Properties

Label 1470.2.g
Level $1470$
Weight $2$
Character orbit 1470.g
Rep. character $\chi_{1470}(589,\cdot)$
Character field $\Q$
Dimension $42$
Newform subspaces $11$
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.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(672\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1470, [\chi])\).

Total New Old
Modular forms 368 42 326
Cusp forms 304 42 262
Eisenstein series 64 0 64

Trace form

\( 42q - 42q^{4} + 4q^{5} - 2q^{6} - 42q^{9} + O(q^{10}) \) \( 42q - 42q^{4} + 4q^{5} - 2q^{6} - 42q^{9} + 2q^{10} - 4q^{11} + 6q^{15} + 42q^{16} - 16q^{19} - 4q^{20} + 2q^{24} + 2q^{25} - 4q^{26} + 16q^{29} + 4q^{30} + 8q^{31} + 20q^{34} + 42q^{36} - 4q^{39} - 2q^{40} - 20q^{41} + 4q^{44} - 4q^{45} + 16q^{50} + 12q^{51} + 2q^{54} - 12q^{59} - 6q^{60} - 4q^{61} - 42q^{64} + 76q^{65} - 12q^{66} - 8q^{69} - 8q^{71} - 68q^{74} + 8q^{75} + 16q^{76} - 8q^{79} + 4q^{80} + 42q^{81} + 68q^{85} - 24q^{86} - 20q^{89} - 2q^{90} + 40q^{94} + 8q^{95} - 2q^{96} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1470, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1470.2.g.a \(2\) \(11.738\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) \(q-iq^{2}+iq^{3}-q^{4}+(-2-i)q^{5}+\cdots\)
1470.2.g.b \(2\) \(11.738\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q-iq^{2}-iq^{3}-q^{4}+(-1-2i)q^{5}+\cdots\)
1470.2.g.c \(2\) \(11.738\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}+iq^{3}-q^{4}+(-1-2i)q^{5}+\cdots\)
1470.2.g.d \(2\) \(11.738\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q-iq^{2}+iq^{3}-q^{4}+(1-2i)q^{5}+q^{6}+\cdots\)
1470.2.g.e \(2\) \(11.738\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+iq^{2}-iq^{3}-q^{4}+(1-2i)q^{5}+q^{6}+\cdots\)
1470.2.g.f \(2\) \(11.738\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q-iq^{2}-iq^{3}-q^{4}+(2+i)q^{5}-q^{6}+\cdots\)
1470.2.g.g \(2\) \(11.738\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q+iq^{2}+iq^{3}-q^{4}+(2-i)q^{5}-q^{6}+\cdots\)
1470.2.g.h \(6\) \(11.738\) 6.0.29160000.2 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}-\beta _{3}q^{3}-q^{4}-\beta _{2}q^{5}-q^{6}+\cdots\)
1470.2.g.i \(6\) \(11.738\) 6.0.29160000.2 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}+\beta _{3}q^{3}-q^{4}+\beta _{2}q^{5}+q^{6}+\cdots\)
1470.2.g.j \(8\) \(11.738\) 8.0.1698758656.6 None \(0\) \(0\) \(-4\) \(0\) \(q-\beta _{5}q^{2}-\beta _{5}q^{3}-q^{4}+(-1+\beta _{7})q^{5}+\cdots\)
1470.2.g.k \(8\) \(11.738\) 8.0.1698758656.6 None \(0\) \(0\) \(4\) \(0\) \(q-\beta _{5}q^{2}+\beta _{5}q^{3}-q^{4}+(1-\beta _{7})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1470, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1470, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(490, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(735, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))
$3$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 20 T^{3} + 125 T^{6} \))(\( 1 - 20 T^{3} + 125 T^{6} \))(\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 20 T^{5} + 150 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 20 T^{5} + 150 T^{6} - 500 T^{7} + 625 T^{8} \))
$7$ 1
$11$ (\( ( 1 + 5 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 5 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 6 T^{2} - 17 T^{3} + 66 T^{4} - 363 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 - 3 T + 6 T^{2} - 17 T^{3} + 66 T^{4} - 363 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 + 26 T^{2} - 16 T^{3} + 362 T^{4} - 176 T^{5} + 3146 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 26 T^{2} - 16 T^{3} + 362 T^{4} - 176 T^{5} + 3146 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( 1 - 25 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 45 T^{2} + 1107 T^{4} - 17390 T^{6} + 187083 T^{8} - 1285245 T^{10} + 4826809 T^{12} \))(\( 1 - 45 T^{2} + 1107 T^{4} - 17390 T^{6} + 187083 T^{8} - 1285245 T^{10} + 4826809 T^{12} \))(\( 1 - 52 T^{2} + 1320 T^{4} - 23228 T^{6} + 329390 T^{8} - 3925532 T^{10} + 37700520 T^{12} - 250994068 T^{14} + 815730721 T^{16} \))(\( 1 - 52 T^{2} + 1320 T^{4} - 23228 T^{6} + 329390 T^{8} - 3925532 T^{10} + 37700520 T^{12} - 250994068 T^{14} + 815730721 T^{16} \))
$17$ (\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 - 18 T^{2} + 289 T^{4} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 2 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 + 30 T^{2} + 927 T^{4} + 15300 T^{6} + 267903 T^{8} + 2505630 T^{10} + 24137569 T^{12} \))(\( 1 + 30 T^{2} + 927 T^{4} + 15300 T^{6} + 267903 T^{8} + 2505630 T^{10} + 24137569 T^{12} \))(\( 1 - 56 T^{2} + 1268 T^{4} - 11176 T^{6} + 45670 T^{8} - 3229864 T^{10} + 105904628 T^{12} - 1351703864 T^{14} + 6975757441 T^{16} \))(\( 1 - 56 T^{2} + 1268 T^{4} - 11176 T^{6} + 45670 T^{8} - 3229864 T^{10} + 105904628 T^{12} - 1351703864 T^{14} + 6975757441 T^{16} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 6 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 3 T + 45 T^{2} - 90 T^{3} + 855 T^{4} - 1083 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 3 T + 45 T^{2} + 90 T^{3} + 855 T^{4} + 1083 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 - 12 T + 88 T^{2} - 460 T^{3} + 2174 T^{4} - 8740 T^{5} + 31768 T^{6} - 82308 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 + 12 T + 88 T^{2} + 460 T^{3} + 2174 T^{4} + 8740 T^{5} + 31768 T^{6} + 82308 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 37 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 37 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 81 T^{2} + 2979 T^{4} - 75606 T^{6} + 1575891 T^{8} - 22667121 T^{10} + 148035889 T^{12} \))(\( 1 - 81 T^{2} + 2979 T^{4} - 75606 T^{6} + 1575891 T^{8} - 22667121 T^{10} + 148035889 T^{12} \))(\( 1 - 96 T^{2} + 4220 T^{4} - 116640 T^{6} + 2693446 T^{8} - 61702560 T^{10} + 1180929020 T^{12} - 14211445344 T^{14} + 78310985281 T^{16} \))(\( 1 - 96 T^{2} + 4220 T^{4} - 116640 T^{6} + 2693446 T^{8} - 61702560 T^{10} + 1180929020 T^{12} - 14211445344 T^{14} + 78310985281 T^{16} \))
$29$ (\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 12 T + 120 T^{2} + 720 T^{3} + 3480 T^{4} + 10092 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 + 12 T + 120 T^{2} + 720 T^{3} + 3480 T^{4} + 10092 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 - 8 T + 50 T^{2} - 360 T^{3} + 2786 T^{4} - 10440 T^{5} + 42050 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 - 8 T + 50 T^{2} - 360 T^{3} + 2786 T^{4} - 10440 T^{5} + 42050 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 78 T^{2} + 10 T^{3} + 2418 T^{4} + 29791 T^{6} )^{2} \))(\( ( 1 + 78 T^{2} - 10 T^{3} + 2418 T^{4} + 29791 T^{6} )^{2} \))(\( ( 1 + 16 T + 202 T^{2} + 1616 T^{3} + 10666 T^{4} + 50096 T^{5} + 194122 T^{6} + 476656 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 - 16 T + 202 T^{2} - 1616 T^{3} + 10666 T^{4} - 50096 T^{5} + 194122 T^{6} - 476656 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( 1 - 49 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( 1 - 49 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 165 T^{2} + 12387 T^{4} - 566110 T^{6} + 16957803 T^{8} - 309236565 T^{10} + 2565726409 T^{12} \))(\( 1 - 165 T^{2} + 12387 T^{4} - 566110 T^{6} + 16957803 T^{8} - 309236565 T^{10} + 2565726409 T^{12} \))(\( 1 - 148 T^{2} + 9560 T^{4} - 370620 T^{6} + 12535566 T^{8} - 507378780 T^{10} + 17916979160 T^{12} - 379727508532 T^{14} + 3512479453921 T^{16} \))(\( 1 - 148 T^{2} + 9560 T^{4} - 370620 T^{6} + 12535566 T^{8} - 507378780 T^{10} + 17916979160 T^{12} - 379727508532 T^{14} + 3512479453921 T^{16} \))
$41$ (\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 75 T^{2} - 610 T^{3} + 3075 T^{4} - 15129 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 9 T + 75 T^{2} + 610 T^{3} + 3075 T^{4} + 15129 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 12 T + 190 T^{2} + 1356 T^{3} + 11842 T^{4} + 55596 T^{5} + 319390 T^{6} + 827052 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 12 T + 190 T^{2} - 1356 T^{3} + 11842 T^{4} - 55596 T^{5} + 319390 T^{6} - 827052 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{3} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{3} \))(\( 1 - 108 T^{2} + 7400 T^{4} - 450340 T^{6} + 22241646 T^{8} - 832678660 T^{10} + 25299127400 T^{12} - 682707209292 T^{14} + 11688200277601 T^{16} \))(\( 1 - 108 T^{2} + 7400 T^{4} - 450340 T^{6} + 22241646 T^{8} - 832678660 T^{10} + 25299127400 T^{12} - 682707209292 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 + 75 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 - 58 T^{2} + 2209 T^{4} \))(\( 1 - 58 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 + 75 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 - 9 T^{2} + 1659 T^{4} - 160614 T^{6} + 3664731 T^{8} - 43917129 T^{10} + 10779215329 T^{12} \))(\( 1 - 9 T^{2} + 1659 T^{4} - 160614 T^{6} + 3664731 T^{8} - 43917129 T^{10} + 10779215329 T^{12} \))(\( 1 - 172 T^{2} + 15464 T^{4} - 948164 T^{6} + 47811790 T^{8} - 2094494276 T^{10} + 75459386984 T^{12} - 1854025036588 T^{14} + 23811286661761 T^{16} \))(\( 1 - 172 T^{2} + 15464 T^{4} - 948164 T^{6} + 47811790 T^{8} - 2094494276 T^{10} + 75459386984 T^{12} - 1854025036588 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 - 105 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( 1 - 105 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 51 T^{2} + 8814 T^{4} - 284551 T^{6} + 24758526 T^{8} - 402414531 T^{10} + 22164361129 T^{12} \))(\( 1 - 51 T^{2} + 8814 T^{4} - 284551 T^{6} + 24758526 T^{8} - 402414531 T^{10} + 22164361129 T^{12} \))(\( 1 - 240 T^{2} + 25532 T^{4} - 1715600 T^{6} + 94098918 T^{8} - 4819120400 T^{10} + 201459760892 T^{12} - 5319446670960 T^{14} + 62259690411361 T^{16} \))(\( 1 - 240 T^{2} + 25532 T^{4} - 1715600 T^{6} + 94098918 T^{8} - 4819120400 T^{10} + 201459760892 T^{12} - 5319446670960 T^{14} + 62259690411361 T^{16} \))
$59$ (\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 8 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 8 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( ( 1 - 10 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 150 T^{2} + 980 T^{3} + 8850 T^{4} + 41772 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 - 12 T + 150 T^{2} - 980 T^{3} + 8850 T^{4} - 41772 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 - 20 T + 288 T^{2} - 2804 T^{3} + 24014 T^{4} - 165436 T^{5} + 1002528 T^{6} - 4107580 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 20 T + 288 T^{2} + 2804 T^{3} + 24014 T^{4} + 165436 T^{5} + 1002528 T^{6} + 4107580 T^{7} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 6 T + 75 T^{2} + 20 T^{3} + 4575 T^{4} + 22326 T^{5} + 226981 T^{6} )^{2} \))(\( ( 1 - 6 T + 75 T^{2} - 20 T^{3} + 4575 T^{4} - 22326 T^{5} + 226981 T^{6} )^{2} \))(\( ( 1 + 12 T + 118 T^{2} + 684 T^{3} + 6306 T^{4} + 41724 T^{5} + 439078 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 12 T + 118 T^{2} - 684 T^{3} + 6306 T^{4} - 41724 T^{5} + 439078 T^{6} - 2723772 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 - 98 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 - 98 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 150 T^{2} + 16167 T^{4} - 1267540 T^{6} + 72573663 T^{8} - 3022668150 T^{10} + 90458382169 T^{12} \))(\( 1 - 150 T^{2} + 16167 T^{4} - 1267540 T^{6} + 72573663 T^{8} - 3022668150 T^{10} + 90458382169 T^{12} \))(\( 1 - 492 T^{2} + 108392 T^{4} - 13982820 T^{6} + 1156204078 T^{8} - 62768878980 T^{10} + 2184220307432 T^{12} - 44505524027148 T^{14} + 406067677556641 T^{16} \))(\( 1 - 492 T^{2} + 108392 T^{4} - 13982820 T^{6} + 1156204078 T^{8} - 62768878980 T^{10} + 2184220307432 T^{12} - 44505524027148 T^{14} + 406067677556641 T^{16} \))
$71$ (\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 14 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{6} \))(\( ( 1 - 6 T + 71 T^{2} )^{6} \))(\( ( 1 + 20 T + 248 T^{2} + 2788 T^{3} + 28078 T^{4} + 197948 T^{5} + 1250168 T^{6} + 7158220 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 20 T + 248 T^{2} + 2788 T^{3} + 28078 T^{4} + 197948 T^{5} + 1250168 T^{6} + 7158220 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 + 50 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( ( 1 - 130 T^{2} + 5329 T^{4} )^{3} \))(\( ( 1 - 130 T^{2} + 5329 T^{4} )^{3} \))(\( 1 - 268 T^{2} + 30568 T^{4} - 2051940 T^{6} + 125567630 T^{8} - 10934788260 T^{10} + 868077430888 T^{12} - 40557572645452 T^{14} + 806460091894081 T^{16} \))(\( 1 - 268 T^{2} + 30568 T^{4} - 2051940 T^{6} + 125567630 T^{8} - 10934788260 T^{10} + 868077430888 T^{12} - 40557572645452 T^{14} + 806460091894081 T^{16} \))
$79$ (\( ( 1 - 14 T + 79 T^{2} )^{2} \))(\( ( 1 - 12 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 14 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 24 T + 414 T^{2} + 4194 T^{3} + 32706 T^{4} + 149784 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 24 T + 414 T^{2} + 4194 T^{3} + 32706 T^{4} + 149784 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 - 8 T + 140 T^{2} - 104 T^{3} + 6054 T^{4} - 8216 T^{5} + 873740 T^{6} - 3944312 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 8 T + 140 T^{2} - 104 T^{3} + 6054 T^{4} - 8216 T^{5} + 873740 T^{6} - 3944312 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 66 T^{2} + 6889 T^{4} \))(\( 1 - 102 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 + 90 T^{2} + 6889 T^{4} \))(\( 1 - 66 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 300 T^{2} + 40392 T^{4} - 3707850 T^{6} + 278260488 T^{8} - 14237496300 T^{10} + 326940373369 T^{12} \))(\( 1 - 300 T^{2} + 40392 T^{4} - 3707850 T^{6} + 278260488 T^{8} - 14237496300 T^{10} + 326940373369 T^{12} \))(\( 1 - 424 T^{2} + 80348 T^{4} - 9509080 T^{6} + 861447654 T^{8} - 65508052120 T^{10} + 3813181175708 T^{12} - 138622718308456 T^{14} + 2252292232139041 T^{16} \))(\( 1 - 424 T^{2} + 80348 T^{4} - 9509080 T^{6} + 861447654 T^{8} - 65508052120 T^{10} + 3813181175708 T^{12} - 138622718308456 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 14 T + 89 T^{2} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 159 T^{2} + 676 T^{3} + 14151 T^{4} + 47526 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 - 6 T + 159 T^{2} - 676 T^{3} + 14151 T^{4} - 47526 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 - 44 T + 1014 T^{2} - 15436 T^{3} + 169698 T^{4} - 1373804 T^{5} + 8031894 T^{6} - 31018636 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 44 T + 1014 T^{2} + 15436 T^{3} + 169698 T^{4} + 1373804 T^{5} + 8031894 T^{6} + 31018636 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( 1 - 94 T^{2} + 9409 T^{4} \))(\( 1 + 2 T^{2} + 9409 T^{4} \))(\( 1 + 2 T^{2} + 9409 T^{4} \))(\( 1 - 94 T^{2} + 9409 T^{4} \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( 1 - 240 T^{2} + 16752 T^{4} - 546370 T^{6} + 157619568 T^{8} - 21247027440 T^{10} + 832972004929 T^{12} \))(\( 1 - 240 T^{2} + 16752 T^{4} - 546370 T^{6} + 157619568 T^{8} - 21247027440 T^{10} + 832972004929 T^{12} \))(\( 1 - 348 T^{2} + 81160 T^{4} - 12144532 T^{6} + 1394593550 T^{8} - 114267901588 T^{10} + 7185036445960 T^{12} - 289874257715292 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 348 T^{2} + 81160 T^{4} - 12144532 T^{6} + 1394593550 T^{8} - 114267901588 T^{10} + 7185036445960 T^{12} - 289874257715292 T^{14} + 7837433594376961 T^{16} \))
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