Properties

Label 201.3.o.a
Level 201
Weight 3
Character orbit 201.o
Analytic conductor 5.477
Analytic rank 0
Dimension 20
CM discriminant -3
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.47685331364\)
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.

\(f(q)\) \(=\) \( q + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{3} + 4 \zeta_{33}^{8} q^{4} + ( -5 \zeta_{33} + 3 \zeta_{33}^{7} + 3 \zeta_{33}^{12} - 5 \zeta_{33}^{18} ) q^{7} + 9 \zeta_{33}^{15} q^{9} +O(q^{10})\) \( q + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{3} + 4 \zeta_{33}^{8} q^{4} + ( -5 \zeta_{33} + 3 \zeta_{33}^{7} + 3 \zeta_{33}^{12} - 5 \zeta_{33}^{18} ) q^{7} + 9 \zeta_{33}^{15} q^{9} + ( -12 + 12 \zeta_{33}^{2} - 12 \zeta_{33}^{3} + 12 \zeta_{33}^{5} - 12 \zeta_{33}^{6} + 12 \zeta_{33}^{8} - 12 \zeta_{33}^{9} + 12 \zeta_{33}^{10} - 12 \zeta_{33}^{12} + 12 \zeta_{33}^{13} - 12 \zeta_{33}^{15} + 12 \zeta_{33}^{16} - 12 \zeta_{33}^{18} + 12 \zeta_{33}^{19} ) q^{12} + ( -15 \zeta_{33}^{3} - 8 \zeta_{33}^{6} - 8 \zeta_{33}^{14} - 15 \zeta_{33}^{17} ) q^{13} + 16 \zeta_{33}^{16} q^{16} + ( 21 - 21 \zeta_{33} + 21 \zeta_{33}^{3} - 21 \zeta_{33}^{4} + 16 \zeta_{33}^{5} + 21 \zeta_{33}^{6} - 21 \zeta_{33}^{7} + 16 \zeta_{33}^{9} - 21 \zeta_{33}^{10} + 21 \zeta_{33}^{11} - 21 \zeta_{33}^{13} + 21 \zeta_{33}^{14} + 21 \zeta_{33}^{17} - 21 \zeta_{33}^{19} ) q^{19} + ( -9 + 9 \zeta_{33} - 33 \zeta_{33}^{3} + 9 \zeta_{33}^{4} - 9 \zeta_{33}^{6} + 9 \zeta_{33}^{7} + 15 \zeta_{33}^{9} + 9 \zeta_{33}^{10} - 9 \zeta_{33}^{11} + 9 \zeta_{33}^{13} - 24 \zeta_{33}^{14} + 9 \zeta_{33}^{16} - 9 \zeta_{33}^{17} + 9 \zeta_{33}^{19} ) q^{21} + ( -25 + 25 \zeta_{33}^{2} - 25 \zeta_{33}^{3} + 25 \zeta_{33}^{5} - 25 \zeta_{33}^{6} + 25 \zeta_{33}^{8} - 25 \zeta_{33}^{9} - 25 \zeta_{33}^{12} + 25 \zeta_{33}^{13} - 25 \zeta_{33}^{15} + 25 \zeta_{33}^{16} - 25 \zeta_{33}^{18} + 25 \zeta_{33}^{19} ) q^{25} -27 \zeta_{33}^{6} q^{27} + ( -12 + 12 \zeta_{33} - 12 \zeta_{33}^{3} + 32 \zeta_{33}^{4} - 12 \zeta_{33}^{6} + 12 \zeta_{33}^{7} - 32 \zeta_{33}^{9} + 12 \zeta_{33}^{10} - 12 \zeta_{33}^{11} + 12 \zeta_{33}^{13} - 12 \zeta_{33}^{14} + 32 \zeta_{33}^{15} + 12 \zeta_{33}^{16} - 12 \zeta_{33}^{17} + 12 \zeta_{33}^{19} ) q^{28} + ( 35 \zeta_{33}^{6} + 24 \zeta_{33}^{7} + 11 \zeta_{33}^{17} - 11 \zeta_{33}^{18} ) q^{31} + ( -36 \zeta_{33} - 36 \zeta_{33}^{12} ) q^{36} + ( 40 + 33 \zeta_{33} - 40 \zeta_{33}^{2} + 40 \zeta_{33}^{3} - 40 \zeta_{33}^{5} + 40 \zeta_{33}^{6} - 40 \zeta_{33}^{8} + 40 \zeta_{33}^{9} - 7 \zeta_{33}^{10} + 80 \zeta_{33}^{12} - 40 \zeta_{33}^{13} + 40 \zeta_{33}^{15} - 40 \zeta_{33}^{16} + 40 \zeta_{33}^{18} - 40 \zeta_{33}^{19} ) q^{37} + ( -21 \zeta_{33}^{5} + 21 \zeta_{33}^{8} - 45 \zeta_{33}^{16} - 24 \zeta_{33}^{19} ) q^{39} + ( 13 \zeta_{33}^{2} - 35 \zeta_{33}^{4} + 48 \zeta_{33}^{13} - 48 \zeta_{33}^{15} ) q^{43} -48 \zeta_{33}^{7} q^{48} + ( 16 \zeta_{33}^{2} + 55 \zeta_{33}^{3} - 39 \zeta_{33}^{13} + 39 \zeta_{33}^{14} + 49 \zeta_{33}^{19} ) q^{49} + ( 32 + 60 \zeta_{33}^{3} - 28 \zeta_{33}^{11} + 28 \zeta_{33}^{14} ) q^{52} + ( 15 - 15 \zeta_{33}^{7} + 63 \zeta_{33}^{11} + 48 \zeta_{33}^{18} ) q^{57} + ( -9 \zeta_{33}^{4} - 65 \zeta_{33}^{8} - 65 \zeta_{33}^{15} - 9 \zeta_{33}^{19} ) q^{61} + ( -72 - 27 \zeta_{33}^{5} - 27 \zeta_{33}^{11} - 72 \zeta_{33}^{16} ) q^{63} + ( -64 \zeta_{33}^{2} - 64 \zeta_{33}^{13} ) q^{64} + ( 77 - 77 \zeta_{33}^{2} + 77 \zeta_{33}^{3} - 77 \zeta_{33}^{5} + 77 \zeta_{33}^{6} - 77 \zeta_{33}^{8} + 77 \zeta_{33}^{9} - 45 \zeta_{33}^{10} + 77 \zeta_{33}^{12} - 77 \zeta_{33}^{13} + 77 \zeta_{33}^{15} - 77 \zeta_{33}^{16} + 77 \zeta_{33}^{18} - 77 \zeta_{33}^{19} ) q^{67} + ( -80 + 80 \zeta_{33} - 63 \zeta_{33}^{11} + 17 \zeta_{33}^{12} ) q^{73} -75 \zeta_{33}^{12} q^{75} + ( -84 \zeta_{33}^{2} + 84 \zeta_{33}^{6} - 20 \zeta_{33}^{13} + 64 \zeta_{33}^{17} ) q^{76} + ( 11 + 40 \zeta_{33} - 40 \zeta_{33}^{3} + 40 \zeta_{33}^{4} - 40 \zeta_{33}^{6} + 40 \zeta_{33}^{7} - 91 \zeta_{33}^{9} + 40 \zeta_{33}^{10} + 51 \zeta_{33}^{11} + 40 \zeta_{33}^{13} - 40 \zeta_{33}^{14} + 40 \zeta_{33}^{16} - 40 \zeta_{33}^{17} + 40 \zeta_{33}^{19} ) q^{79} + ( -81 \zeta_{33}^{8} - 81 \zeta_{33}^{19} ) q^{81} + ( 60 - 36 \zeta_{33}^{6} - 36 \zeta_{33}^{11} + 60 \zeta_{33}^{17} ) q^{84} + ( -11 + 91 \zeta_{33}^{2} - 11 \zeta_{33}^{3} + 99 \zeta_{33}^{4} + 11 \zeta_{33}^{5} - 11 \zeta_{33}^{6} + 85 \zeta_{33}^{7} + 11 \zeta_{33}^{8} - 11 \zeta_{33}^{9} - 85 \zeta_{33}^{10} - 11 \zeta_{33}^{12} - 8 \zeta_{33}^{13} + 8 \zeta_{33}^{15} + 11 \zeta_{33}^{16} + 85 \zeta_{33}^{18} + 11 \zeta_{33}^{19} ) q^{91} + ( -72 + 72 \zeta_{33} - 72 \zeta_{33}^{3} + 72 \zeta_{33}^{4} - 72 \zeta_{33}^{6} + 72 \zeta_{33}^{7} + 72 \zeta_{33}^{8} + 33 \zeta_{33}^{9} + 72 \zeta_{33}^{10} - 72 \zeta_{33}^{11} + 72 \zeta_{33}^{13} - 72 \zeta_{33}^{14} + 72 \zeta_{33}^{16} - 72 \zeta_{33}^{17} + 177 \zeta_{33}^{19} ) q^{93} + ( 112 - 112 \zeta_{33} + 57 \zeta_{33}^{2} + 112 \zeta_{33}^{3} - 112 \zeta_{33}^{4} + 112 \zeta_{33}^{6} - 112 \zeta_{33}^{7} + 57 \zeta_{33}^{9} - 112 \zeta_{33}^{10} + 112 \zeta_{33}^{11} + 112 \zeta_{33}^{14} - 112 \zeta_{33}^{16} + 112 \zeta_{33}^{17} - 112 \zeta_{33}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{3} + 4q^{4} + 2q^{7} - 18q^{9} + O(q^{10}) \) \( 20q + 6q^{3} + 4q^{4} + 2q^{7} - 18q^{9} - 12q^{12} + 23q^{13} + 16q^{16} + 26q^{19} - 6q^{21} - 50q^{25} + 54q^{27} + 8q^{28} - 13q^{31} + 36q^{36} + 26q^{37} - 69q^{39} + 122q^{43} - 48q^{48} - 45q^{49} + 828q^{52} - 441q^{57} + 47q^{61} - 1269q^{63} - 128q^{64} + 109q^{67} - 924q^{73} + 150q^{75} - 208q^{76} + 252q^{79} - 162q^{81} + 1692q^{84} - 92q^{91} + 39q^{93} + 167q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(\zeta_{33}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.723734 + 0.690079i
−0.327068 + 0.945001i
0.235759 0.971812i
−0.327068 0.945001i
0.981929 0.189251i
−0.888835 + 0.458227i
0.928368 0.371662i
0.723734 0.690079i
0.981929 + 0.189251i
0.0475819 0.998867i
−0.786053 + 0.618159i
0.235759 + 0.971812i
−0.995472 0.0950560i
−0.888835 0.458227i
0.0475819 + 0.998867i
0.928368 + 0.371662i
0.580057 + 0.814576i
−0.995472 + 0.0950560i
0.580057 0.814576i
−0.786053 0.618159i
0 −2.52376 + 1.62192i 3.92771 0.757005i 0 0 −6.83405 9.59709i 0 3.73874 8.18669i 0
23.1 0 0.426945 2.96946i −3.55534 + 1.83291i 0 0 6.63942 6.33068i 0 −8.63544 2.53559i 0
26.1 0 −2.52376 + 1.62192i −1.30827 + 3.78000i 0 0 −9.12076 + 0.870927i 0 3.73874 8.18669i 0
35.1 0 0.426945 + 2.96946i −3.55534 1.83291i 0 0 6.63942 + 6.33068i 0 −8.63544 + 2.53559i 0
47.1 0 0.426945 2.96946i 0.190328 3.99547i 0 0 −1.36948 5.64509i 0 −8.63544 2.53559i 0
56.1 0 −1.24625 2.72890i −3.14421 + 2.47264i 0 0 13.1880 + 2.54178i 0 −5.89375 + 6.80175i 0
65.1 0 2.87848 + 0.845198i −3.98189 0.380224i 0 0 −11.9416 + 6.15630i 0 7.57128 + 4.86577i 0
71.1 0 −2.52376 1.62192i 3.92771 + 0.757005i 0 0 −6.83405 + 9.59709i 0 3.73874 + 8.18669i 0
77.1 0 0.426945 + 2.96946i 0.190328 + 3.99547i 0 0 −1.36948 + 5.64509i 0 −8.63544 + 2.53559i 0
83.1 0 −1.24625 2.72890i 3.71347 + 1.48665i 0 0 4.57895 + 13.2300i 0 −5.89375 + 6.80175i 0
86.1 0 2.87848 0.845198i 2.32023 + 3.25830i 0 0 −0.560201 11.7601i 0 7.57128 4.86577i 0
116.1 0 −2.52376 1.62192i −1.30827 3.78000i 0 0 −9.12076 0.870927i 0 3.73874 + 8.18669i 0
122.1 0 1.96458 2.26725i 2.89494 + 2.76032i 0 0 4.57702 3.59941i 0 −1.28083 8.90839i 0
140.1 0 −1.24625 + 2.72890i −3.14421 2.47264i 0 0 13.1880 2.54178i 0 −5.89375 6.80175i 0
155.1 0 −1.24625 + 2.72890i 3.71347 1.48665i 0 0 4.57895 13.2300i 0 −5.89375 6.80175i 0
167.1 0 2.87848 0.845198i −3.98189 + 0.380224i 0 0 −11.9416 6.15630i 0 7.57128 4.86577i 0
170.1 0 1.96458 + 2.26725i 0.943036 + 3.88725i 0 0 1.84264 0.737681i 0 −1.28083 + 8.90839i 0
173.1 0 1.96458 + 2.26725i 2.89494 2.76032i 0 0 4.57702 + 3.59941i 0 −1.28083 + 8.90839i 0
188.1 0 1.96458 2.26725i 0.943036 3.88725i 0 0 1.84264 + 0.737681i 0 −1.28083 8.90839i 0
194.1 0 2.87848 + 0.845198i 2.32023 3.25830i 0 0 −0.560201 + 11.7601i 0 7.57128 + 4.86577i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 194.1
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.g even 33 1 inner
201.o odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.o.a 20
3.b odd 2 1 CM 201.3.o.a 20
67.g even 33 1 inner 201.3.o.a 20
201.o odd 66 1 inner 201.3.o.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.o.a 20 1.a even 1 1 trivial
201.3.o.a 20 3.b odd 2 1 CM
201.3.o.a 20 67.g even 33 1 inner
201.3.o.a 20 201.o odd 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(201, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 8 T^{3} - 16 T^{4} + 64 T^{6} - 128 T^{7} + 512 T^{9} - 1024 T^{10} + 2048 T^{11} - 8192 T^{13} + 16384 T^{14} - 65536 T^{16} + 131072 T^{17} - 524288 T^{19} + 1048576 T^{20} )( 1 + 2 T - 8 T^{3} - 16 T^{4} + 64 T^{6} + 128 T^{7} - 512 T^{9} - 1024 T^{10} - 2048 T^{11} + 8192 T^{13} + 16384 T^{14} - 65536 T^{16} - 131072 T^{17} + 524288 T^{19} + 1048576 T^{20} ) \)
$3$ \( ( 1 - 3 T + 9 T^{2} - 27 T^{3} + 81 T^{4} - 243 T^{5} + 729 T^{6} - 2187 T^{7} + 6561 T^{8} - 19683 T^{9} + 59049 T^{10} )^{2} \)
$5$ \( ( 1 - 5 T + 25 T^{2} - 125 T^{3} + 625 T^{4} - 3125 T^{5} + 15625 T^{6} - 78125 T^{7} + 390625 T^{8} - 1953125 T^{9} + 9765625 T^{10} )^{2}( 1 + 5 T + 25 T^{2} + 125 T^{3} + 625 T^{4} + 3125 T^{5} + 15625 T^{6} + 78125 T^{7} + 390625 T^{8} + 1953125 T^{9} + 9765625 T^{10} )^{2} \)
$7$ \( ( 1 - 13 T + 120 T^{2} - 923 T^{3} + 6119 T^{4} - 34320 T^{5} + 146329 T^{6} - 220597 T^{7} - 4302360 T^{8} + 66739933 T^{9} - 656803489 T^{10} + 3270256717 T^{11} - 10329966360 T^{12} - 25953016453 T^{13} + 843557565529 T^{14} - 9694550545680 T^{15} + 84694836382919 T^{16} - 625999896239627 T^{17} + 3987951668352120 T^{18} - 21169376772835837 T^{19} + 79792266297612001 T^{20} )( 1 + 11 T + 72 T^{2} + 253 T^{3} - 745 T^{4} - 20592 T^{5} - 190007 T^{6} - 1081069 T^{7} - 2581416 T^{8} + 24576805 T^{9} + 396834239 T^{10} + 1204263445 T^{11} - 6197979816 T^{12} - 127186686781 T^{13} - 1095352543607 T^{14} - 5816730327408 T^{15} - 10311758964745 T^{16} + 171590437430797 T^{17} + 2392771001011272 T^{18} + 17912549577014939 T^{19} + 79792266297612001 T^{20} ) \)
$11$ \( ( 1 - 11 T + 1331 T^{3} - 14641 T^{4} + 1771561 T^{6} - 19487171 T^{7} + 2357947691 T^{9} - 25937424601 T^{10} + 285311670611 T^{11} - 34522712143931 T^{13} + 379749833583241 T^{14} - 45949729863572161 T^{16} + 505447028499293771 T^{17} - 61159090448414546291 T^{19} + \)\(67\!\cdots\!01\)\( T^{20} )( 1 + 11 T - 1331 T^{3} - 14641 T^{4} + 1771561 T^{6} + 19487171 T^{7} - 2357947691 T^{9} - 25937424601 T^{10} - 285311670611 T^{11} + 34522712143931 T^{13} + 379749833583241 T^{14} - 45949729863572161 T^{16} - 505447028499293771 T^{17} + 61159090448414546291 T^{19} + \)\(67\!\cdots\!01\)\( T^{20} ) \)
$13$ \( ( 1 - 22 T + 315 T^{2} - 3212 T^{3} + 17429 T^{4} + 159390 T^{5} - 6452081 T^{6} + 115008872 T^{7} - 1439793495 T^{8} + 12238957522 T^{9} - 25931964829 T^{10} + 2068383821218 T^{11} - 41121942010695 T^{12} + 555125858449448 T^{13} - 5263160686080401 T^{14} + 21973265015812110 T^{15} + 406062325599721349 T^{16} - 12646852950866116268 T^{17} + \)\(20\!\cdots\!15\)\( T^{18} - \)\(24\!\cdots\!38\)\( T^{19} + \)\(19\!\cdots\!01\)\( T^{20} )( 1 - T - 168 T^{2} + 337 T^{3} + 28055 T^{4} - 85008 T^{5} - 4656287 T^{6} + 19022639 T^{7} + 767889864 T^{8} - 3982715855 T^{9} - 125790671161 T^{10} - 673078979495 T^{11} + 21931702405704 T^{12} + 91818645128951 T^{13} - 3798276351692927 T^{14} - 11719074675099792 T^{15} + 653627778111204455 T^{16} + 1326895841980660393 T^{17} - \)\(11\!\cdots\!88\)\( T^{18} - \)\(11\!\cdots\!29\)\( T^{19} + \)\(19\!\cdots\!01\)\( T^{20} ) \)
$17$ \( ( 1 - 17 T + 4913 T^{3} - 83521 T^{4} + 24137569 T^{6} - 410338673 T^{7} + 118587876497 T^{9} - 2015993900449 T^{10} + 34271896307633 T^{11} - 9904578032905937 T^{13} + 168377826559400929 T^{14} - 48661191875666868481 T^{16} + \)\(82\!\cdots\!77\)\( T^{17} - \)\(23\!\cdots\!53\)\( T^{19} + \)\(40\!\cdots\!01\)\( T^{20} )( 1 + 17 T - 4913 T^{3} - 83521 T^{4} + 24137569 T^{6} + 410338673 T^{7} - 118587876497 T^{9} - 2015993900449 T^{10} - 34271896307633 T^{11} + 9904578032905937 T^{13} + 168377826559400929 T^{14} - 48661191875666868481 T^{16} - \)\(82\!\cdots\!77\)\( T^{17} + \)\(23\!\cdots\!53\)\( T^{19} + \)\(40\!\cdots\!01\)\( T^{20} ) \)
$19$ \( ( 1 - 37 T + 1008 T^{2} - 23939 T^{3} + 521855 T^{4} - 10666656 T^{5} + 206276617 T^{6} - 3781572013 T^{7} + 65452305744 T^{8} - 1056587815835 T^{9} + 15465466812311 T^{10} - 381428201516435 T^{11} + 8529809936863824 T^{12} - 177907386916528453 T^{13} + 3503311928703712297 T^{14} - 65397974685170583456 T^{15} + \)\(11\!\cdots\!55\)\( T^{16} - \)\(19\!\cdots\!19\)\( T^{17} + \)\(29\!\cdots\!48\)\( T^{18} - \)\(38\!\cdots\!17\)\( T^{19} + \)\(37\!\cdots\!01\)\( T^{20} )( 1 + 11 T - 240 T^{2} - 6611 T^{3} + 13919 T^{4} + 2539680 T^{5} + 22911721 T^{6} - 664795549 T^{7} - 15583882320 T^{8} + 68568487669 T^{9} + 6380034881879 T^{10} + 24753224048509 T^{11} - 2030907127824720 T^{12} - 31275892287583669 T^{13} + 389122657981303561 T^{14} + 15570946353612043680 T^{15} + 30807130358481894959 T^{16} - \)\(52\!\cdots\!31\)\( T^{17} - \)\(69\!\cdots\!40\)\( T^{18} + \)\(11\!\cdots\!51\)\( T^{19} + \)\(37\!\cdots\!01\)\( T^{20} ) \)
$23$ \( ( 1 - 23 T + 12167 T^{3} - 279841 T^{4} + 148035889 T^{6} - 3404825447 T^{7} + 1801152661463 T^{9} - 41426511213649 T^{10} + 952809757913927 T^{11} - 504036361936467383 T^{13} + 11592836324538749809 T^{14} - \)\(61\!\cdots\!61\)\( T^{16} + \)\(14\!\cdots\!03\)\( T^{17} - \)\(74\!\cdots\!87\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} )( 1 + 23 T - 12167 T^{3} - 279841 T^{4} + 148035889 T^{6} + 3404825447 T^{7} - 1801152661463 T^{9} - 41426511213649 T^{10} - 952809757913927 T^{11} + 504036361936467383 T^{13} + 11592836324538749809 T^{14} - \)\(61\!\cdots\!61\)\( T^{16} - \)\(14\!\cdots\!03\)\( T^{17} + \)\(74\!\cdots\!87\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} ) \)
$29$ \( ( 1 - 29 T + 841 T^{2} )^{10}( 1 + 29 T + 841 T^{2} )^{10} \)
$31$ \( ( 1 - 46 T + 1155 T^{2} - 8924 T^{3} - 699451 T^{4} + 40750710 T^{5} - 1202360249 T^{6} + 16147139144 T^{7} + 412699798665 T^{8} - 34501591455974 T^{9} + 1190468700457739 T^{10} - 33156029389191014 T^{11} + 381136930762899465 T^{12} + 14330645427919189064 T^{13} - \)\(10\!\cdots\!09\)\( T^{14} + \)\(33\!\cdots\!10\)\( T^{15} - \)\(55\!\cdots\!11\)\( T^{16} - \)\(67\!\cdots\!04\)\( T^{17} + \)\(84\!\cdots\!55\)\( T^{18} - \)\(32\!\cdots\!86\)\( T^{19} + \)\(67\!\cdots\!01\)\( T^{20} )( 1 + 59 T + 2520 T^{2} + 91981 T^{3} + 3005159 T^{4} + 88910640 T^{5} + 2357769961 T^{6} + 53665302659 T^{7} + 900435924360 T^{8} + 1553363681941 T^{9} - 773670466075441 T^{10} + 1492782498345301 T^{11} + 831571485300871560 T^{12} + 47628153651841587779 T^{13} + \)\(20\!\cdots\!01\)\( T^{14} + \)\(72\!\cdots\!40\)\( T^{15} + \)\(23\!\cdots\!99\)\( T^{16} + \)\(69\!\cdots\!01\)\( T^{17} + \)\(18\!\cdots\!20\)\( T^{18} + \)\(41\!\cdots\!19\)\( T^{19} + \)\(67\!\cdots\!01\)\( T^{20} ) \)
$37$ \( ( 1 - 73 T + 3960 T^{2} - 189143 T^{3} + 8386199 T^{4} - 353255760 T^{5} + 14306964049 T^{6} - 560801240137 T^{7} + 21352256746920 T^{8} - 790977844777607 T^{9} + 28510143182231831 T^{10} - 1082848669500543983 T^{11} + 40017566857064334120 T^{12} - \)\(14\!\cdots\!33\)\( T^{13} + \)\(50\!\cdots\!29\)\( T^{14} - \)\(16\!\cdots\!40\)\( T^{15} + \)\(55\!\cdots\!19\)\( T^{16} - \)\(17\!\cdots\!27\)\( T^{17} + \)\(48\!\cdots\!60\)\( T^{18} - \)\(12\!\cdots\!17\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} )( 1 + 47 T + 840 T^{2} - 24863 T^{3} - 2318521 T^{4} - 74933040 T^{5} - 347797631 T^{6} + 86236843103 T^{7} + 4529266582680 T^{8} + 94817291177953 T^{9} - 1744153266325129 T^{10} + 129804871622617657 T^{11} + 8488574787862131480 T^{12} + \)\(22\!\cdots\!27\)\( T^{13} - \)\(12\!\cdots\!51\)\( T^{14} - \)\(36\!\cdots\!60\)\( T^{15} - \)\(15\!\cdots\!01\)\( T^{16} - \)\(22\!\cdots\!07\)\( T^{17} + \)\(10\!\cdots\!40\)\( T^{18} + \)\(79\!\cdots\!63\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} ) \)
$41$ \( ( 1 - 41 T + 68921 T^{3} - 2825761 T^{4} + 4750104241 T^{6} - 194754273881 T^{7} + 327381934393961 T^{9} - 13422659310152401 T^{10} + 550329031716248441 T^{11} - \)\(92\!\cdots\!21\)\( T^{13} + \)\(37\!\cdots\!61\)\( T^{14} - \)\(63\!\cdots\!41\)\( T^{16} + \)\(26\!\cdots\!81\)\( T^{17} - \)\(43\!\cdots\!61\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} )( 1 + 41 T - 68921 T^{3} - 2825761 T^{4} + 4750104241 T^{6} + 194754273881 T^{7} - 327381934393961 T^{9} - 13422659310152401 T^{10} - 550329031716248441 T^{11} + \)\(92\!\cdots\!21\)\( T^{13} + \)\(37\!\cdots\!61\)\( T^{14} - \)\(63\!\cdots\!41\)\( T^{16} - \)\(26\!\cdots\!81\)\( T^{17} + \)\(43\!\cdots\!61\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} ) \)
$43$ \( ( 1 - 61 T + 1872 T^{2} - 1403 T^{3} - 3375745 T^{4} + 208514592 T^{5} - 6477637607 T^{6} + 9592413419 T^{7} + 11392014716784 T^{8} - 712649270135555 T^{9} + 22407770266935239 T^{10} - 1317688500480641195 T^{11} + 38947031305755855984 T^{12} + 60637127737598354531 T^{13} - \)\(75\!\cdots\!07\)\( T^{14} + \)\(45\!\cdots\!08\)\( T^{15} - \)\(13\!\cdots\!45\)\( T^{16} - \)\(10\!\cdots\!47\)\( T^{17} + \)\(25\!\cdots\!72\)\( T^{18} - \)\(15\!\cdots\!89\)\( T^{19} + \)\(46\!\cdots\!01\)\( T^{20} )^{2} \)
$47$ \( ( 1 - 47 T + 103823 T^{3} - 4879681 T^{4} + 10779215329 T^{6} - 506623120463 T^{7} + 1119130473102767 T^{9} - 52599132235830049 T^{10} + 2472159215084012303 T^{11} - \)\(54\!\cdots\!27\)\( T^{13} + \)\(25\!\cdots\!69\)\( T^{14} - \)\(56\!\cdots\!21\)\( T^{16} + \)\(26\!\cdots\!87\)\( T^{17} - \)\(58\!\cdots\!83\)\( T^{19} + \)\(27\!\cdots\!01\)\( T^{20} )( 1 + 47 T - 103823 T^{3} - 4879681 T^{4} + 10779215329 T^{6} + 506623120463 T^{7} - 1119130473102767 T^{9} - 52599132235830049 T^{10} - 2472159215084012303 T^{11} + \)\(54\!\cdots\!27\)\( T^{13} + \)\(25\!\cdots\!69\)\( T^{14} - \)\(56\!\cdots\!21\)\( T^{16} - \)\(26\!\cdots\!87\)\( T^{17} + \)\(58\!\cdots\!83\)\( T^{19} + \)\(27\!\cdots\!01\)\( T^{20} ) \)
$53$ \( ( 1 - 53 T + 2809 T^{2} - 148877 T^{3} + 7890481 T^{4} - 418195493 T^{5} + 22164361129 T^{6} - 1174711139837 T^{7} + 62259690411361 T^{8} - 3299763591802133 T^{9} + 174887470365513049 T^{10} )^{2}( 1 + 53 T + 2809 T^{2} + 148877 T^{3} + 7890481 T^{4} + 418195493 T^{5} + 22164361129 T^{6} + 1174711139837 T^{7} + 62259690411361 T^{8} + 3299763591802133 T^{9} + 174887470365513049 T^{10} )^{2} \)
$59$ \( ( 1 - 59 T + 3481 T^{2} - 205379 T^{3} + 12117361 T^{4} - 714924299 T^{5} + 42180533641 T^{6} - 2488651484819 T^{7} + 146830437604321 T^{8} - 8662995818654939 T^{9} + 511116753300641401 T^{10} )^{2}( 1 + 59 T + 3481 T^{2} + 205379 T^{3} + 12117361 T^{4} + 714924299 T^{5} + 42180533641 T^{6} + 2488651484819 T^{7} + 146830437604321 T^{8} + 8662995818654939 T^{9} + 511116753300641401 T^{10} )^{2} \)
$61$ \( ( 1 - 121 T + 10920 T^{2} - 871079 T^{3} + 64767239 T^{4} - 4595550960 T^{5} + 315062769841 T^{6} - 21022550028601 T^{7} + 1371379986882360 T^{8} - 87712069756341239 T^{9} + 5510255509328028359 T^{10} - \)\(32\!\cdots\!19\)\( T^{11} + \)\(18\!\cdots\!60\)\( T^{12} - \)\(10\!\cdots\!61\)\( T^{13} + \)\(60\!\cdots\!21\)\( T^{14} - \)\(32\!\cdots\!60\)\( T^{15} + \)\(17\!\cdots\!19\)\( T^{16} - \)\(86\!\cdots\!39\)\( T^{17} + \)\(40\!\cdots\!20\)\( T^{18} - \)\(16\!\cdots\!01\)\( T^{19} + \)\(50\!\cdots\!01\)\( T^{20} )( 1 + 74 T + 1755 T^{2} - 145484 T^{3} - 17296171 T^{4} - 738570690 T^{5} + 9704821231 T^{6} + 3466378308584 T^{7} + 220400355034665 T^{8} + 3411232586324146 T^{9} - 567678509696001661 T^{10} + 12693196453712147266 T^{11} + \)\(30\!\cdots\!65\)\( T^{12} + \)\(17\!\cdots\!24\)\( T^{13} + \)\(18\!\cdots\!11\)\( T^{14} - \)\(52\!\cdots\!90\)\( T^{15} - \)\(45\!\cdots\!91\)\( T^{16} - \)\(14\!\cdots\!44\)\( T^{17} + \)\(64\!\cdots\!55\)\( T^{18} + \)\(10\!\cdots\!94\)\( T^{19} + \)\(50\!\cdots\!01\)\( T^{20} ) \)
$67$ \( 1 - 109 T + 7392 T^{2} - 316427 T^{3} + 1307855 T^{4} + 1277884608 T^{5} - 145160383367 T^{6} + 10086057781691 T^{7} - 447755337269856 T^{8} + 3529018380403405 T^{9} + 1625310705540412439 T^{10} + 15841763509630885045 T^{11} - \)\(90\!\cdots\!76\)\( T^{12} + \)\(91\!\cdots\!79\)\( T^{13} - \)\(58\!\cdots\!47\)\( T^{14} + \)\(23\!\cdots\!92\)\( T^{15} + \)\(10\!\cdots\!55\)\( T^{16} - \)\(11\!\cdots\!83\)\( T^{17} + \)\(12\!\cdots\!52\)\( T^{18} - \)\(80\!\cdots\!81\)\( T^{19} + \)\(33\!\cdots\!01\)\( T^{20} \)
$71$ \( ( 1 - 71 T + 357911 T^{3} - 25411681 T^{4} + 128100283921 T^{6} - 9095120158391 T^{7} + 45848500718449031 T^{9} - 3255243551009881201 T^{10} + \)\(23\!\cdots\!71\)\( T^{11} - \)\(11\!\cdots\!11\)\( T^{13} + \)\(82\!\cdots\!81\)\( T^{14} - \)\(41\!\cdots\!21\)\( T^{16} + \)\(29\!\cdots\!91\)\( T^{17} - \)\(14\!\cdots\!31\)\( T^{19} + \)\(10\!\cdots\!01\)\( T^{20} )( 1 + 71 T - 357911 T^{3} - 25411681 T^{4} + 128100283921 T^{6} + 9095120158391 T^{7} - 45848500718449031 T^{9} - 3255243551009881201 T^{10} - \)\(23\!\cdots\!71\)\( T^{11} + \)\(11\!\cdots\!11\)\( T^{13} + \)\(82\!\cdots\!81\)\( T^{14} - \)\(41\!\cdots\!21\)\( T^{16} - \)\(29\!\cdots\!91\)\( T^{17} + \)\(14\!\cdots\!31\)\( T^{19} + \)\(10\!\cdots\!01\)\( T^{20} ) \)
$73$ \( ( 1 + 97 T + 5329 T^{2} )^{10}( 1 - 46 T - 3213 T^{2} + 392932 T^{3} - 952795 T^{4} - 2050106058 T^{5} + 99382323223 T^{6} + 6353428314824 T^{7} - 821866102937271 T^{8} + 3948421245417370 T^{9} + 4198097085263518139 T^{10} + 21041136816829164730 T^{11} - \)\(23\!\cdots\!11\)\( T^{12} + \)\(96\!\cdots\!36\)\( T^{13} + \)\(80\!\cdots\!63\)\( T^{14} - \)\(88\!\cdots\!42\)\( T^{15} - \)\(21\!\cdots\!95\)\( T^{16} + \)\(47\!\cdots\!88\)\( T^{17} - \)\(20\!\cdots\!93\)\( T^{18} - \)\(15\!\cdots\!74\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} ) \)
$79$ \( ( 1 - 11 T + 6241 T^{2} )^{10}( 1 - 142 T + 13923 T^{2} - 1090844 T^{3} + 68006405 T^{4} - 2848952106 T^{5} - 19876774553 T^{6} + 20602812080072 T^{7} - 2801548365384951 T^{8} + 269237717692933690 T^{9} - 20747292564029104789 T^{10} + \)\(16\!\cdots\!90\)\( T^{11} - \)\(10\!\cdots\!31\)\( T^{12} + \)\(50\!\cdots\!12\)\( T^{13} - \)\(30\!\cdots\!33\)\( T^{14} - \)\(26\!\cdots\!06\)\( T^{15} + \)\(40\!\cdots\!05\)\( T^{16} - \)\(40\!\cdots\!64\)\( T^{17} + \)\(32\!\cdots\!83\)\( T^{18} - \)\(20\!\cdots\!62\)\( T^{19} + \)\(89\!\cdots\!01\)\( T^{20} ) \)
$83$ \( ( 1 - 83 T + 571787 T^{3} - 47458321 T^{4} + 326940373369 T^{6} - 27136050989627 T^{7} + 186940255267540403 T^{9} - 15516041187205853449 T^{10} + \)\(12\!\cdots\!67\)\( T^{11} - \)\(88\!\cdots\!63\)\( T^{13} + \)\(73\!\cdots\!29\)\( T^{14} - \)\(50\!\cdots\!81\)\( T^{16} + \)\(42\!\cdots\!23\)\( T^{17} - \)\(29\!\cdots\!47\)\( T^{19} + \)\(24\!\cdots\!01\)\( T^{20} )( 1 + 83 T - 571787 T^{3} - 47458321 T^{4} + 326940373369 T^{6} + 27136050989627 T^{7} - 186940255267540403 T^{9} - 15516041187205853449 T^{10} - \)\(12\!\cdots\!67\)\( T^{11} + \)\(88\!\cdots\!63\)\( T^{13} + \)\(73\!\cdots\!29\)\( T^{14} - \)\(50\!\cdots\!81\)\( T^{16} - \)\(42\!\cdots\!23\)\( T^{17} + \)\(29\!\cdots\!47\)\( T^{19} + \)\(24\!\cdots\!01\)\( T^{20} ) \)
$89$ \( ( 1 - 89 T + 7921 T^{2} - 704969 T^{3} + 62742241 T^{4} - 5584059449 T^{5} + 496981290961 T^{6} - 44231334895529 T^{7} + 3936588805702081 T^{8} - 350356403707485209 T^{9} + 31181719929966183601 T^{10} )^{2}( 1 + 89 T + 7921 T^{2} + 704969 T^{3} + 62742241 T^{4} + 5584059449 T^{5} + 496981290961 T^{6} + 44231334895529 T^{7} + 3936588805702081 T^{8} + 350356403707485209 T^{9} + 31181719929966183601 T^{10} )^{2} \)
$97$ \( ( 1 - 169 T + 19152 T^{2} - 1646567 T^{3} + 98068655 T^{4} - 1081053792 T^{5} - 740029884047 T^{6} + 135236685532871 T^{7} - 15892058676056976 T^{8} + 1413315942074845705 T^{9} - 89322014127628836961 T^{10} + \)\(13\!\cdots\!45\)\( T^{11} - \)\(14\!\cdots\!56\)\( T^{12} + \)\(11\!\cdots\!59\)\( T^{13} - \)\(57\!\cdots\!67\)\( T^{14} - \)\(79\!\cdots\!08\)\( T^{15} + \)\(68\!\cdots\!55\)\( T^{16} - \)\(10\!\cdots\!23\)\( T^{17} + \)\(11\!\cdots\!92\)\( T^{18} - \)\(97\!\cdots\!41\)\( T^{19} + \)\(54\!\cdots\!01\)\( T^{20} )( 1 + 2 T - 9405 T^{2} - 37628 T^{3} + 88416389 T^{4} + 530874630 T^{5} - 830848054841 T^{6} - 6656695503352 T^{7} + 7804135956992265 T^{8} + 78241119905023498 T^{9} - 73272632979530174389 T^{10} + \)\(73\!\cdots\!82\)\( T^{11} + \)\(69\!\cdots\!65\)\( T^{12} - \)\(55\!\cdots\!08\)\( T^{13} - \)\(65\!\cdots\!01\)\( T^{14} + \)\(39\!\cdots\!70\)\( T^{15} + \)\(61\!\cdots\!49\)\( T^{16} - \)\(24\!\cdots\!32\)\( T^{17} - \)\(57\!\cdots\!05\)\( T^{18} + \)\(11\!\cdots\!78\)\( T^{19} + \)\(54\!\cdots\!01\)\( T^{20} ) \)
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