Properties

Label 2010.2.a
Level 2010
Weight 2
Character orbit a
Rep. character \(\chi_{2010}(1,\cdot)\)
Character field \(\Q\)
Dimension 45
Newform subspaces 21
Sturm bound 816
Trace bound 7

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Defining parameters

Level: \( N \) = \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2010.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 21 \)
Sturm bound: \(816\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2010))\).

Total New Old
Modular forms 416 45 371
Cusp forms 401 45 356
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)\(67\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(6\)
Plus space\(+\)\(15\)
Minus space\(-\)\(30\)

Trace form

\( 45q + q^{2} + q^{3} + 45q^{4} + q^{5} + q^{6} + 8q^{7} + q^{8} + 45q^{9} + O(q^{10}) \) \( 45q + q^{2} + q^{3} + 45q^{4} + q^{5} + q^{6} + 8q^{7} + q^{8} + 45q^{9} - 3q^{10} + 4q^{11} + q^{12} + 14q^{13} + q^{15} + 45q^{16} + 10q^{17} + q^{18} + 4q^{19} + q^{20} + 8q^{21} + 4q^{22} + q^{24} + 45q^{25} - 10q^{26} + q^{27} + 8q^{28} + 22q^{29} + q^{30} + q^{32} - 12q^{33} + 2q^{34} + 45q^{36} - 10q^{37} - 12q^{38} + 14q^{39} - 3q^{40} + 2q^{41} - 8q^{42} + 12q^{43} + 4q^{44} + q^{45} + 8q^{47} + q^{48} + 77q^{49} + q^{50} + 2q^{51} + 14q^{52} + 22q^{53} + q^{54} + 12q^{55} + 20q^{57} - 2q^{58} + 44q^{59} + q^{60} + 30q^{61} + 16q^{62} + 8q^{63} + 45q^{64} + 6q^{65} + 12q^{66} - 3q^{67} + 10q^{68} + 8q^{69} + 8q^{70} - 16q^{71} + q^{72} - 30q^{73} - 2q^{74} + q^{75} + 4q^{76} - 32q^{77} + 14q^{78} + 32q^{79} + q^{80} + 45q^{81} - 22q^{82} - 68q^{83} + 8q^{84} + 18q^{85} - 20q^{86} + 14q^{87} + 4q^{88} - 22q^{89} - 3q^{90} + 16q^{91} + 24q^{93} + 24q^{94} + 20q^{95} + q^{96} - 14q^{97} - 7q^{98} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2010))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 67
2010.2.a.a \(1\) \(16.050\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
2010.2.a.b \(1\) \(16.050\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-3\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-3q^{7}+\cdots\)
2010.2.a.c \(1\) \(16.050\) \(\Q\) None \(-1\) \(-1\) \(1\) \(2\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+2q^{7}+\cdots\)
2010.2.a.d \(1\) \(16.050\) \(\Q\) None \(-1\) \(1\) \(-1\) \(4\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
2010.2.a.e \(1\) \(16.050\) \(\Q\) None \(-1\) \(1\) \(1\) \(-2\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
2010.2.a.f \(1\) \(16.050\) \(\Q\) None \(1\) \(-1\) \(1\) \(-1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2010.2.a.g \(1\) \(16.050\) \(\Q\) None \(1\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
2010.2.a.h \(1\) \(16.050\) \(\Q\) None \(1\) \(1\) \(-1\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-4q^{7}+\cdots\)
2010.2.a.i \(1\) \(16.050\) \(\Q\) None \(1\) \(1\) \(-1\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
2010.2.a.j \(1\) \(16.050\) \(\Q\) None \(1\) \(1\) \(-1\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
2010.2.a.k \(1\) \(16.050\) \(\Q\) None \(1\) \(1\) \(1\) \(4\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)
2010.2.a.l \(2\) \(16.050\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
2010.2.a.m \(2\) \(16.050\) \(\Q(\sqrt{17}) \) None \(-2\) \(2\) \(-2\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+\beta q^{7}+\cdots\)
2010.2.a.n \(3\) \(16.050\) 3.3.568.1 None \(3\) \(-3\) \(-3\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-\beta _{1}q^{7}+\cdots\)
2010.2.a.o \(3\) \(16.050\) 3.3.316.1 None \(3\) \(-3\) \(-3\) \(2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+(\beta _{1}-\beta _{2})q^{7}+\cdots\)
2010.2.a.p \(3\) \(16.050\) 3.3.316.1 None \(3\) \(-3\) \(3\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
2010.2.a.q \(3\) \(16.050\) 3.3.3132.1 None \(3\) \(3\) \(-3\) \(3\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
2010.2.a.r \(4\) \(16.050\) 4.4.70292.1 None \(-4\) \(-4\) \(-4\) \(1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+\beta _{1}q^{7}+\cdots\)
2010.2.a.s \(4\) \(16.050\) 4.4.11324.1 None \(-4\) \(-4\) \(4\) \(-2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
2010.2.a.t \(5\) \(16.050\) 5.5.31460256.1 None \(-5\) \(5\) \(5\) \(5\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)
2010.2.a.u \(5\) \(16.050\) 5.5.6517908.1 None \(5\) \(5\) \(5\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+\beta _{2}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2010))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2010)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(402))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(670))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{5} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{5} \))(\( ( 1 - T )^{5} \))
$5$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{5} \))(\( ( 1 - T )^{5} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 2 T + 12 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - T + 10 T^{2} - 7 T^{3} + 49 T^{4} \))(\( 1 + T + 15 T^{2} + 16 T^{3} + 105 T^{4} + 49 T^{5} + 343 T^{6} \))(\( 1 - 2 T + 5 T^{2} - 12 T^{3} + 35 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 - 2 T + 7 T^{2} + 4 T^{3} + 49 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 - 3 T + 9 T^{2} - 34 T^{3} + 63 T^{4} - 147 T^{5} + 343 T^{6} \))(\( 1 - T + 8 T^{2} - 9 T^{3} + 66 T^{4} - 63 T^{5} + 392 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 + 2 T + 8 T^{2} + 10 T^{3} + 46 T^{4} + 70 T^{5} + 392 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 - 5 T + 17 T^{2} - 26 T^{3} + 40 T^{4} - 18 T^{5} + 280 T^{6} - 1274 T^{7} + 5831 T^{8} - 12005 T^{9} + 16807 T^{10} \))(\( 1 + T + 15 T^{2} + 36 T^{3} + 114 T^{4} + 390 T^{5} + 798 T^{6} + 1764 T^{7} + 5145 T^{8} + 2401 T^{9} + 16807 T^{10} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 10 T^{2} + 121 T^{4} \))(\( 1 + 7 T + 30 T^{2} + 77 T^{3} + 121 T^{4} \))(\( 1 - T + 9 T^{2} - 42 T^{3} + 99 T^{4} - 121 T^{5} + 1331 T^{6} \))(\( 1 + 2 T + 27 T^{2} + 36 T^{3} + 297 T^{4} + 242 T^{5} + 1331 T^{6} \))(\( 1 - 2 T + 5 T^{2} - 52 T^{3} + 55 T^{4} - 242 T^{5} + 1331 T^{6} \))(\( 1 - 3 T + 9 T^{2} - 36 T^{3} + 99 T^{4} - 363 T^{5} + 1331 T^{6} \))(\( 1 + 3 T + 24 T^{2} + 67 T^{3} + 398 T^{4} + 737 T^{5} + 2904 T^{6} + 3993 T^{7} + 14641 T^{8} \))(\( 1 - 10 T + 58 T^{2} - 242 T^{3} + 874 T^{4} - 2662 T^{5} + 7018 T^{6} - 13310 T^{7} + 14641 T^{8} \))(\( 1 - T + 27 T^{2} - 34 T^{3} + 374 T^{4} - 578 T^{5} + 4114 T^{6} - 4114 T^{7} + 35937 T^{8} - 14641 T^{9} + 161051 T^{10} \))(\( 1 - T + 29 T^{2} - 4 T^{3} + 432 T^{4} + 122 T^{5} + 4752 T^{6} - 484 T^{7} + 38599 T^{8} - 14641 T^{9} + 161051 T^{10} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 26 T^{3} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( 1 + 6 T + 41 T^{2} + 136 T^{3} + 533 T^{4} + 1014 T^{5} + 2197 T^{6} \))(\( 1 + 11 T^{2} - 16 T^{3} + 143 T^{4} + 2197 T^{6} \))(\( 1 - 8 T + 53 T^{2} - 204 T^{3} + 689 T^{4} - 1352 T^{5} + 2197 T^{6} \))(\( ( 1 - 2 T + 13 T^{2} )^{3} \))(\( 1 - 2 T + 32 T^{2} - 50 T^{3} + 486 T^{4} - 650 T^{5} + 5408 T^{6} - 4394 T^{7} + 28561 T^{8} \))(\( 1 + 2 T + 20 T^{2} + 54 T^{3} + 390 T^{4} + 702 T^{5} + 3380 T^{6} + 4394 T^{7} + 28561 T^{8} \))(\( 1 - 2 T - 5 T^{2} + 60 T^{3} + 56 T^{4} - 820 T^{5} + 728 T^{6} + 10140 T^{7} - 10985 T^{8} - 57122 T^{9} + 371293 T^{10} \))(\( 1 + 29 T^{2} - 60 T^{3} + 382 T^{4} - 1512 T^{5} + 4966 T^{6} - 10140 T^{7} + 63713 T^{8} + 371293 T^{10} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 14 T^{2} + 289 T^{4} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( 1 + 10 T + 59 T^{2} + 260 T^{3} + 1003 T^{4} + 2890 T^{5} + 4913 T^{6} \))(\( 1 - 8 T + 65 T^{2} - 268 T^{3} + 1105 T^{4} - 2312 T^{5} + 4913 T^{6} \))(\( ( 1 - 4 T + 17 T^{2} )^{3} \))(\( 1 + 33 T^{2} + 20 T^{3} + 561 T^{4} + 4913 T^{6} \))(\( 1 + 2 T + 36 T^{2} + 22 T^{3} + 678 T^{4} + 374 T^{5} + 10404 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( 1 + 6 T + 26 T^{2} + 154 T^{3} + 914 T^{4} + 2618 T^{5} + 7514 T^{6} + 29478 T^{7} + 83521 T^{8} \))(\( 1 - 4 T + 39 T^{2} - 4 T^{3} + 176 T^{4} + 2272 T^{5} + 2992 T^{6} - 1156 T^{7} + 191607 T^{8} - 334084 T^{9} + 1419857 T^{10} \))(\( 1 - 10 T + 77 T^{2} - 472 T^{3} + 2514 T^{4} - 10780 T^{5} + 42738 T^{6} - 136408 T^{7} + 378301 T^{8} - 835210 T^{9} + 1419857 T^{10} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 8 T + 42 T^{2} + 152 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} \))(\( ( 1 + 2 T + 19 T^{2} )^{3} \))(\( 1 + 29 T^{2} + 16 T^{3} + 551 T^{4} + 6859 T^{6} \))(\( 1 + 4 T + 45 T^{2} + 136 T^{3} + 855 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( ( 1 + 2 T + 19 T^{2} )^{3} \))(\( 1 - 2 T + 44 T^{2} - 34 T^{3} + 982 T^{4} - 646 T^{5} + 15884 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 - 8 T + 64 T^{2} - 312 T^{3} + 1646 T^{4} - 5928 T^{5} + 23104 T^{6} - 54872 T^{7} + 130321 T^{8} \))(\( ( 1 - 2 T + 19 T^{2} )^{5} \))(\( 1 - 10 T + 87 T^{2} - 552 T^{3} + 3186 T^{4} - 14652 T^{5} + 60534 T^{6} - 199272 T^{7} + 596733 T^{8} - 1303210 T^{9} + 2476099 T^{10} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 6 T + 52 T^{2} - 138 T^{3} + 529 T^{4} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 + 4 T + 7 T^{2} - 48 T^{3} + 161 T^{4} + 2116 T^{5} + 12167 T^{6} \))(\( 1 - 6 T + 53 T^{2} - 212 T^{3} + 1219 T^{4} - 3174 T^{5} + 12167 T^{6} \))(\( 1 + 6 T + 47 T^{2} + 140 T^{3} + 1081 T^{4} + 3174 T^{5} + 12167 T^{6} \))(\( ( 1 - 4 T + 23 T^{2} )^{3} \))(\( 1 + 12 T + 100 T^{2} + 624 T^{3} + 3094 T^{4} + 14352 T^{5} + 52900 T^{6} + 146004 T^{7} + 279841 T^{8} \))(\( 1 - 6 T + 56 T^{2} - 158 T^{3} + 1262 T^{4} - 3634 T^{5} + 29624 T^{6} - 73002 T^{7} + 279841 T^{8} \))(\( 1 + 57 T^{2} - 24 T^{3} + 2024 T^{4} + 48 T^{5} + 46552 T^{6} - 12696 T^{7} + 693519 T^{8} + 6436343 T^{10} \))(\( 1 - 8 T + 71 T^{2} - 260 T^{3} + 1326 T^{4} - 2984 T^{5} + 30498 T^{6} - 137540 T^{7} + 863857 T^{8} - 2238728 T^{9} + 6436343 T^{10} \))
$29$ (\( 1 + 10 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 64 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 50 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 77 T^{2} + 8 T^{3} + 2233 T^{4} + 24389 T^{6} \))(\( 1 - 8 T + 79 T^{2} - 432 T^{3} + 2291 T^{4} - 6728 T^{5} + 24389 T^{6} \))(\( 1 + 4 T + 37 T^{2} + 300 T^{3} + 1073 T^{4} + 3364 T^{5} + 24389 T^{6} \))(\( 1 + 15 T^{2} + 160 T^{3} + 435 T^{4} + 24389 T^{6} \))(\( 1 - 2 T + 64 T^{2} - 322 T^{3} + 1918 T^{4} - 9338 T^{5} + 53824 T^{6} - 48778 T^{7} + 707281 T^{8} \))(\( 1 - 14 T + 120 T^{2} - 770 T^{3} + 4478 T^{4} - 22330 T^{5} + 100920 T^{6} - 341446 T^{7} + 707281 T^{8} \))(\( 1 + 27 T^{2} + 96 T^{3} - 304 T^{4} + 3840 T^{5} - 8816 T^{6} + 80736 T^{7} + 658503 T^{8} + 20511149 T^{10} \))(\( 1 - 4 T + 29 T^{2} - 4 T^{3} + 1158 T^{4} - 4672 T^{5} + 33582 T^{6} - 3364 T^{7} + 707281 T^{8} - 2829124 T^{9} + 20511149 T^{10} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 6 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 6 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 6 T + 31 T^{2} \))(\( 1 + 14 T + 108 T^{2} + 434 T^{3} + 961 T^{4} \))(\( 1 - 4 T - 2 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 3 T^{2} - 152 T^{3} + 93 T^{4} + 3844 T^{5} + 29791 T^{6} \))(\( 1 + 59 T^{2} - 76 T^{3} + 1829 T^{4} + 29791 T^{6} \))(\( 1 + 4 T + 83 T^{2} + 252 T^{3} + 2573 T^{4} + 3844 T^{5} + 29791 T^{6} \))(\( 1 - 18 T + 183 T^{2} - 1244 T^{3} + 5673 T^{4} - 17298 T^{5} + 29791 T^{6} \))(\( 1 - 10 T + 140 T^{2} - 878 T^{3} + 6646 T^{4} - 27218 T^{5} + 134540 T^{6} - 297910 T^{7} + 923521 T^{8} \))(\( 1 + 26 T^{2} - 12 T^{3} + 1954 T^{4} - 372 T^{5} + 24986 T^{6} + 923521 T^{8} \))(\( 1 - 14 T + 211 T^{2} - 1788 T^{3} + 14798 T^{4} - 83932 T^{5} + 458738 T^{6} - 1718268 T^{7} + 6285901 T^{8} - 12929294 T^{9} + 28629151 T^{10} \))(\( 1 - 12 T + 107 T^{2} - 540 T^{3} + 3346 T^{4} - 15792 T^{5} + 103726 T^{6} - 518940 T^{7} + 3187637 T^{8} - 11082252 T^{9} + 28629151 T^{10} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( ( 1 + 10 T + 37 T^{2} )^{2} \))(\( 1 - 11 T + 100 T^{2} - 407 T^{3} + 1369 T^{4} \))(\( 1 + T + 87 T^{2} + 94 T^{3} + 3219 T^{4} + 1369 T^{5} + 50653 T^{6} \))(\( 1 + 8 T + 37 T^{2} + 68 T^{3} + 1369 T^{4} + 10952 T^{5} + 50653 T^{6} \))(\( 1 + 83 T^{2} + 16 T^{3} + 3071 T^{4} + 50653 T^{6} \))(\( 1 - 9 T + 99 T^{2} - 664 T^{3} + 3663 T^{4} - 12321 T^{5} + 50653 T^{6} \))(\( 1 - 3 T + 122 T^{2} - 273 T^{3} + 6346 T^{4} - 10101 T^{5} + 167018 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 10 T + 110 T^{2} + 582 T^{3} + 4434 T^{4} + 21534 T^{5} + 150590 T^{6} + 506530 T^{7} + 1874161 T^{8} \))(\( 1 - 9 T + 81 T^{2} - 690 T^{3} + 5178 T^{4} - 34394 T^{5} + 191586 T^{6} - 944610 T^{7} + 4102893 T^{8} - 16867449 T^{9} + 69343957 T^{10} \))(\( 1 + T + 135 T^{2} + 96 T^{3} + 8484 T^{4} + 4110 T^{5} + 313908 T^{6} + 131424 T^{7} + 6838155 T^{8} + 1874161 T^{9} + 69343957 T^{10} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 - 12 T + 106 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 14 T^{2} + 1681 T^{4} \))(\( 1 + 14 T + 155 T^{2} + 1084 T^{3} + 6355 T^{4} + 23534 T^{5} + 68921 T^{6} \))(\( 1 - 4 T + 121 T^{2} - 324 T^{3} + 4961 T^{4} - 6724 T^{5} + 68921 T^{6} \))(\( 1 - 16 T + 179 T^{2} - 1280 T^{3} + 7339 T^{4} - 26896 T^{5} + 68921 T^{6} \))(\( 1 - 12 T + 153 T^{2} - 956 T^{3} + 6273 T^{4} - 20172 T^{5} + 68921 T^{6} \))(\( 1 + 92 T^{2} + 16 T^{3} + 4358 T^{4} + 656 T^{5} + 154652 T^{6} + 2825761 T^{8} \))(\( 1 - 2 T + 142 T^{2} - 190 T^{3} + 8290 T^{4} - 7790 T^{5} + 238702 T^{6} - 137842 T^{7} + 2825761 T^{8} \))(\( 1 - 4 T + 103 T^{2} - 420 T^{3} + 5288 T^{4} - 21184 T^{5} + 216808 T^{6} - 706020 T^{7} + 7098863 T^{8} - 11303044 T^{9} + 115856201 T^{10} \))(\( 1 + 85 T^{2} + 528 T^{3} + 2290 T^{4} + 40608 T^{5} + 93890 T^{6} + 887568 T^{7} + 5858285 T^{8} + 115856201 T^{10} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 8 T + 54 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4} \))(\( 1 + 10 T + 129 T^{2} + 796 T^{3} + 5547 T^{4} + 18490 T^{5} + 79507 T^{6} \))(\( 1 + 101 T^{2} - 16 T^{3} + 4343 T^{4} + 79507 T^{6} \))(\( 1 - 4 T + 65 T^{2} - 216 T^{3} + 2795 T^{4} - 7396 T^{5} + 79507 T^{6} \))(\( ( 1 + 2 T + 43 T^{2} )^{3} \))(\( 1 + 6 T + 92 T^{2} + 518 T^{3} + 6006 T^{4} + 22274 T^{5} + 170108 T^{6} + 477042 T^{7} + 3418801 T^{8} \))(\( 1 + 4 T + 80 T^{2} + 196 T^{3} + 4206 T^{4} + 8428 T^{5} + 147920 T^{6} + 318028 T^{7} + 3418801 T^{8} \))(\( 1 - 18 T + 243 T^{2} - 2448 T^{3} + 19926 T^{4} - 143708 T^{5} + 856818 T^{6} - 4526352 T^{7} + 19320201 T^{8} - 61538418 T^{9} + 147008443 T^{10} \))(\( 1 + 2 T + 111 T^{2} + 24 T^{3} + 6354 T^{4} - 4308 T^{5} + 273222 T^{6} + 44376 T^{7} + 8825277 T^{8} + 6837602 T^{9} + 147008443 T^{10} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 6 T + 100 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 14 T + 126 T^{2} - 658 T^{3} + 2209 T^{4} \))(\( 1 + 18 T + 239 T^{2} + 1856 T^{3} + 11233 T^{4} + 39762 T^{5} + 103823 T^{6} \))(\( 1 - 26 T + 349 T^{2} - 2940 T^{3} + 16403 T^{4} - 57434 T^{5} + 103823 T^{6} \))(\( 1 - 2 T + 127 T^{2} - 156 T^{3} + 5969 T^{4} - 4418 T^{5} + 103823 T^{6} \))(\( 1 + 6 T + 45 T^{2} + 388 T^{3} + 2115 T^{4} + 13254 T^{5} + 103823 T^{6} \))(\( 1 + 14 T + 116 T^{2} + 762 T^{3} + 4862 T^{4} + 35814 T^{5} + 256244 T^{6} + 1453522 T^{7} + 4879681 T^{8} \))(\( 1 - 18 T + 288 T^{2} - 2698 T^{3} + 22622 T^{4} - 126806 T^{5} + 636192 T^{6} - 1868814 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T + 73 T^{2} - 228 T^{3} + 4304 T^{4} - 9244 T^{5} + 202288 T^{6} - 503652 T^{7} + 7579079 T^{8} + 9759362 T^{9} + 229345007 T^{10} \))(\( 1 - 10 T + 143 T^{2} - 1204 T^{3} + 8646 T^{4} - 69988 T^{5} + 406362 T^{6} - 2659636 T^{7} + 14846689 T^{8} - 48796810 T^{9} + 229345007 T^{10} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( 1 + 8 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( 1 - 6 T + 98 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 11 T^{2} - 424 T^{3} + 583 T^{4} + 33708 T^{5} + 148877 T^{6} \))(\( 1 - 22 T + 303 T^{2} - 2604 T^{3} + 16059 T^{4} - 61798 T^{5} + 148877 T^{6} \))(\( 1 - 10 T + 163 T^{2} - 972 T^{3} + 8639 T^{4} - 28090 T^{5} + 148877 T^{6} \))(\( 1 + 12 T + 135 T^{2} + 888 T^{3} + 7155 T^{4} + 33708 T^{5} + 148877 T^{6} \))(\( 1 + 10 T + 168 T^{2} + 1230 T^{3} + 12542 T^{4} + 65190 T^{5} + 471912 T^{6} + 1488770 T^{7} + 7890481 T^{8} \))(\( 1 + 4 T + 40 T^{2} + 332 T^{3} + 5950 T^{4} + 17596 T^{5} + 112360 T^{6} + 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 61 T^{2} + 432 T^{3} + 6310 T^{4} + 4320 T^{5} + 334430 T^{6} + 1213488 T^{7} + 9081497 T^{8} + 418195493 T^{10} \))(\( 1 - 12 T + 241 T^{2} - 2256 T^{3} + 24418 T^{4} - 172296 T^{5} + 1294154 T^{6} - 6337104 T^{7} + 35879357 T^{8} - 94685772 T^{9} + 418195493 T^{10} \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 2 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 12 T + 106 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 150 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 + 2 T + 101 T^{2} + 436 T^{3} + 5959 T^{4} + 6962 T^{5} + 205379 T^{6} \))(\( 1 - 20 T + 281 T^{2} - 2488 T^{3} + 16579 T^{4} - 69620 T^{5} + 205379 T^{6} \))(\( 1 + 8 T + 69 T^{2} + 96 T^{3} + 4071 T^{4} + 27848 T^{5} + 205379 T^{6} \))(\( 1 - 6 T + 33 T^{2} + 300 T^{3} + 1947 T^{4} - 20886 T^{5} + 205379 T^{6} \))(\( 1 + 164 T^{2} + 16 T^{3} + 12566 T^{4} + 944 T^{5} + 570884 T^{6} + 12117361 T^{8} \))(\( 1 - 28 T + 480 T^{2} - 5516 T^{3} + 48942 T^{4} - 325444 T^{5} + 1670880 T^{6} - 5750612 T^{7} + 12117361 T^{8} \))(\( 1 - 10 T + 75 T^{2} - 112 T^{3} + 5534 T^{4} - 44780 T^{5} + 326506 T^{6} - 389872 T^{7} + 15403425 T^{8} - 121173610 T^{9} + 714924299 T^{10} \))(\( 1 + 175 T^{2} - 528 T^{3} + 13810 T^{4} - 59616 T^{5} + 814790 T^{6} - 1837968 T^{7} + 35941325 T^{8} + 714924299 T^{10} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 + 11 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 10 T + 120 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 5 T + 124 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( 1 - 3 T + 41 T^{2} + 344 T^{3} + 2501 T^{4} - 11163 T^{5} + 226981 T^{6} \))(\( 1 - 10 T + 161 T^{2} - 1212 T^{3} + 9821 T^{4} - 37210 T^{5} + 226981 T^{6} \))(\( 1 - 10 T + 25 T^{2} + 356 T^{3} + 1525 T^{4} - 37210 T^{5} + 226981 T^{6} \))(\( 1 + 3 T + 147 T^{2} + 240 T^{3} + 8967 T^{4} + 11163 T^{5} + 226981 T^{6} \))(\( 1 - 9 T + 254 T^{2} - 1599 T^{3} + 23518 T^{4} - 97539 T^{5} + 945134 T^{6} - 2042829 T^{7} + 13845841 T^{8} \))(\( 1 + 28 T + 466 T^{2} + 5400 T^{3} + 47978 T^{4} + 329400 T^{5} + 1733986 T^{6} + 6355468 T^{7} + 13845841 T^{8} \))(\( 1 - 19 T + 351 T^{2} - 3744 T^{3} + 40596 T^{4} - 310374 T^{5} + 2476356 T^{6} - 13931424 T^{7} + 79670331 T^{8} - 263070979 T^{9} + 844596301 T^{10} \))(\( 1 - 7 T + 195 T^{2} - 1608 T^{3} + 18348 T^{4} - 146034 T^{5} + 1119228 T^{6} - 5983368 T^{7} + 44261295 T^{8} - 96920887 T^{9} + 844596301 T^{10} \))
$67$ (\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{5} \))(\( ( 1 + T )^{5} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 6 T + 124 T^{2} - 426 T^{3} + 5041 T^{4} \))(\( 1 - 9 T + 158 T^{2} - 639 T^{3} + 5041 T^{4} \))(\( 1 - 9 T + 191 T^{2} - 1036 T^{3} + 13561 T^{4} - 45369 T^{5} + 357911 T^{6} \))(\( 1 + 6 T + 113 T^{2} + 764 T^{3} + 8023 T^{4} + 30246 T^{5} + 357911 T^{6} \))(\( 1 - 38 T + 687 T^{2} - 7332 T^{3} + 48777 T^{4} - 191558 T^{5} + 357911 T^{6} \))(\( 1 + 15 T + 45 T^{2} - 402 T^{3} + 3195 T^{4} + 75615 T^{5} + 357911 T^{6} \))(\( 1 + 9 T + 242 T^{2} + 1313 T^{3} + 22814 T^{4} + 93223 T^{5} + 1219922 T^{6} + 3221199 T^{7} + 25411681 T^{8} \))(\( 1 - 6 T + 156 T^{2} - 1078 T^{3} + 12086 T^{4} - 76538 T^{5} + 786396 T^{6} - 2147466 T^{7} + 25411681 T^{8} \))(\( 1 + 3 T + 285 T^{2} + 630 T^{3} + 36044 T^{4} + 60486 T^{5} + 2559124 T^{6} + 3175830 T^{7} + 102004635 T^{8} + 76235043 T^{9} + 1804229351 T^{10} \))(\( 1 + 21 T + 229 T^{2} + 1356 T^{3} + 5944 T^{4} + 28542 T^{5} + 422024 T^{6} + 6835596 T^{7} + 81961619 T^{8} + 533645301 T^{9} + 1804229351 T^{10} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( 1 + 10 T + 154 T^{2} + 730 T^{3} + 5329 T^{4} \))(\( 1 + 16 T + 207 T^{2} + 1744 T^{3} + 15111 T^{4} + 85264 T^{5} + 389017 T^{6} \))(\( 1 - 10 T + 223 T^{2} - 1372 T^{3} + 16279 T^{4} - 53290 T^{5} + 389017 T^{6} \))(\( ( 1 + 2 T + 73 T^{2} )^{3} \))(\( 1 - 12 T + 159 T^{2} - 1352 T^{3} + 11607 T^{4} - 63948 T^{5} + 389017 T^{6} \))(\( 1 + 14 T + 248 T^{2} + 2274 T^{3} + 25134 T^{4} + 166002 T^{5} + 1321592 T^{6} + 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 12 T - 4 T^{2} + 628 T^{3} + 15846 T^{4} + 45844 T^{5} - 21316 T^{6} + 4668204 T^{7} + 28398241 T^{8} \))(\( 1 - 12 T + 309 T^{2} - 2880 T^{3} + 40914 T^{4} - 296552 T^{5} + 2986722 T^{6} - 15347520 T^{7} + 120206253 T^{8} - 340778892 T^{9} + 2073071593 T^{10} \))(\( 1 + 12 T + 221 T^{2} + 2016 T^{3} + 23338 T^{4} + 166056 T^{5} + 1703674 T^{6} + 10743264 T^{7} + 85972757 T^{8} + 340778892 T^{9} + 2073071593 T^{10} \))
$79$ (\( 1 + 12 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))(\( 1 - 14 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 2 T + 12 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 30 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 10 T + 75 T^{2} - 80 T^{3} + 5925 T^{4} + 62410 T^{5} + 493039 T^{6} \))(\( 1 - 4 T + 187 T^{2} - 700 T^{3} + 14773 T^{4} - 24964 T^{5} + 493039 T^{6} \))(\( 1 - 4 T + 227 T^{2} - 636 T^{3} + 17933 T^{4} - 24964 T^{5} + 493039 T^{6} \))(\( 1 - 12 T + 195 T^{2} - 1308 T^{3} + 15405 T^{4} - 74892 T^{5} + 493039 T^{6} \))(\( 1 - 12 T + 200 T^{2} - 992 T^{3} + 13686 T^{4} - 78368 T^{5} + 1248200 T^{6} - 5916468 T^{7} + 38950081 T^{8} \))(\( 1 + 4 T + 202 T^{2} + 480 T^{3} + 19394 T^{4} + 37920 T^{5} + 1260682 T^{6} + 1972156 T^{7} + 38950081 T^{8} \))(\( 1 - 8 T + 239 T^{2} - 2036 T^{3} + 28066 T^{4} - 219600 T^{5} + 2217214 T^{6} - 12706676 T^{7} + 117836321 T^{8} - 311600648 T^{9} + 3077056399 T^{10} \))(\( 1 - 10 T + 195 T^{2} - 1932 T^{3} + 17874 T^{4} - 195492 T^{5} + 1412046 T^{6} - 12057612 T^{7} + 96142605 T^{8} - 389500810 T^{9} + 3077056399 T^{10} \))
$83$ (\( 1 - 8 T + 83 T^{2} \))(\( 1 + 7 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( 1 + 5 T + 66 T^{2} + 415 T^{3} + 6889 T^{4} \))(\( 1 + 23 T + 297 T^{2} + 2730 T^{3} + 24651 T^{4} + 158447 T^{5} + 571787 T^{6} \))(\( 1 - 14 T + 219 T^{2} - 1692 T^{3} + 18177 T^{4} - 96446 T^{5} + 571787 T^{6} \))(\( 1 - 2 T + 185 T^{2} - 460 T^{3} + 15355 T^{4} - 13778 T^{5} + 571787 T^{6} \))(\( 1 - 3 T + 213 T^{2} - 548 T^{3} + 17679 T^{4} - 20667 T^{5} + 571787 T^{6} \))(\( 1 + 25 T + 324 T^{2} + 3793 T^{3} + 40262 T^{4} + 314819 T^{5} + 2232036 T^{6} + 14294675 T^{7} + 47458321 T^{8} \))(\( 1 - 14 T + 306 T^{2} - 3150 T^{3} + 36890 T^{4} - 261450 T^{5} + 2108034 T^{6} - 8005018 T^{7} + 47458321 T^{8} \))(\( 1 - 7 T + 307 T^{2} - 1962 T^{3} + 41918 T^{4} - 229342 T^{5} + 3479194 T^{6} - 13516218 T^{7} + 175538609 T^{8} - 332208247 T^{9} + 3939040643 T^{10} \))(\( 1 + 27 T + 361 T^{2} + 4092 T^{3} + 49780 T^{4} + 525762 T^{5} + 4131740 T^{6} + 28189788 T^{7} + 206415107 T^{8} + 1281374667 T^{9} + 3939040643 T^{10} \))
$89$ (\( 1 - 2 T + 89 T^{2} \))(\( 1 + 7 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 3 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 12 T + 166 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 + T + 140 T^{2} + 89 T^{3} + 7921 T^{4} \))(\( 1 + 11 T + 235 T^{2} + 1506 T^{3} + 20915 T^{4} + 87131 T^{5} + 704969 T^{6} \))(\( 1 - 34 T + 591 T^{2} - 6636 T^{3} + 52599 T^{4} - 269314 T^{5} + 704969 T^{6} \))(\( 1 - 2 T + 23 T^{2} + 1220 T^{3} + 2047 T^{4} - 15842 T^{5} + 704969 T^{6} \))(\( 1 + 15 T + 327 T^{2} + 2714 T^{3} + 29103 T^{4} + 118815 T^{5} + 704969 T^{6} \))(\( 1 + 9 T + 122 T^{2} + 783 T^{3} + 10410 T^{4} + 69687 T^{5} + 966362 T^{6} + 6344721 T^{7} + 62742241 T^{8} \))(\( 1 - 4 T + 268 T^{2} - 1148 T^{3} + 32518 T^{4} - 102172 T^{5} + 2122828 T^{6} - 2819876 T^{7} + 62742241 T^{8} \))(\( 1 + 11 T + 225 T^{2} + 2180 T^{3} + 30518 T^{4} + 254530 T^{5} + 2716102 T^{6} + 17267780 T^{7} + 158618025 T^{8} + 690164651 T^{9} + 5584059449 T^{10} \))(\( 1 - 5 T + 245 T^{2} - 764 T^{3} + 35490 T^{4} - 107486 T^{5} + 3158610 T^{6} - 6051644 T^{7} + 172717405 T^{8} - 313711205 T^{9} + 5584059449 T^{10} \))
$97$ (\( 1 - 6 T + 97 T^{2} \))(\( 1 + 3 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 12 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 16 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( ( 1 + 16 T + 97 T^{2} )^{2} \))(\( 1 - 27 T + 372 T^{2} - 2619 T^{3} + 9409 T^{4} \))(\( 1 + 5 T + 291 T^{2} + 962 T^{3} + 28227 T^{4} + 47045 T^{5} + 912673 T^{6} \))(\( 1 + 6 T + 191 T^{2} + 820 T^{3} + 18527 T^{4} + 56454 T^{5} + 912673 T^{6} \))(\( 1 + 2 T - 45 T^{2} - 220 T^{3} - 4365 T^{4} + 18818 T^{5} + 912673 T^{6} \))(\( 1 - 15 T + 267 T^{2} - 2698 T^{3} + 25899 T^{4} - 141135 T^{5} + 912673 T^{6} \))(\( 1 + 3 T + 368 T^{2} + 841 T^{3} + 52686 T^{4} + 81577 T^{5} + 3462512 T^{6} + 2738019 T^{7} + 88529281 T^{8} \))(\( 1 + 8 T + 200 T^{2} + 2120 T^{3} + 21582 T^{4} + 205640 T^{5} + 1881800 T^{6} + 7301384 T^{7} + 88529281 T^{8} \))(\( 1 + 15 T + 453 T^{2} + 5616 T^{3} + 85002 T^{4} + 807778 T^{5} + 8245194 T^{6} + 52840944 T^{7} + 413440869 T^{8} + 1327939215 T^{9} + 8587340257 T^{10} \))(\( 1 + 3 T + 47 T^{2} + 132 T^{3} + 4888 T^{4} - 13662 T^{5} + 474136 T^{6} + 1241988 T^{7} + 42895631 T^{8} + 265587843 T^{9} + 8587340257 T^{10} \))
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