \( 1 - 17 T^{2} + 4913 T^{6} - 83521 T^{8} + 24137569 T^{12} - 410338673 T^{14} + 118587876497 T^{18} - 2015993900449 T^{20} + 34271896307633 T^{22} - 9904578032905937 T^{26} + 168377826559400929 T^{28} - 48661191875666868481 T^{32} + \)\(82\!\cdots\!77\)\( T^{34} - \)\(23\!\cdots\!53\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} + 1008 T^{5} - 4787 T^{6} - 23939 T^{7} + 67014 T^{8} + 521855 T^{9} - 751411 T^{10} + 9915245 T^{11} + 24192054 T^{12} - 164197601 T^{13} - 623846627 T^{14} + 2495907792 T^{15} + 14348993705 T^{16} - 33073254343 T^{17} - 305704134738 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20} )( 1 + 7 T + 30 T^{2} + 77 T^{3} - 31 T^{4} - 1680 T^{5} - 11171 T^{6} - 46277 T^{7} - 111690 T^{8} + 97433 T^{9} + 2804141 T^{10} + 1851227 T^{11} - 40320090 T^{12} - 317413943 T^{13} - 1455815891 T^{14} - 4159846320 T^{15} - 1458422311 T^{16} + 68828123903 T^{17} + 509506891230 T^{18} + 2258813884453 T^{19} + 6131066257801 T^{20} ) \)
$23$ \( 1 - 23 T^{2} + 12167 T^{6} - 279841 T^{8} + 148035889 T^{12} - 3404825447 T^{14} + 1801152661463 T^{18} - 41426511213649 T^{20} + 952809757913927 T^{22} - 504036361936467383 T^{26} + 11592836324538749809 T^{28} - \)\(61\!\cdots\!61\)\( T^{32} + \)\(14\!\cdots\!03\)\( T^{34} - \)\(74\!\cdots\!87\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( ( 1 + 29 T^{2} + 841 T^{4} )^{10} \)
$31$ \( ( 1 - 11 T + 90 T^{2} - 649 T^{3} + 4349 T^{4} - 27720 T^{5} + 170101 T^{6} - 1011791 T^{7} + 5856570 T^{8} - 33056749 T^{9} + 182070569 T^{10} - 1024759219 T^{11} + 5628163770 T^{12} - 30142265681 T^{13} + 157091845621 T^{14} - 793600065720 T^{15} + 3859753508669 T^{16} - 17855686558039 T^{17} + 76760193369690 T^{18} - 290835843767381 T^{19} + 819628286980801 T^{20} )( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 86731924 T^{11} - 938027295 T^{12} + 1063419536 T^{13} + 24825168001 T^{14} - 132266677620 T^{15} - 240513497551 T^{16} + 5062320996424 T^{17} - 12793365561615 T^{18} - 105758488642684 T^{19} + 819628286980801 T^{20} ) \)
$37$ \( ( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 9240 T^{5} + 6217 T^{6} + 273493 T^{7} - 3238452 T^{8} + 25503731 T^{9} - 160718317 T^{10} + 943638047 T^{11} - 4433440788 T^{12} + 13853240929 T^{13} + 11651658937 T^{14} - 640738162680 T^{15} + 6617008408811 T^{16} - 49079780477761 T^{17} + 295048274129364 T^{18} - 1429579137745847 T^{19} + 4808584372417849 T^{20} )( 1 + T - 36 T^{2} - 73 T^{3} + 1259 T^{4} + 3960 T^{5} - 42623 T^{6} - 189143 T^{7} + 1387908 T^{8} + 8386199 T^{9} - 42966397 T^{10} + 310289363 T^{11} + 1900046052 T^{12} - 9580660379 T^{13} - 79882364303 T^{14} + 274602069720 T^{15} + 3230249548931 T^{16} - 6930027030709 T^{17} - 126449260341156 T^{18} + 129961739795077 T^{19} + 4808584372417849 T^{20} ) \)
$41$ \( 1 + 41 T^{2} - 68921 T^{6} - 2825761 T^{8} + 4750104241 T^{12} + 194754273881 T^{14} - 327381934393961 T^{18} - 13422659310152401 T^{20} - 550329031716248441 T^{22} + \)\(92\!\cdots\!21\)\( T^{26} + \)\(37\!\cdots\!61\)\( T^{28} - \)\(63\!\cdots\!41\)\( T^{32} - \)\(26\!\cdots\!81\)\( T^{34} + \)\(43\!\cdots\!61\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 - 5 T - 18 T^{2} + 305 T^{3} - 751 T^{4} - 9360 T^{5} + 79093 T^{6} + 7015 T^{7} - 3436074 T^{8} + 16878725 T^{9} + 63357557 T^{10} + 725785175 T^{11} - 6353300826 T^{12} + 557741605 T^{13} + 270403227493 T^{14} - 1375999026480 T^{15} - 4747343649799 T^{16} + 82904676387635 T^{17} - 210387604996818 T^{18} - 2512963059684215 T^{19} + 21611482313284249 T^{20} )( 1 + 5 T - 18 T^{2} - 305 T^{3} - 751 T^{4} + 9360 T^{5} + 79093 T^{6} - 7015 T^{7} - 3436074 T^{8} - 16878725 T^{9} + 63357557 T^{10} - 725785175 T^{11} - 6353300826 T^{12} - 557741605 T^{13} + 270403227493 T^{14} + 1375999026480 T^{15} - 4747343649799 T^{16} - 82904676387635 T^{17} - 210387604996818 T^{18} + 2512963059684215 T^{19} + 21611482313284249 T^{20} ) \)
$47$ \( 1 - 47 T^{2} + 103823 T^{6} - 4879681 T^{8} + 10779215329 T^{12} - 506623120463 T^{14} + 1119130473102767 T^{18} - 52599132235830049 T^{20} + 2472159215084012303 T^{22} - \)\(54\!\cdots\!27\)\( T^{26} + \)\(25\!\cdots\!69\)\( T^{28} - \)\(56\!\cdots\!21\)\( T^{32} + \)\(26\!\cdots\!87\)\( T^{34} - \)\(58\!\cdots\!83\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( ( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} - 418195493 T^{10} + 22164361129 T^{12} - 1174711139837 T^{14} + 62259690411361 T^{16} - 3299763591802133 T^{18} + 174887470365513049 T^{20} )^{2} \)
$59$ \( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2} \)
$61$ \( ( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 14770930034 T^{11} - 97381156095 T^{12} + 462309453256 T^{13} - 532081823789 T^{14} - 20751731115570 T^{15} + 322981226869109 T^{16} - 3255881578117756 T^{17} + 25880487254632935 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20} )( 1 - T - 60 T^{2} + 121 T^{3} + 3539 T^{4} - 10920 T^{5} - 204959 T^{6} + 871079 T^{7} + 11631420 T^{8} - 64767239 T^{9} - 644749381 T^{10} - 3950801579 T^{11} + 43280513820 T^{12} + 197718382499 T^{13} - 2837829725519 T^{14} - 9222991606920 T^{15} + 182330604863579 T^{16} + 380271883158541 T^{17} - 11502438779836860 T^{18} - 11694146092834141 T^{19} + 713342911662882601 T^{20} ) \)
$67$ \( 1 - 5 T - 42 T^{2} + 545 T^{3} + 89 T^{4} - 36960 T^{5} + 178837 T^{6} + 1582135 T^{7} - 19892754 T^{8} - 6539275 T^{9} + 1365510893 T^{10} - 438131425 T^{11} - 89298572706 T^{12} + 475847669005 T^{13} + 3603766026277 T^{14} - 49900623954720 T^{15} + 8050796013041 T^{16} + 3303087824901035 T^{17} - 17054842457378922 T^{18} - 136032671981474735 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 71 T^{2} + 357911 T^{6} - 25411681 T^{8} + 128100283921 T^{12} - 9095120158391 T^{14} + 45848500718449031 T^{18} - 3255243551009881201 T^{20} + \)\(23\!\cdots\!71\)\( T^{22} - \)\(11\!\cdots\!11\)\( T^{26} + \)\(82\!\cdots\!81\)\( T^{28} - \)\(41\!\cdots\!21\)\( T^{32} + \)\(29\!\cdots\!91\)\( T^{34} - \)\(14\!\cdots\!31\)\( T^{38} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 695540350 T^{11} + 147779783289 T^{12} + 1528572278440 T^{13} + 4497798604303 T^{14} - 66607790283090 T^{15} - 994417200945019 T^{16} - 5081803318784620 T^{17} + 21774422481140187 T^{18} + 588715867082679130 T^{19} + 4297625829703557649 T^{20} ) \)
$79$ \( ( 1 - 13 T + 79 T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 21490023980 T^{11} - 113400661311 T^{12} + 2151314539664 T^{13} + 353394084913 T^{14} - 171367424973108 T^{15} + 657551567184305 T^{16} + 10907820304138312 T^{17} - 95577855024113343 T^{18} - 479406383930473276 T^{19} + 9468276082626847201 T^{20} ) \)
$83$ \( 1 - 83 T^{2} + 571787 T^{6} - 47458321 T^{8} + 326940373369 T^{12} - 27136050989627 T^{14} + 186940255267540403 T^{18} - 15516041187205853449 T^{20} + \)\(12\!\cdots\!67\)\( T^{22} - \)\(88\!\cdots\!63\)\( T^{26} + \)\(73\!\cdots\!29\)\( T^{28} - \)\(50\!\cdots\!81\)\( T^{32} + \)\(42\!\cdots\!23\)\( T^{34} - \)\(29\!\cdots\!47\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 + 89 T^{2} + 7921 T^{4} + 704969 T^{6} + 62742241 T^{8} + 5584059449 T^{10} + 496981290961 T^{12} + 44231334895529 T^{14} + 3936588805702081 T^{16} + 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2} \)
$97$ \( ( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 120069456262 T^{11} + 797567744457 T^{12} + 480788835016 T^{13} - 84095114902553 T^{14} + 1130695091639190 T^{15} - 7672505137401019 T^{16} - 2262351965387164 T^{17} + 775905925843319139 T^{18} - 10643234821163913038 T^{19} + 73742412689492826049 T^{20} )( 1 + 5 T - 72 T^{2} - 845 T^{3} + 2759 T^{4} + 95760 T^{5} + 211177 T^{6} - 8232835 T^{7} - 61648344 T^{8} + 490343275 T^{9} + 8431605743 T^{10} + 47563297675 T^{11} - 580049268696 T^{12} - 7513886217955 T^{13} + 18695347973737 T^{14} + 822323703010320 T^{15} + 2298169761599111 T^{16} - 68274550384005485 T^{17} - 564295218795141192 T^{18} + 3801155293272826085 T^{19} + 73742412689492826049 T^{20} ) \)
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