Properties

Label 784.2.i
Level 784
Weight 2
Character orbit i
Rep. character \(\chi_{784}(177,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 36
Newform subspaces 14
Sturm bound 224
Trace bound 11

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Defining parameters

Level: \( N \) = \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 784.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(224\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(784, [\chi])\).

Total New Old
Modular forms 272 44 228
Cusp forms 176 36 140
Eisenstein series 96 8 88

Trace form

\( 36q - 3q^{3} + q^{5} - 13q^{9} + O(q^{10}) \) \( 36q - 3q^{3} + q^{5} - 13q^{9} + 3q^{11} + 4q^{13} + 14q^{15} + q^{17} - 5q^{19} + 9q^{23} - 13q^{25} + 18q^{27} + 16q^{29} + 13q^{31} - 9q^{33} - 7q^{37} - 26q^{39} + 12q^{41} - 16q^{43} - 10q^{45} + 3q^{47} - 47q^{51} - 11q^{53} - 22q^{55} - 54q^{57} - 27q^{59} - 3q^{61} + 18q^{65} + 15q^{67} - 2q^{69} + 64q^{71} + 13q^{73} + 4q^{75} + 33q^{79} + 22q^{81} + 40q^{83} - 50q^{85} + 26q^{87} + 13q^{89} + 21q^{93} - 33q^{95} + 12q^{97} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(784, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
784.2.i.a \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-1\) \(0\) \(q+(-3+3\zeta_{6})q^{3}-\zeta_{6}q^{5}-6\zeta_{6}q^{9}+\cdots\)
784.2.i.b \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(-4\) \(0\) \(q+(-2+2\zeta_{6})q^{3}-4\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
784.2.i.c \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(0\) \(0\) \(q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}-4q^{13}+\cdots\)
784.2.i.d \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(3\) \(0\) \(q+(-1+\zeta_{6})q^{3}+3\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
784.2.i.e \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(0\) \(q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
784.2.i.f \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+8\zeta_{6}q^{23}+\cdots\)
784.2.i.g \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(0\) \(q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
784.2.i.h \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(0\) \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+2\zeta_{6}q^{9}+(3+\cdots)q^{11}+\cdots\)
784.2.i.i \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(0\) \(0\) \(q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+4q^{13}+(6+\cdots)q^{17}+\cdots\)
784.2.i.j \(2\) \(6.260\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(4\) \(0\) \(q+(2-2\zeta_{6})q^{3}+4\zeta_{6}q^{5}-\zeta_{6}q^{9}+8q^{15}+\cdots\)
784.2.i.k \(4\) \(6.260\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{5}+5\beta _{2}q^{9}+(-4+\cdots)q^{11}+\cdots\)
784.2.i.l \(4\) \(6.260\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+5\beta _{2}q^{9}+\cdots\)
784.2.i.m \(4\) \(6.260\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(-2\beta _{1}-2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots\)
784.2.i.n \(4\) \(6.260\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}+(2\beta _{1}+2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(784, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \))(\( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \))(\( 1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8} \))(\( 1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8} \))
$5$ (\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} ) \))(\( 1 - 8 T^{2} + 39 T^{4} - 200 T^{6} + 625 T^{8} \))(\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} ) \))(\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} ) \))
$7$ 1
$11$ (\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( ( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 18 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 8 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 - 6 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \))(\( 1 - 32 T^{2} + 735 T^{4} - 9248 T^{6} + 83521 T^{8} \))(\( 1 - 32 T^{2} + 735 T^{4} - 9248 T^{6} + 83521 T^{8} \))(\( 1 - 32 T^{2} + 735 T^{4} - 9248 T^{6} + 83521 T^{8} \))
$19$ (\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( 1 - 19 T^{2} + 361 T^{4} \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( 1 + 12 T^{2} - 217 T^{4} + 4332 T^{6} + 130321 T^{8} \))(\( 1 - 20 T^{2} + 39 T^{4} - 7220 T^{6} + 130321 T^{8} \))
$23$ (\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{4} \))(\( ( 1 - 2 T + 29 T^{2} )^{4} \))(\( ( 1 + 6 T + 29 T^{2} )^{4} \))
$31$ (\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 - 30 T^{2} - 61 T^{4} - 28830 T^{6} + 923521 T^{8} \))(\( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \))(\( 1 + 10 T^{2} - 861 T^{4} + 9610 T^{6} + 923521 T^{8} \))(\( 1 - 54 T^{2} + 1955 T^{4} - 51894 T^{6} + 923521 T^{8} \))
$37$ (\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \))(\( ( 1 - T + 37 T^{2} )^{2}( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2}( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 32 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 16 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 80 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{4} \))(\( ( 1 - 4 T + 43 T^{2} )^{4} \))(\( ( 1 + 2 T + 43 T^{2} )^{4} \))(\( ( 1 + 10 T + 43 T^{2} )^{4} \))
$47$ (\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 - 4 T - 31 T^{2} - 188 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 + 5 T - 22 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 4 T - 31 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 - 62 T^{2} + 1635 T^{4} - 136958 T^{6} + 4879681 T^{8} \))(\( 1 - 62 T^{2} + 1635 T^{4} - 136958 T^{6} + 4879681 T^{8} \))(\( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \))(\( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \))(\( ( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 10 T + 47 T^{2} + 530 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 110 T^{2} + 8619 T^{4} - 382910 T^{6} + 12117361 T^{8} \))(\( 1 + 82 T^{2} + 3243 T^{4} + 285442 T^{6} + 12117361 T^{8} \))(\( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} \))(\( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 + 78 T^{2} + 2363 T^{4} + 290238 T^{6} + 13845841 T^{8} \))(\( 1 - 72 T^{2} + 1463 T^{4} - 267912 T^{6} + 13845841 T^{8} \))(\( 1 - 114 T^{2} + 9275 T^{4} - 424194 T^{6} + 13845841 T^{8} \))(\( 1 - 50 T^{2} - 1221 T^{4} - 186050 T^{6} + 13845841 T^{8} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 14 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( ( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 67 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 - 12 T + 71 T^{2} )^{4} \))(\( ( 1 - 12 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 73 T^{2} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( ( 1 - 73 T^{2} + 5329 T^{4} )^{2} \))(\( 1 - 96 T^{2} + 3887 T^{4} - 511584 T^{6} + 28398241 T^{8} \))(\( 1 - 144 T^{2} + 15407 T^{4} - 767376 T^{6} + 28398241 T^{8} \))(\( 1 - 48 T^{2} - 3025 T^{4} - 255792 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( ( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))
$83$ (\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 8 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 8 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 34 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 68 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 164 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \))(\( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 128 T^{2} + 8463 T^{4} - 1013888 T^{6} + 62742241 T^{8} \))(\( 1 - 128 T^{2} + 8463 T^{4} - 1013888 T^{6} + 62742241 T^{8} \))(\( 1 - 160 T^{2} + 17679 T^{4} - 1267360 T^{6} + 62742241 T^{8} \))
$97$ (\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 162 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 192 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 96 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 32 T^{2} + 9409 T^{4} )^{2} \))
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