Properties

Label 1470.2.a
Level $1470$
Weight $2$
Character orbit 1470.a
Rep. character $\chi_{1470}(1,\cdot)$
Character field $\Q$
Dimension $26$
Newform subspaces $22$
Sturm bound $672$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(672\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1470))\).

Total New Old
Modular forms 368 26 342
Cusp forms 305 26 279
Eisenstein series 63 0 63

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

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(20\)

Trace form

\( 26q - 2q^{3} + 26q^{4} + 26q^{9} + O(q^{10}) \) \( 26q - 2q^{3} + 26q^{4} + 26q^{9} - 2q^{10} + 4q^{11} - 2q^{12} + 26q^{16} + 8q^{17} - 4q^{22} + 24q^{23} + 26q^{25} + 4q^{26} - 2q^{27} - 8q^{29} + 2q^{30} + 16q^{31} + 4q^{33} - 4q^{34} + 26q^{36} + 56q^{37} + 8q^{38} + 48q^{39} - 2q^{40} + 12q^{41} + 56q^{43} + 4q^{44} + 16q^{47} - 2q^{48} - 8q^{51} - 4q^{55} + 48q^{57} + 4q^{58} + 4q^{59} + 12q^{61} + 8q^{62} + 26q^{64} + 4q^{65} + 12q^{66} + 48q^{67} + 8q^{68} - 8q^{69} - 8q^{71} + 4q^{73} - 20q^{74} - 2q^{75} - 4q^{78} + 40q^{79} + 26q^{81} - 16q^{82} - 32q^{83} - 4q^{85} + 24q^{87} - 4q^{88} - 44q^{89} - 2q^{90} + 24q^{92} + 40q^{93} + 24q^{94} - 24q^{95} - 20q^{97} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1470))\) 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 7
1470.2.a.a \(1\) \(11.738\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.b \(1\) \(11.738\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.c \(1\) \(11.738\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.d \(1\) \(11.738\) \(\Q\) None \(-1\) \(-1\) \(1\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.e \(1\) \(11.738\) \(\Q\) None \(-1\) \(-1\) \(1\) \(0\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.f \(1\) \(11.738\) \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.g \(1\) \(11.738\) \(\Q\) None \(-1\) \(1\) \(1\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.h \(1\) \(11.738\) \(\Q\) None \(-1\) \(1\) \(1\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.i \(1\) \(11.738\) \(\Q\) None \(-1\) \(1\) \(1\) \(0\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.j \(1\) \(11.738\) \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.k \(1\) \(11.738\) \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.l \(1\) \(11.738\) \(\Q\) None \(1\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.m \(1\) \(11.738\) \(\Q\) None \(1\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.n \(1\) \(11.738\) \(\Q\) None \(1\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.o \(1\) \(11.738\) \(\Q\) None \(1\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
1470.2.a.p \(1\) \(11.738\) \(\Q\) None \(1\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
1470.2.a.q \(1\) \(11.738\) \(\Q\) None \(1\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
1470.2.a.r \(1\) \(11.738\) \(\Q\) None \(1\) \(1\) \(1\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)
1470.2.a.s \(2\) \(11.738\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(2\) \(0\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
1470.2.a.t \(2\) \(11.738\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(-2\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
1470.2.a.u \(2\) \(11.738\) \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(-2\) \(0\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
1470.2.a.v \(2\) \(11.738\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(2\) \(0\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1470))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1470)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(735))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$5$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$7$ 1
$11$ (\( 1 + T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 - T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 7 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 7 T + 13 T^{2} \))(\( 1 - 6 T^{2} + 169 T^{4} \))(\( 1 - 6 T^{2} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 8 T + 48 T^{2} - 136 T^{3} + 289 T^{4} \))(\( 1 + 8 T + 48 T^{2} + 136 T^{3} + 289 T^{4} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( 1 + 32 T^{2} + 289 T^{4} \))
$19$ (\( 1 + 3 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 3 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 7 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 8 T + 46 T^{2} + 152 T^{3} + 361 T^{4} \))(\( 1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))
$23$ (\( 1 - 7 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 12 T + 74 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 - 12 T + 74 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 42 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 42 T^{2} - 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 8 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 8 T + 72 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 72 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 2 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 10 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 6 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 12 T + 80 T^{2} + 372 T^{3} + 961 T^{4} \))(\( 1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 16 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 16 T^{2} + 124 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 11 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - 8 T + 72 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 72 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 72 T^{2} + 1369 T^{4} \))(\( 1 + 72 T^{2} + 1369 T^{4} \))
$41$ (\( 1 + 11 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 11 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 5 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 - 4 T + 14 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 14 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 110 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 110 T^{2} + 492 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 40 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 4 T + 40 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 12 T + 120 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 - 12 T + 120 T^{2} - 516 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 5 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 5 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 11 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 - 11 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 4 T + 80 T^{2} - 188 T^{3} + 2209 T^{4} \))(\( 1 + 4 T + 80 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 - 4 T + 48 T^{2} - 188 T^{3} + 2209 T^{4} \))(\( 1 + 4 T + 48 T^{2} + 188 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 11 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 11 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 - 12 T + 110 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 110 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 4 T - 18 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 4 T - 18 T^{2} - 212 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 146 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 146 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 82 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 82 T^{2} + 708 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 + 12 T + 126 T^{2} + 732 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 126 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 126 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 + 12 T + 126 T^{2} + 732 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 2 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 2 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 4 T + 88 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 4 T + 88 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 40 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 40 T^{2} + 268 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 - 8 T + 86 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 86 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( 1 - 16 T + 198 T^{2} - 1136 T^{3} + 5041 T^{4} \))(\( 1 - 16 T + 198 T^{2} - 1136 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( 1 + 4 T + 118 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 4 T + 118 T^{2} - 292 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( 1 + 30 T^{2} + 6241 T^{4} \))(\( 1 + 30 T^{2} + 6241 T^{4} \))
$83$ (\( 1 - 8 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 16 T + 222 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( 1 + 16 T + 222 T^{2} + 1328 T^{3} + 6889 T^{4} \))(\( 1 - 16 T + 222 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( 1 + 16 T + 222 T^{2} + 1328 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 10 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 - 4 T + 174 T^{2} - 356 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 174 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 110 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 - 4 T + 110 T^{2} - 356 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 16 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 16 T + 97 T^{2} \))(\( 1 - 4 T + 126 T^{2} - 388 T^{3} + 9409 T^{4} \))(\( 1 + 4 T + 126 T^{2} + 388 T^{3} + 9409 T^{4} \))(\( 1 + 12 T + 222 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 - 12 T + 222 T^{2} - 1164 T^{3} + 9409 T^{4} \))
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