Properties

Label 1568.2.a
Level 1568
Weight 2
Character orbit a
Rep. character \(\chi_{1568}(1,\cdot)\)
Character field \(\Q\)
Dimension 41
Newform subspaces 24
Sturm bound 448
Trace bound 25

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Defining parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(448\)
Trace bound: \(25\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1568))\).

Total New Old
Modular forms 256 41 215
Cusp forms 193 41 152
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.

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(8\)
\(+\)\(-\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(12\)
\(-\)\(-\)\(+\)\(9\)
Plus space\(+\)\(17\)
Minus space\(-\)\(24\)

Trace form

\( 41q - 2q^{5} + 37q^{9} + O(q^{10}) \) \( 41q - 2q^{5} + 37q^{9} - 10q^{13} + 2q^{17} + 55q^{25} + 6q^{29} + 16q^{33} - 18q^{37} + 26q^{41} + 22q^{45} + 30q^{53} + 16q^{57} - 10q^{61} + 36q^{65} + 48q^{69} - 6q^{73} + 33q^{81} - 20q^{85} - 22q^{89} + 48q^{93} - 46q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
1568.2.a.a \(1\) \(12.521\) \(\Q\) None \(0\) \(-2\) \(-2\) \(0\) \(+\) \(-\) \(q-2q^{3}-2q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1568.2.a.b \(1\) \(12.521\) \(\Q\) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(q-2q^{3}+q^{9}+4q^{11}+4q^{13}+2q^{17}+\cdots\)
1568.2.a.c \(1\) \(12.521\) \(\Q\) None \(0\) \(-2\) \(2\) \(0\) \(-\) \(-\) \(q-2q^{3}+2q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots\)
1568.2.a.d \(1\) \(12.521\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-4\) \(0\) \(+\) \(-\) \(q-4q^{5}-3q^{9}+4q^{13}-8q^{17}+11q^{25}+\cdots\)
1568.2.a.e \(1\) \(12.521\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(q+2q^{5}-3q^{9}-6q^{13}-2q^{17}-q^{25}+\cdots\)
1568.2.a.f \(1\) \(12.521\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(4\) \(0\) \(+\) \(-\) \(q+4q^{5}-3q^{9}-4q^{13}+8q^{17}+11q^{25}+\cdots\)
1568.2.a.g \(1\) \(12.521\) \(\Q\) None \(0\) \(2\) \(-2\) \(0\) \(-\) \(-\) \(q+2q^{3}-2q^{5}+q^{9}-4q^{11}-6q^{13}+\cdots\)
1568.2.a.h \(1\) \(12.521\) \(\Q\) None \(0\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+2q^{3}+q^{9}-4q^{11}+4q^{13}+2q^{17}+\cdots\)
1568.2.a.i \(1\) \(12.521\) \(\Q\) None \(0\) \(2\) \(2\) \(0\) \(+\) \(-\) \(q+2q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
1568.2.a.j \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) \(-\) \(-\) \(q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}-2\beta q^{9}+\cdots\)
1568.2.a.k \(2\) \(12.521\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-2\) \(0\) \(-\) \(-\) \(q+(-1-\beta )q^{3}+(-1+\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
1568.2.a.l \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(2\) \(0\) \(+\) \(+\) \(q+(-1+\beta )q^{3}+(1-2\beta )q^{5}-2\beta q^{9}+\cdots\)
1568.2.a.m \(2\) \(12.521\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(-6\) \(0\) \(+\) \(+\) \(q+\beta q^{3}-3q^{5}+4q^{9}-\beta q^{11}-4q^{13}+\cdots\)
1568.2.a.n \(2\) \(12.521\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(q+\beta q^{3}-q^{5}-3\beta q^{11}-\beta q^{15}-5q^{17}+\cdots\)
1568.2.a.o \(2\) \(12.521\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{5}-3q^{9}-\beta q^{13}-5\beta q^{17}-3q^{25}+\cdots\)
1568.2.a.p \(2\) \(12.521\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+3\beta q^{5}-3q^{9}+5\beta q^{13}-3\beta q^{17}+\cdots\)
1568.2.a.q \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{3}-q^{9}-2q^{11}-2\beta q^{13}-3\beta q^{17}+\cdots\)
1568.2.a.r \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta q^{3}-q^{9}+2q^{11}+2\beta q^{13}+3\beta q^{17}+\cdots\)
1568.2.a.s \(2\) \(12.521\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(0\) \(-\) \(+\) \(q+\beta q^{3}+q^{5}+3\beta q^{11}+\beta q^{15}+5q^{17}+\cdots\)
1568.2.a.t \(2\) \(12.521\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(6\) \(0\) \(+\) \(-\) \(q+\beta q^{3}+3q^{5}+4q^{9}+\beta q^{11}+4q^{13}+\cdots\)
1568.2.a.u \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-2\) \(0\) \(+\) \(-\) \(q+(1+\beta )q^{3}+(-1-2\beta )q^{5}+2\beta q^{9}+\cdots\)
1568.2.a.v \(2\) \(12.521\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-2\) \(0\) \(+\) \(-\) \(q+(1+\beta )q^{3}+(-1+\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
1568.2.a.w \(2\) \(12.521\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) \(-\) \(+\) \(q+(1+\beta )q^{3}+(1+2\beta )q^{5}+2\beta q^{9}+(-1+\cdots)q^{11}+\cdots\)
1568.2.a.x \(4\) \(12.521\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta _{2}q^{3}-2\beta _{1}q^{5}+7q^{9}-\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1568))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1568)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 - T^{2} + 9 T^{4} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 2 T + 3 T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 10 T^{3} + 25 T^{4} \))(\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 + T + 5 T^{2} )^{2} \))(\( 1 + 8 T^{2} + 25 T^{4} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - T + 5 T^{2} )^{2} \))(\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( 1 + 2 T + 3 T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 10 T^{3} + 25 T^{4} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))
$7$ 1
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 21 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 2 T + 21 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 15 T^{2} + 121 T^{4} \))(\( 1 - 5 T^{2} + 121 T^{4} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( 1 - 5 T^{2} + 121 T^{4} \))(\( 1 + 15 T^{2} + 121 T^{4} \))(\( 1 - 2 T + 21 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 21 T^{2} + 22 T^{3} + 121 T^{4} \))(\( ( 1 + 2 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 18 T^{2} + 169 T^{4} \))(\( 1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 + 18 T^{2} + 169 T^{4} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 + 24 T^{2} + 169 T^{4} \))(\( 1 - 24 T^{2} + 169 T^{4} \))(\( 1 + 18 T^{2} + 169 T^{4} \))(\( 1 + 18 T^{2} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( 1 + 18 T^{2} + 169 T^{4} \))(\( 1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 + 18 T^{2} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))
$17$ (\( 1 - 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 6 T + 35 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 + 6 T + 35 T^{2} + 102 T^{3} + 289 T^{4} \))(\( ( 1 - T + 17 T^{2} )^{2} \))(\( ( 1 + 5 T + 17 T^{2} )^{2} \))(\( 1 - 16 T^{2} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( ( 1 - 5 T + 17 T^{2} )^{2} \))(\( ( 1 + T + 17 T^{2} )^{2} \))(\( 1 - 6 T + 35 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 + 6 T + 35 T^{2} + 102 T^{3} + 289 T^{4} \))(\( ( 1 + 16 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 10 T + 61 T^{2} + 190 T^{3} + 361 T^{4} \))(\( 1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 10 T + 61 T^{2} + 190 T^{3} + 361 T^{4} \))(\( 1 - 25 T^{2} + 361 T^{4} \))(\( 1 + 35 T^{2} + 361 T^{4} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( 1 + 20 T^{2} + 361 T^{4} \))(\( 1 + 20 T^{2} + 361 T^{4} \))(\( 1 + 35 T^{2} + 361 T^{4} \))(\( 1 - 25 T^{2} + 361 T^{4} \))(\( 1 - 10 T + 61 T^{2} - 190 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 10 T + 61 T^{2} - 190 T^{3} + 361 T^{4} \))(\( ( 1 + 28 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 2 T + 29 T^{2} + 46 T^{3} + 529 T^{4} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( 1 - 2 T + 29 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 + 39 T^{2} + 529 T^{4} \))(\( 1 + 43 T^{2} + 529 T^{4} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{2} \))(\( 1 + 43 T^{2} + 529 T^{4} \))(\( 1 + 39 T^{2} + 529 T^{4} \))(\( 1 - 2 T + 29 T^{2} - 46 T^{3} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( 1 + 2 T + 29 T^{2} + 46 T^{3} + 529 T^{4} \))(\( ( 1 - 34 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( 1 + 38 T^{2} + 841 T^{4} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( 1 + 38 T^{2} + 841 T^{4} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( ( 1 + 6 T + 29 T^{2} )^{4} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 14 T + 109 T^{2} + 434 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + 14 T + 109 T^{2} + 434 T^{3} + 961 T^{4} \))(\( 1 + 55 T^{2} + 961 T^{4} \))(\( 1 - 13 T^{2} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( 1 - 13 T^{2} + 961 T^{4} \))(\( 1 + 55 T^{2} + 961 T^{4} \))(\( 1 - 14 T + 109 T^{2} - 434 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 14 T + 109 T^{2} - 434 T^{3} + 961 T^{4} \))(\( ( 1 + 22 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( 1 + 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 6 T + 51 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 - 6 T + 51 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( ( 1 + 5 T + 37 T^{2} )^{2} \))(\( ( 1 + 5 T + 37 T^{2} )^{2} \))(\( ( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 5 T + 37 T^{2} )^{2} \))(\( ( 1 + 5 T + 37 T^{2} )^{2} \))(\( 1 - 6 T + 51 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 - 6 T + 51 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( ( 1 - 2 T + 37 T^{2} )^{4} \))
$41$ (\( 1 + 4 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 - 8 T + 90 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 - 8 T + 78 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( ( 1 - 8 T + 41 T^{2} )^{2} \))(\( ( 1 + 4 T + 41 T^{2} )^{2} \))(\( 1 - 80 T^{2} + 1681 T^{4} \))(\( 1 + 80 T^{2} + 1681 T^{4} \))(\( 1 + 64 T^{2} + 1681 T^{4} \))(\( 1 + 64 T^{2} + 1681 T^{4} \))(\( ( 1 - 4 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( 1 - 8 T + 90 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 - 8 T + 78 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( ( 1 + 80 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 12 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 - 26 T^{2} + 1849 T^{4} \))(\( 1 + 38 T^{2} + 1849 T^{4} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} )^{2} \))(\( 1 + 38 T^{2} + 1849 T^{4} \))(\( 1 - 26 T^{2} + 1849 T^{4} \))(\( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 - 4 T + 70 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( ( 1 + 66 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 18 T + 173 T^{2} + 846 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 110 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 18 T + 173 T^{2} + 846 T^{3} + 2209 T^{4} \))(\( 1 + 87 T^{2} + 2209 T^{4} \))(\( 1 + 19 T^{2} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 19 T^{2} + 2209 T^{4} \))(\( 1 + 87 T^{2} + 2209 T^{4} \))(\( 1 - 18 T + 173 T^{2} - 846 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 110 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 18 T + 173 T^{2} - 846 T^{3} + 2209 T^{4} \))(\( ( 1 + 54 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 14 T + 53 T^{2} \))(\( 1 - 14 T + 53 T^{2} \))(\( 1 - 14 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( ( 1 + T + 53 T^{2} )^{2} \))(\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( ( 1 + T + 53 T^{2} )^{2} \))(\( ( 1 - 7 T + 53 T^{2} )^{2} \))(\( ( 1 + T + 53 T^{2} )^{2} \))(\( ( 1 + 14 T + 53 T^{2} )^{2} \))(\( ( 1 + 14 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + T + 53 T^{2} )^{2} \))(\( ( 1 - 7 T + 53 T^{2} )^{2} \))(\( ( 1 + T + 53 T^{2} )^{2} \))(\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( ( 1 + T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{4} \))
$59$ (\( 1 + 6 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 2 T + 21 T^{2} + 118 T^{3} + 3481 T^{4} \))(\( 1 + 14 T + 162 T^{2} + 826 T^{3} + 3481 T^{4} \))(\( 1 + 2 T + 21 T^{2} + 118 T^{3} + 3481 T^{4} \))(\( 1 + 111 T^{2} + 3481 T^{4} \))(\( 1 + 115 T^{2} + 3481 T^{4} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 - 44 T^{2} + 3481 T^{4} \))(\( 1 - 44 T^{2} + 3481 T^{4} \))(\( 1 + 115 T^{2} + 3481 T^{4} \))(\( 1 + 111 T^{2} + 3481 T^{4} \))(\( 1 - 2 T + 21 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 162 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 - 2 T + 21 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( ( 1 + 108 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 6 T + 99 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 18 T + 198 T^{2} + 1098 T^{3} + 3721 T^{4} \))(\( 1 - 6 T + 99 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 + 5 T + 61 T^{2} )^{2} \))(\( ( 1 + 11 T + 61 T^{2} )^{2} \))(\( 1 - 120 T^{2} + 3721 T^{4} \))(\( 1 + 120 T^{2} + 3721 T^{4} \))(\( 1 + 90 T^{2} + 3721 T^{4} \))(\( 1 + 90 T^{2} + 3721 T^{4} \))(\( ( 1 - 11 T + 61 T^{2} )^{2} \))(\( ( 1 - 5 T + 61 T^{2} )^{2} \))(\( 1 + 6 T + 99 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 18 T + 198 T^{2} + 1098 T^{3} + 3721 T^{4} \))(\( 1 - 6 T + 99 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 + 50 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 14 T + 165 T^{2} + 938 T^{3} + 4489 T^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( 1 - 14 T + 165 T^{2} - 938 T^{3} + 4489 T^{4} \))(\( 1 + 127 T^{2} + 4489 T^{4} \))(\( 1 - 13 T^{2} + 4489 T^{4} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} )^{2} \))(\( 1 - 13 T^{2} + 4489 T^{4} \))(\( 1 + 127 T^{2} + 4489 T^{4} \))(\( 1 - 14 T + 165 T^{2} - 938 T^{3} + 4489 T^{4} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( 1 + 14 T + 165 T^{2} + 938 T^{3} + 4489 T^{4} \))(\( ( 1 + 67 T^{2} )^{4} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 16 T + 174 T^{2} - 1136 T^{3} + 5041 T^{4} \))(\( 1 + 8 T + 78 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 + 16 T + 174 T^{2} + 1136 T^{3} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( 1 - 50 T^{2} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( 1 - 50 T^{2} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( 1 + 16 T + 174 T^{2} + 1136 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 78 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( 1 - 16 T + 174 T^{2} - 1136 T^{3} + 5041 T^{4} \))(\( ( 1 + 62 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - 16 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 16 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - 18 T + 195 T^{2} - 1314 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 102 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 + 18 T + 195 T^{2} + 1314 T^{3} + 5329 T^{4} \))(\( ( 1 + 9 T + 73 T^{2} )^{2} \))(\( ( 1 + 15 T + 73 T^{2} )^{2} \))(\( 1 - 96 T^{2} + 5329 T^{4} \))(\( 1 + 96 T^{2} + 5329 T^{4} \))(\( 1 + 144 T^{2} + 5329 T^{4} \))(\( 1 + 144 T^{2} + 5329 T^{4} \))(\( ( 1 - 15 T + 73 T^{2} )^{2} \))(\( ( 1 - 9 T + 73 T^{2} )^{2} \))(\( 1 - 18 T + 195 T^{2} - 1314 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 102 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 + 18 T + 195 T^{2} + 1314 T^{3} + 5329 T^{4} \))(\( ( 1 + 96 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 2 T + 109 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 2 T + 109 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 + 151 T^{2} + 6241 T^{4} \))(\( 1 + 155 T^{2} + 6241 T^{4} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 12 T + 79 T^{2} )^{2} \))(\( ( 1 - 12 T + 79 T^{2} )^{2} \))(\( 1 + 155 T^{2} + 6241 T^{4} \))(\( 1 + 151 T^{2} + 6241 T^{4} \))(\( 1 - 2 T + 109 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 2 T + 109 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( 1 + 14 T + 210 T^{2} + 1162 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( 1 + 54 T^{2} + 6889 T^{4} \))(\( 1 + 118 T^{2} + 6889 T^{4} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 + 68 T^{2} + 6889 T^{4} \))(\( 1 + 68 T^{2} + 6889 T^{4} \))(\( 1 + 118 T^{2} + 6889 T^{4} \))(\( 1 + 54 T^{2} + 6889 T^{4} \))(\( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 - 14 T + 210 T^{2} - 1162 T^{3} + 6889 T^{4} \))(\( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( ( 1 + 76 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 - 16 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 - 16 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 - 16 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( ( 1 - 9 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 7 T + 89 T^{2} )^{2} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( 1 - 160 T^{2} + 7921 T^{4} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( ( 1 - 7 T + 89 T^{2} )^{2} \))(\( ( 1 - 9 T + 89 T^{2} )^{2} \))(\( ( 1 - 9 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 80 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 12 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 12 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 12 T + 97 T^{2} \))(\( 1 - 8 T + 202 T^{2} - 776 T^{3} + 9409 T^{4} \))(\( 1 + 16 T + 238 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 202 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 + 12 T + 97 T^{2} )^{2} \))(\( 1 + 144 T^{2} + 9409 T^{4} \))(\( 1 - 144 T^{2} + 9409 T^{4} \))(\( 1 - 144 T^{2} + 9409 T^{4} \))(\( 1 - 144 T^{2} + 9409 T^{4} \))(\( ( 1 - 12 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( 1 - 8 T + 202 T^{2} - 776 T^{3} + 9409 T^{4} \))(\( 1 + 16 T + 238 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 202 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( ( 1 + 176 T^{2} + 9409 T^{4} )^{2} \))
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