Properties

Label 1792.2.b
Level 1792
Weight 2
Character orbit b
Rep. character \(\chi_{1792}(897,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 16
Sturm bound 512
Trace bound 17

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Defining parameters

Level: \( N \) = \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1792.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(512\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\), \(23\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1792, [\chi])\).

Total New Old
Modular forms 280 48 232
Cusp forms 232 48 184
Eisenstein series 48 0 48

Trace form

\( 48q - 48q^{9} + O(q^{10}) \) \( 48q - 48q^{9} - 48q^{25} + 48q^{49} - 32q^{65} + 32q^{73} + 48q^{81} + 32q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1792, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1792.2.b.a \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+iq^{3}+2iq^{5}-q^{7}-q^{9}-8q^{15}+\cdots\)
1792.2.b.b \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+iq^{3}-q^{7}-q^{9}-2iq^{11}-2iq^{13}+\cdots\)
1792.2.b.c \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+iq^{3}-q^{7}-q^{9}+2iq^{13}+6q^{17}+\cdots\)
1792.2.b.d \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q-iq^{5}-q^{7}+3q^{9}-2iq^{11}+iq^{13}+\cdots\)
1792.2.b.e \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q-q^{7}+3q^{9}-iq^{11}+2iq^{13}-2q^{17}+\cdots\)
1792.2.b.f \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+iq^{3}+q^{7}-q^{9}-2iq^{11}+2iq^{13}+\cdots\)
1792.2.b.g \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+iq^{3}+q^{7}-q^{9}-2iq^{13}+6q^{17}+\cdots\)
1792.2.b.h \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+iq^{3}-2iq^{5}+q^{7}-q^{9}+8q^{15}+\cdots\)
1792.2.b.i \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q-iq^{5}+q^{7}+3q^{9}+2iq^{11}+iq^{13}+\cdots\)
1792.2.b.j \(2\) \(14.309\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+q^{7}+3q^{9}+iq^{11}+2iq^{13}-2q^{17}+\cdots\)
1792.2.b.k \(4\) \(14.309\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(-4\) \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}-q^{7}+(-3+\cdots)q^{9}+\cdots\)
1792.2.b.l \(4\) \(14.309\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{12}^{2}q^{3}-\zeta_{12}q^{5}-q^{7}+(-1+\zeta_{12}^{3})q^{9}+\cdots\)
1792.2.b.m \(4\) \(14.309\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{5}+q^{7}+(-3+\cdots)q^{9}+\cdots\)
1792.2.b.n \(4\) \(14.309\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{12}^{2}q^{3}+\zeta_{12}q^{5}+q^{7}+(-1+\zeta_{12}^{3})q^{9}+\cdots\)
1792.2.b.o \(6\) \(14.309\) 6.0.399424.1 None \(0\) \(0\) \(0\) \(-6\) \(q+\beta _{5}q^{3}+\beta _{4}q^{5}-q^{7}+(-2-\beta _{3}+\cdots)q^{9}+\cdots\)
1792.2.b.p \(6\) \(14.309\) 6.0.399424.1 None \(0\) \(0\) \(0\) \(6\) \(q+\beta _{5}q^{3}-\beta _{4}q^{5}+q^{7}+(-2-\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1792, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1792, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(448, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(896, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} )( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} ) \))(\( 1 - 4 T^{2} + 10 T^{4} - 36 T^{6} + 81 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} )( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} ) \))(\( 1 - 4 T^{2} + 10 T^{4} - 36 T^{6} + 81 T^{8} \))(\( 1 - 2 T^{2} + 11 T^{4} - 20 T^{6} + 99 T^{8} - 162 T^{10} + 729 T^{12} \))(\( 1 - 2 T^{2} + 11 T^{4} - 20 T^{6} + 99 T^{8} - 162 T^{10} + 729 T^{12} \))
$5$ (\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 - 8 T^{2} + 46 T^{4} - 200 T^{6} + 625 T^{8} \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 - 8 T^{2} + 46 T^{4} - 200 T^{6} + 625 T^{8} \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 + 2 T^{2} + 59 T^{4} + 84 T^{6} + 1475 T^{8} + 1250 T^{10} + 15625 T^{12} \))(\( 1 + 2 T^{2} + 59 T^{4} + 84 T^{6} + 1475 T^{8} + 1250 T^{10} + 15625 T^{12} \))
$7$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{6} \))(\( ( 1 - T )^{6} \))
$11$ (\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 + 4 T^{2} - 74 T^{4} + 484 T^{6} + 14641 T^{8} \))(\( 1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8} \))(\( 1 + 4 T^{2} - 74 T^{4} + 484 T^{6} + 14641 T^{8} \))(\( 1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8} \))(\( 1 - 30 T^{2} + 551 T^{4} - 7268 T^{6} + 66671 T^{8} - 439230 T^{10} + 1771561 T^{12} \))(\( 1 - 30 T^{2} + 551 T^{4} - 7268 T^{6} + 66671 T^{8} - 439230 T^{10} + 1771561 T^{12} \))
$13$ (\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 24 T^{2} + 302 T^{4} - 4056 T^{6} + 28561 T^{8} \))(\( 1 - 44 T^{2} + 810 T^{4} - 7436 T^{6} + 28561 T^{8} \))(\( 1 - 24 T^{2} + 302 T^{4} - 4056 T^{6} + 28561 T^{8} \))(\( 1 - 44 T^{2} + 810 T^{4} - 7436 T^{6} + 28561 T^{8} \))(\( 1 + 2 T^{2} + 491 T^{4} + 660 T^{6} + 82979 T^{8} + 57122 T^{10} + 4826809 T^{12} \))(\( 1 + 2 T^{2} + 491 T^{4} + 660 T^{6} + 82979 T^{8} + 57122 T^{10} + 4826809 T^{12} \))
$17$ (\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 8 T + 38 T^{2} - 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 8 T + 38 T^{2} - 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 2 T + 23 T^{2} + 76 T^{3} + 391 T^{4} + 578 T^{5} + 4913 T^{6} )^{2} \))(\( ( 1 + 2 T + 23 T^{2} + 76 T^{3} + 391 T^{4} + 578 T^{5} + 4913 T^{6} )^{2} \))
$19$ (\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 - 2 T^{2} + 361 T^{4} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 2 T^{2} + 361 T^{4} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 - 64 T^{2} + 1726 T^{4} - 23104 T^{6} + 130321 T^{8} \))(\( 1 - 20 T^{2} + 714 T^{4} - 7220 T^{6} + 130321 T^{8} \))(\( 1 - 64 T^{2} + 1726 T^{4} - 23104 T^{6} + 130321 T^{8} \))(\( 1 - 20 T^{2} + 714 T^{4} - 7220 T^{6} + 130321 T^{8} \))(\( 1 - 34 T^{2} + 907 T^{4} - 19540 T^{6} + 327427 T^{8} - 4430914 T^{10} + 47045881 T^{12} \))(\( 1 - 34 T^{2} + 907 T^{4} - 19540 T^{6} + 327427 T^{8} - 4430914 T^{10} + 47045881 T^{12} \))
$23$ (\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{4} \))(\( ( 1 - 4 T + 2 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 4 T + 2 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} - 16 T^{3} + 943 T^{4} + 12167 T^{6} )^{2} \))(\( ( 1 + 41 T^{2} + 16 T^{3} + 943 T^{4} + 12167 T^{6} )^{2} \))
$29$ (\( 1 - 54 T^{2} + 841 T^{4} \))(\( 1 - 54 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 54 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 54 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 82 T^{2} + 3719 T^{4} - 118812 T^{6} + 3127679 T^{8} - 57997042 T^{10} + 594823321 T^{12} \))(\( 1 - 82 T^{2} + 3719 T^{4} - 118812 T^{6} + 3127679 T^{8} - 57997042 T^{10} + 594823321 T^{12} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 12 T + 86 T^{2} + 372 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 12 T + 86 T^{2} - 372 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} + 8 T^{3} + 1147 T^{4} - 3844 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} - 8 T^{3} + 1147 T^{4} + 3844 T^{5} + 29791 T^{6} )^{2} \))
$37$ (\( 1 - 38 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 54 T^{2} + 1369 T^{4} )^{2} \))(\( 1 + 4 T^{2} - 330 T^{4} + 5476 T^{6} + 1874161 T^{8} \))(\( ( 1 - 54 T^{2} + 1369 T^{4} )^{2} \))(\( 1 + 4 T^{2} - 330 T^{4} + 5476 T^{6} + 1874161 T^{8} \))(\( 1 - 98 T^{2} + 5239 T^{4} - 202172 T^{6} + 7172191 T^{8} - 183667778 T^{10} + 2565726409 T^{12} \))(\( 1 - 98 T^{2} + 5239 T^{4} - 202172 T^{6} + 7172191 T^{8} - 183667778 T^{10} + 2565726409 T^{12} \))
$41$ (\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 8 T + 78 T^{2} - 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 8 T + 78 T^{2} - 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 14 T + 159 T^{2} + 1140 T^{3} + 6519 T^{4} + 23534 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 14 T + 159 T^{2} + 1140 T^{3} + 6519 T^{4} + 23534 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 124 T^{2} + 7222 T^{4} - 229276 T^{6} + 3418801 T^{8} \))(\( 1 - 76 T^{2} + 3414 T^{4} - 140524 T^{6} + 3418801 T^{8} \))(\( 1 - 124 T^{2} + 7222 T^{4} - 229276 T^{6} + 3418801 T^{8} \))(\( 1 - 76 T^{2} + 3414 T^{4} - 140524 T^{6} + 3418801 T^{8} \))(\( 1 - 62 T^{2} + 2023 T^{4} - 11300 T^{6} + 3740527 T^{8} - 211965662 T^{10} + 6321363049 T^{12} \))(\( 1 - 62 T^{2} + 2023 T^{4} - 11300 T^{6} + 3740527 T^{8} - 211965662 T^{10} + 6321363049 T^{12} \))
$47$ (\( ( 1 + 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 110 T^{2} - 564 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 4 T + 86 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 12 T + 110 T^{2} + 564 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 86 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 117 T^{2} - 312 T^{3} + 5499 T^{4} - 8836 T^{5} + 103823 T^{6} )^{2} \))(\( ( 1 + 4 T + 117 T^{2} + 312 T^{3} + 5499 T^{4} + 8836 T^{5} + 103823 T^{6} )^{2} \))
$53$ (\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( ( 1 - 6 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 6 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 102 T^{2} + 2809 T^{4} )^{3} \))(\( ( 1 - 102 T^{2} + 2809 T^{4} )^{3} \))
$59$ (\( 1 - 82 T^{2} + 3481 T^{4} \))(\( 1 - 18 T^{2} + 3481 T^{4} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( 1 - 54 T^{2} + 3481 T^{4} \))(\( 1 - 18 T^{2} + 3481 T^{4} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( 1 - 54 T^{2} + 3481 T^{4} \))(\( 1 - 128 T^{2} + 10078 T^{4} - 445568 T^{6} + 12117361 T^{8} \))(\( 1 - 180 T^{2} + 14954 T^{4} - 626580 T^{6} + 12117361 T^{8} \))(\( 1 - 128 T^{2} + 10078 T^{4} - 445568 T^{6} + 12117361 T^{8} \))(\( 1 - 180 T^{2} + 14954 T^{4} - 626580 T^{6} + 12117361 T^{8} \))(\( 1 + 62 T^{2} + 10523 T^{4} + 399660 T^{6} + 36630563 T^{8} + 751276382 T^{10} + 42180533641 T^{12} \))(\( 1 + 62 T^{2} + 10523 T^{4} + 399660 T^{6} + 36630563 T^{8} + 751276382 T^{10} + 42180533641 T^{12} \))
$61$ (\( 1 - 106 T^{2} + 3721 T^{4} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 86 T^{2} + 3721 T^{4} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 106 T^{2} + 3721 T^{4} \))(\( 1 - 86 T^{2} + 3721 T^{4} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 72 T^{2} + 7118 T^{4} - 267912 T^{6} + 13845841 T^{8} \))(\( 1 + 52 T^{2} + 7530 T^{4} + 193492 T^{6} + 13845841 T^{8} \))(\( 1 - 72 T^{2} + 7118 T^{4} - 267912 T^{6} + 13845841 T^{8} \))(\( 1 + 52 T^{2} + 7530 T^{4} + 193492 T^{6} + 13845841 T^{8} \))(\( 1 - 110 T^{2} + 11659 T^{4} - 811724 T^{6} + 43383139 T^{8} - 1523042510 T^{10} + 51520374361 T^{12} \))(\( 1 - 110 T^{2} + 11659 T^{4} - 811724 T^{6} + 43383139 T^{8} - 1523042510 T^{10} + 51520374361 T^{12} \))
$67$ (\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( ( 1 - 118 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 44 T^{2} - 2826 T^{4} - 197516 T^{6} + 20151121 T^{8} \))(\( ( 1 - 118 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 44 T^{2} - 2826 T^{4} - 197516 T^{6} + 20151121 T^{8} \))(\( 1 - 238 T^{2} + 28631 T^{4} - 2300292 T^{6} + 128524559 T^{8} - 4795966798 T^{10} + 90458382169 T^{12} \))(\( 1 - 238 T^{2} + 28631 T^{4} - 2300292 T^{6} + 128524559 T^{8} - 4795966798 T^{10} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 78 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 8 T + 78 T^{2} - 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{6} \))(\( ( 1 + 71 T^{2} )^{6} \))
$73$ (\( ( 1 - 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{2} \))(\( ( 1 + 12 T + 102 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 4 T + 102 T^{2} + 292 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 12 T + 102 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 4 T + 102 T^{2} + 292 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 2 T + 151 T^{2} - 284 T^{3} + 11023 T^{4} - 10658 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 - 2 T + 151 T^{2} - 284 T^{3} + 11023 T^{4} - 10658 T^{5} + 389017 T^{6} )^{2} \))
$79$ (\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 16 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 16 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 12 T + 173 T^{2} - 1384 T^{3} + 13667 T^{4} - 74892 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 12 T + 173 T^{2} + 1384 T^{3} + 13667 T^{4} + 74892 T^{5} + 493039 T^{6} )^{2} \))
$83$ (\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 162 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 102 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 162 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 102 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 224 T^{2} + 25342 T^{4} - 1543136 T^{6} + 47458321 T^{8} \))(\( 1 - 164 T^{2} + 19530 T^{4} - 1129796 T^{6} + 47458321 T^{8} \))(\( 1 - 224 T^{2} + 25342 T^{4} - 1543136 T^{6} + 47458321 T^{8} \))(\( 1 - 164 T^{2} + 19530 T^{4} - 1129796 T^{6} + 47458321 T^{8} \))(\( 1 - 274 T^{2} + 39467 T^{4} - 3784500 T^{6} + 271888163 T^{8} - 13003579954 T^{10} + 326940373369 T^{12} \))(\( 1 - 274 T^{2} + 39467 T^{4} - 3784500 T^{6} + 271888163 T^{8} - 13003579954 T^{10} + 326940373369 T^{12} \))
$89$ (\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{4} \))(\( ( 1 - 4 T + 134 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{4} \))(\( ( 1 - 4 T + 134 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 2 T + 151 T^{2} - 60 T^{3} + 13439 T^{4} - 15842 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 - 2 T + 151 T^{2} - 60 T^{3} + 13439 T^{4} - 15842 T^{5} + 704969 T^{6} )^{2} \))
$97$ (\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 198 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 198 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 10 T + 103 T^{2} + 124 T^{3} + 9991 T^{4} + 94090 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 + 10 T + 103 T^{2} + 124 T^{3} + 9991 T^{4} + 94090 T^{5} + 912673 T^{6} )^{2} \))
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