Properties

Label 735.2.a
Level 735
Weight 2
Character orbit a
Rep. character \(\chi_{735}(1,\cdot)\)
Character field \(\Q\)
Dimension 28
Newform subspaces 15
Sturm bound 224
Trace bound 4

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Defining parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(224\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(735))\).

Total New Old
Modular forms 128 28 100
Cusp forms 97 28 69
Eisenstein series 31 0 31

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

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(7\)
Minus space\(-\)\(21\)

Trace form

\( 28q + 4q^{2} + 2q^{3} + 32q^{4} - 2q^{6} + 12q^{8} + 28q^{9} + O(q^{10}) \) \( 28q + 4q^{2} + 2q^{3} + 32q^{4} - 2q^{6} + 12q^{8} + 28q^{9} + 8q^{11} + 6q^{12} + 8q^{13} - 2q^{15} + 48q^{16} + 4q^{18} + 8q^{20} + 24q^{22} - 8q^{23} + 6q^{24} + 28q^{25} + 24q^{26} + 2q^{27} + 24q^{29} - 2q^{30} - 16q^{31} + 28q^{32} + 32q^{36} - 4q^{37} - 8q^{38} - 24q^{39} - 20q^{43} + 8q^{44} + 8q^{46} - 24q^{47} - 2q^{48} + 4q^{50} + 4q^{51} - 8q^{52} + 32q^{53} - 2q^{54} + 8q^{55} - 12q^{57} + 16q^{58} - 6q^{60} + 8q^{61} - 24q^{62} + 72q^{64} + 8q^{65} - 16q^{66} - 20q^{67} + 16q^{68} + 16q^{71} + 12q^{72} + 8q^{73} - 32q^{74} + 2q^{75} - 16q^{76} - 4q^{78} - 28q^{79} + 28q^{81} + 16q^{82} + 8q^{83} - 8q^{85} - 112q^{86} - 4q^{87} - 112q^{88} + 16q^{89} - 160q^{92} - 20q^{93} + 40q^{94} + 16q^{95} - 10q^{96} + 8q^{97} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
735.2.a.a \(1\) \(5.869\) \(\Q\) None \(-2\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(-\) \(q-2q^{2}-q^{3}+2q^{4}-q^{5}+2q^{6}+q^{9}+\cdots\)
735.2.a.b \(1\) \(5.869\) \(\Q\) None \(-2\) \(1\) \(1\) \(0\) \(-\) \(-\) \(+\) \(q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}+q^{9}+\cdots\)
735.2.a.c \(1\) \(5.869\) \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+3q^{8}+\cdots\)
735.2.a.d \(1\) \(5.869\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(+\) \(-\) \(-\) \(q-q^{3}-2q^{4}+q^{5}+q^{9}+2q^{12}+q^{13}+\cdots\)
735.2.a.e \(1\) \(5.869\) \(\Q\) None \(0\) \(1\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(q+q^{3}-2q^{4}-q^{5}+q^{9}-2q^{12}-q^{13}+\cdots\)
735.2.a.f \(1\) \(5.869\) \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}-3q^{8}+\cdots\)
735.2.a.g \(2\) \(5.869\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(0\) \(+\) \(-\) \(+\) \(q+(-1+\beta )q^{2}-q^{3}+(2-2\beta )q^{4}+q^{5}+\cdots\)
735.2.a.h \(2\) \(5.869\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+q^{3}+(2-2\beta )q^{4}-q^{5}+\cdots\)
735.2.a.i \(2\) \(5.869\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) \(+\) \(+\) \(+\) \(q+\beta q^{2}-q^{3}-q^{5}-\beta q^{6}-2\beta q^{8}+\cdots\)
735.2.a.j \(2\) \(5.869\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+\beta q^{2}+q^{3}+q^{5}+\beta q^{6}-2\beta q^{8}+\cdots\)
735.2.a.k \(2\) \(5.869\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q-\beta q^{2}+q^{3}+3q^{4}+q^{5}-\beta q^{6}-\beta q^{8}+\cdots\)
735.2.a.l \(2\) \(5.869\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(-2\) \(0\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
735.2.a.m \(2\) \(5.869\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+q^{5}+\cdots\)
735.2.a.n \(4\) \(5.869\) 4.4.4352.1 None \(4\) \(-4\) \(4\) \(0\) \(+\) \(-\) \(+\) \(q+(1+\beta _{1}-\beta _{3})q^{2}-q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
735.2.a.o \(4\) \(5.869\) 4.4.4352.1 None \(4\) \(4\) \(-4\) \(0\) \(-\) \(+\) \(+\) \(q+(1+\beta _{1}-\beta _{3})q^{2}+q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(735))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(735)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 - T^{2} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} + 17 T^{4} - 24 T^{5} + 32 T^{6} - 32 T^{7} + 16 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} + 17 T^{4} - 24 T^{5} + 32 T^{6} - 32 T^{7} + 16 T^{8} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))
$5$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))
$7$ 1
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 2 T + 20 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 20 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 14 T^{2} + 121 T^{4} \))(\( 1 + 14 T^{2} + 121 T^{4} \))(\( 1 - 8 T + 36 T^{2} - 104 T^{3} + 326 T^{4} - 1144 T^{5} + 4356 T^{6} - 10648 T^{7} + 14641 T^{8} \))(\( 1 - 8 T + 36 T^{2} - 104 T^{3} + 326 T^{4} - 1144 T^{5} + 4356 T^{6} - 10648 T^{7} + 14641 T^{8} \))
$13$ (\( 1 - 3 T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 8 T + 39 T^{2} - 104 T^{3} + 169 T^{4} \))(\( 1 + 8 T + 39 T^{2} + 104 T^{3} + 169 T^{4} \))(\( 1 + 6 T + 33 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 33 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 + 6 T^{2} + 169 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 + 20 T^{2} + 64 T^{3} + 198 T^{4} + 832 T^{5} + 3380 T^{6} + 28561 T^{8} \))(\( 1 + 20 T^{2} - 64 T^{3} + 198 T^{4} - 832 T^{5} + 3380 T^{6} + 28561 T^{8} \))
$17$ (\( 1 - 4 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 10 T + 56 T^{2} - 170 T^{3} + 289 T^{4} \))(\( 1 + 10 T + 56 T^{2} + 170 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 20 T^{2} - 68 T^{3} + 289 T^{4} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 8 T + 52 T^{2} + 280 T^{3} + 1318 T^{4} + 4760 T^{5} + 15028 T^{6} + 39304 T^{7} + 83521 T^{8} \))(\( 1 - 8 T + 52 T^{2} - 280 T^{3} + 1318 T^{4} - 4760 T^{5} + 15028 T^{6} - 39304 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 - 2 T + 27 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 27 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 7 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 2 T + 7 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 34 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T + 34 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 8 T + 64 T^{2} - 280 T^{3} + 1426 T^{4} - 5320 T^{5} + 23104 T^{6} - 54872 T^{7} + 130321 T^{8} \))(\( 1 + 8 T + 64 T^{2} + 280 T^{3} + 1426 T^{4} + 5320 T^{5} + 23104 T^{6} + 54872 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 6 T + 52 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 52 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + 4 T + 32 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 4 T + 32 T^{2} + 92 T^{3} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 48 T^{2} - 64 T^{3} + 1346 T^{4} - 1472 T^{5} + 25392 T^{6} + 279841 T^{8} \))(\( 1 + 48 T^{2} - 64 T^{3} + 1346 T^{4} - 1472 T^{5} + 25392 T^{6} + 279841 T^{8} \))
$29$ (\( 1 + 8 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 2 T + 32 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 - 2 T + 32 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( 1 - 8 T + 44 T^{2} - 88 T^{3} + 502 T^{4} - 2552 T^{5} + 37004 T^{6} - 195112 T^{7} + 707281 T^{8} \))(\( 1 - 8 T + 44 T^{2} - 88 T^{3} + 502 T^{4} - 2552 T^{5} + 37004 T^{6} - 195112 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + T + 31 T^{2} \))(\( 1 - T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 6 T + 59 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 59 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 63 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 63 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 + 12 T + 78 T^{2} + 372 T^{3} + 961 T^{4} \))(\( 1 - 4 T - 6 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + 4 T - 6 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 112 T^{2} - 696 T^{3} + 4994 T^{4} - 21576 T^{5} + 107632 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 + 8 T + 112 T^{2} + 696 T^{3} + 4994 T^{4} + 21576 T^{5} + 107632 T^{6} + 238328 T^{7} + 923521 T^{8} \))
$37$ (\( 1 - 7 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 4 T + 51 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 51 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 + 14 T + 121 T^{2} + 518 T^{3} + 1369 T^{4} \))(\( 1 + 14 T + 121 T^{2} + 518 T^{3} + 1369 T^{4} \))(\( 1 - 4 T - 2 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 92 T^{2} - 600 T^{3} + 4854 T^{4} - 22200 T^{5} + 125948 T^{6} - 405224 T^{7} + 1874161 T^{8} \))(\( 1 - 8 T + 92 T^{2} - 600 T^{3} + 4854 T^{4} - 22200 T^{5} + 125948 T^{6} - 405224 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 2 T + 80 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( 1 + 2 T + 80 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( 1 + 12 T + 86 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 108 T^{2} - 128 T^{3} + 5510 T^{4} - 5248 T^{5} + 181548 T^{6} + 2825761 T^{8} \))(\( 1 + 108 T^{2} + 128 T^{3} + 5510 T^{4} + 5248 T^{5} + 181548 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 - T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 63 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 63 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 18 T + 165 T^{2} + 774 T^{3} + 1849 T^{4} \))(\( 1 + 18 T + 165 T^{2} + 774 T^{3} + 1849 T^{4} \))(\( 1 + 6 T^{2} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( 1 + 8 T + 140 T^{2} + 840 T^{3} + 8406 T^{4} + 36120 T^{5} + 258860 T^{6} + 636056 T^{7} + 3418801 T^{8} \))(\( 1 + 8 T + 140 T^{2} + 840 T^{3} + 8406 T^{4} + 36120 T^{5} + 258860 T^{6} + 636056 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 + 2 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( ( 1 - 2 T + 47 T^{2} )^{2} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( 1 + 4 T - 30 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 - 4 T - 30 T^{2} - 188 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 30 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))(\( ( 1 + 4 T + 26 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 26 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 4 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 - 4 T + 2 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 4 T + 2 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 16 T + 150 T^{2} + 848 T^{3} + 2809 T^{4} \))(\( 1 + 34 T^{2} + 2809 T^{4} \))(\( 1 + 34 T^{2} + 2809 T^{4} \))(\( 1 - 8 T + 208 T^{2} - 1192 T^{3} + 16402 T^{4} - 63176 T^{5} + 584272 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( 1 - 8 T + 208 T^{2} - 1192 T^{3} + 16402 T^{4} - 63176 T^{5} + 584272 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 - 8 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 10 T + 116 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 116 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 + 116 T^{2} + 3481 T^{4} \))(\( 1 + 116 T^{2} + 3481 T^{4} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))(\( 1 + 16 T + 150 T^{2} + 944 T^{3} + 3481 T^{4} \))(\( 1 - 16 T + 150 T^{2} - 944 T^{3} + 3481 T^{4} \))(\( 1 + 16 T + 188 T^{2} + 1424 T^{3} + 10678 T^{4} + 84016 T^{5} + 654428 T^{6} + 3286064 T^{7} + 12117361 T^{8} \))(\( 1 - 16 T + 188 T^{2} - 1424 T^{3} + 10678 T^{4} - 84016 T^{5} + 654428 T^{6} - 3286064 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 14 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( ( 1 - 4 T + 61 T^{2} )^{2} \))(\( ( 1 + 4 T + 61 T^{2} )^{2} \))(\( 1 - 8 T + 66 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 66 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 - 16 T + 184 T^{2} - 976 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 16 T + 184 T^{2} + 976 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 7 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 12 T + 95 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 95 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( 1 + 92 T^{2} - 512 T^{3} + 6486 T^{4} - 34304 T^{5} + 412988 T^{6} + 20151121 T^{8} \))(\( 1 + 92 T^{2} - 512 T^{3} + 6486 T^{4} - 34304 T^{5} + 412988 T^{6} + 20151121 T^{8} \))
$71$ (\( 1 - 6 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 2 T + 116 T^{2} - 142 T^{3} + 5041 T^{4} \))(\( 1 - 2 T + 116 T^{2} - 142 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} \))(\( 1 + 134 T^{2} + 5041 T^{4} \))(\( 1 + 134 T^{2} + 5041 T^{4} \))(\( 1 + 8 T + 212 T^{2} + 1608 T^{3} + 20054 T^{4} + 114168 T^{5} + 1068692 T^{6} + 2863288 T^{7} + 25411681 T^{8} \))(\( 1 + 8 T + 212 T^{2} + 1608 T^{3} + 20054 T^{4} + 114168 T^{5} + 1068692 T^{6} + 2863288 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 + T + 73 T^{2} \))(\( 1 - T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 8 T + 87 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 + 8 T + 87 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( 1 + 10 T + 121 T^{2} + 730 T^{3} + 5329 T^{4} \))(\( 1 - 10 T + 121 T^{2} - 730 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 190 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 - 28 T + 334 T^{2} - 2044 T^{3} + 5329 T^{4} \))(\( 1 + 28 T + 334 T^{2} + 2044 T^{3} + 5329 T^{4} \))(\( 1 + 164 T^{2} + 128 T^{3} + 16374 T^{4} + 9344 T^{5} + 873956 T^{6} + 28398241 T^{8} \))(\( 1 + 164 T^{2} - 128 T^{3} + 16374 T^{4} - 9344 T^{5} + 873956 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 13 T + 79 T^{2} \))(\( 1 + 13 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 6 T + 59 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 - 6 T + 59 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 - 2 T + 127 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 - 2 T + 127 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( 1 + 132 T^{2} - 384 T^{3} + 10950 T^{4} - 30336 T^{5} + 823812 T^{6} + 38950081 T^{8} \))(\( 1 + 132 T^{2} - 384 T^{3} + 10950 T^{4} - 30336 T^{5} + 823812 T^{6} + 38950081 T^{8} \))
$83$ (\( 1 + 2 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 6 T + 28 T^{2} - 498 T^{3} + 6889 T^{4} \))(\( 1 + 6 T + 28 T^{2} + 498 T^{3} + 6889 T^{4} \))(\( 1 - 8 T + 180 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 180 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( 1 - 16 T + 150 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 150 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( 1 - 8 T + 150 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( ( 1 + 20 T + 258 T^{2} + 1660 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 20 T + 258 T^{2} - 1660 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( 1 - 12 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 40 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T + 40 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 16 T + 224 T^{2} + 1424 T^{3} + 7921 T^{4} \))(\( 1 - 16 T + 224 T^{2} - 1424 T^{3} + 7921 T^{4} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( 1 + 12 T + 86 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 86 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( ( 1 - 4 T + 174 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 4 T + 174 T^{2} + 356 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 6 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 - 16 T + 210 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 16 T + 210 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 + 16 T + 250 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 250 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 - 4 T + 126 T^{2} - 388 T^{3} + 9409 T^{4} \))(\( 1 + 4 T + 126 T^{2} + 388 T^{3} + 9409 T^{4} \))(\( 1 + 196 T^{2} + 1280 T^{3} + 17174 T^{4} + 124160 T^{5} + 1844164 T^{6} + 88529281 T^{8} \))(\( 1 + 196 T^{2} - 1280 T^{3} + 17174 T^{4} - 124160 T^{5} + 1844164 T^{6} + 88529281 T^{8} \))
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