Properties

Label 3150.2.bp
Level 3150
Weight 2
Character orbit bp
Rep. character \(\chi_{3150}(899,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 96
Newform subspaces 8
Sturm bound 1440
Trace bound 11

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Defining parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.bp (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(1440\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3150, [\chi])\).

Total New Old
Modular forms 1536 96 1440
Cusp forms 1344 96 1248
Eisenstein series 192 0 192

Trace form

\( 96q - 48q^{4} + O(q^{10}) \) \( 96q - 48q^{4} - 48q^{16} + 24q^{19} - 24q^{31} + 16q^{46} + 64q^{49} + 24q^{61} + 96q^{64} + 32q^{79} + 8q^{91} + 144q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3150, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3150.2.bp.a \(8\) \(25.153\) \(\Q(\zeta_{24})\) None \(-4\) \(0\) \(0\) \(0\) \(q+(-1+\zeta_{24}^{4})q^{2}-\zeta_{24}^{4}q^{4}+(3\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
3150.2.bp.b \(8\) \(25.153\) \(\Q(\zeta_{24})\) None \(-4\) \(0\) \(0\) \(0\) \(q+(-1+\zeta_{24}^{2})q^{2}-\zeta_{24}^{2}q^{4}+(-\zeta_{24}+\cdots)q^{7}+\cdots\)
3150.2.bp.c \(8\) \(25.153\) \(\Q(\zeta_{24})\) None \(-4\) \(0\) \(0\) \(0\) \(q+(-1+\zeta_{24}^{4})q^{2}-\zeta_{24}^{4}q^{4}+(3\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
3150.2.bp.d \(8\) \(25.153\) \(\Q(\zeta_{24})\) None \(4\) \(0\) \(0\) \(0\) \(q+(1-\zeta_{24}^{4})q^{2}-\zeta_{24}^{4}q^{4}+(-3\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
3150.2.bp.e \(8\) \(25.153\) \(\Q(\zeta_{24})\) None \(4\) \(0\) \(0\) \(0\) \(q+(1-\zeta_{24}^{2})q^{2}-\zeta_{24}^{2}q^{4}+(\zeta_{24}-\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
3150.2.bp.f \(8\) \(25.153\) \(\Q(\zeta_{24})\) None \(4\) \(0\) \(0\) \(0\) \(q+(1-\zeta_{24}^{4})q^{2}-\zeta_{24}^{4}q^{4}+(-3\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
3150.2.bp.g \(24\) \(25.153\) None \(-12\) \(0\) \(0\) \(0\)
3150.2.bp.h \(24\) \(25.153\) None \(12\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(3150, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3150, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(630, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1050, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1575, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$7$ (\( 1 - 94 T^{4} + 2401 T^{8} \))(\( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \))(\( 1 - 94 T^{4} + 2401 T^{8} \))(\( 1 - 94 T^{4} + 2401 T^{8} \))(\( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \))(\( 1 - 94 T^{4} + 2401 T^{8} \))
$11$ (\( 1 + 24 T + 304 T^{2} + 2688 T^{3} + 18481 T^{4} + 104232 T^{5} + 496816 T^{6} + 2036016 T^{7} + 7239280 T^{8} + 22396176 T^{9} + 60114736 T^{10} + 138732792 T^{11} + 270580321 T^{12} + 432905088 T^{13} + 538554544 T^{14} + 467692104 T^{15} + 214358881 T^{16} \))(\( ( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2} \))(\( 1 - 24 T + 304 T^{2} - 2688 T^{3} + 18481 T^{4} - 104232 T^{5} + 496816 T^{6} - 2036016 T^{7} + 7239280 T^{8} - 22396176 T^{9} + 60114736 T^{10} - 138732792 T^{11} + 270580321 T^{12} - 432905088 T^{13} + 538554544 T^{14} - 467692104 T^{15} + 214358881 T^{16} \))(\( 1 + 24 T + 304 T^{2} + 2688 T^{3} + 18481 T^{4} + 104232 T^{5} + 496816 T^{6} + 2036016 T^{7} + 7239280 T^{8} + 22396176 T^{9} + 60114736 T^{10} + 138732792 T^{11} + 270580321 T^{12} + 432905088 T^{13} + 538554544 T^{14} + 467692104 T^{15} + 214358881 T^{16} \))(\( ( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2} \))(\( 1 - 24 T + 304 T^{2} - 2688 T^{3} + 18481 T^{4} - 104232 T^{5} + 496816 T^{6} - 2036016 T^{7} + 7239280 T^{8} - 22396176 T^{9} + 60114736 T^{10} - 138732792 T^{11} + 270580321 T^{12} - 432905088 T^{13} + 538554544 T^{14} - 467692104 T^{15} + 214358881 T^{16} \))
$13$ (\( ( 1 - 8 T + 72 T^{2} - 328 T^{3} + 1535 T^{4} - 4264 T^{5} + 12168 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 + 8 T + 72 T^{2} + 328 T^{3} + 1535 T^{4} + 4264 T^{5} + 12168 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 8 T + 72 T^{2} + 328 T^{3} + 1535 T^{4} + 4264 T^{5} + 12168 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 - 8 T + 72 T^{2} - 328 T^{3} + 1535 T^{4} - 4264 T^{5} + 12168 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 24 T + 316 T^{2} + 2976 T^{3} + 22330 T^{4} + 141240 T^{5} + 776752 T^{6} + 3781944 T^{7} + 16463059 T^{8} + 64293048 T^{9} + 224481328 T^{10} + 693912120 T^{11} + 1865023930 T^{12} + 4225494432 T^{13} + 7627471804 T^{14} + 9848128152 T^{15} + 6975757441 T^{16} \))(\( ( 1 - 12 T + 88 T^{2} - 480 T^{3} + 2127 T^{4} - 8160 T^{5} + 25432 T^{6} - 58956 T^{7} + 83521 T^{8} )^{2} \))(\( 1 + 24 T + 316 T^{2} + 2976 T^{3} + 22330 T^{4} + 141240 T^{5} + 776752 T^{6} + 3781944 T^{7} + 16463059 T^{8} + 64293048 T^{9} + 224481328 T^{10} + 693912120 T^{11} + 1865023930 T^{12} + 4225494432 T^{13} + 7627471804 T^{14} + 9848128152 T^{15} + 6975757441 T^{16} \))(\( 1 - 24 T + 316 T^{2} - 2976 T^{3} + 22330 T^{4} - 141240 T^{5} + 776752 T^{6} - 3781944 T^{7} + 16463059 T^{8} - 64293048 T^{9} + 224481328 T^{10} - 693912120 T^{11} + 1865023930 T^{12} - 4225494432 T^{13} + 7627471804 T^{14} - 9848128152 T^{15} + 6975757441 T^{16} \))(\( ( 1 + 12 T + 88 T^{2} + 480 T^{3} + 2127 T^{4} + 8160 T^{5} + 25432 T^{6} + 58956 T^{7} + 83521 T^{8} )^{2} \))(\( 1 - 24 T + 316 T^{2} - 2976 T^{3} + 22330 T^{4} - 141240 T^{5} + 776752 T^{6} - 3781944 T^{7} + 16463059 T^{8} - 64293048 T^{9} + 224481328 T^{10} - 693912120 T^{11} + 1865023930 T^{12} - 4225494432 T^{13} + 7627471804 T^{14} - 9848128152 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 58 T^{2} + 1881 T^{4} + 1872 T^{5} + 44906 T^{6} + 71184 T^{7} + 869156 T^{8} + 1352496 T^{9} + 16211066 T^{10} + 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16} \))(\( ( 1 - 12 T + 92 T^{2} - 528 T^{3} + 2487 T^{4} - 10032 T^{5} + 33212 T^{6} - 82308 T^{7} + 130321 T^{8} )^{2} \))(\( 1 + 58 T^{2} + 1881 T^{4} + 1872 T^{5} + 44906 T^{6} + 71184 T^{7} + 869156 T^{8} + 1352496 T^{9} + 16211066 T^{10} + 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16} \))(\( 1 + 58 T^{2} + 1881 T^{4} + 1872 T^{5} + 44906 T^{6} + 71184 T^{7} + 869156 T^{8} + 1352496 T^{9} + 16211066 T^{10} + 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16} \))(\( ( 1 - 12 T + 92 T^{2} - 528 T^{3} + 2487 T^{4} - 10032 T^{5} + 33212 T^{6} - 82308 T^{7} + 130321 T^{8} )^{2} \))(\( 1 + 58 T^{2} + 1881 T^{4} + 1872 T^{5} + 44906 T^{6} + 71184 T^{7} + 869156 T^{8} + 1352496 T^{9} + 16211066 T^{10} + 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16} \))
$23$ (\( ( 1 - 4 T - 31 T^{2} - 4 T^{3} + 1312 T^{4} - 92 T^{5} - 16399 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 28 T^{2} + 255 T^{4} - 14812 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 4 T - 31 T^{2} - 4 T^{3} + 1312 T^{4} - 92 T^{5} - 16399 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 4 T - 31 T^{2} + 4 T^{3} + 1312 T^{4} + 92 T^{5} - 16399 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 28 T^{2} + 255 T^{4} - 14812 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 4 T - 31 T^{2} + 4 T^{3} + 1312 T^{4} + 92 T^{5} - 16399 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16} \))(\( ( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 2046 T^{5} + 22103 T^{6} + 178746 T^{7} + 923521 T^{8} )^{2} \))(\( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16} \))(\( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16} \))(\( ( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 2046 T^{5} + 22103 T^{6} + 178746 T^{7} + 923521 T^{8} )^{2} \))(\( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 + 72 T^{2} + 1441 T^{4} + 2520 T^{5} + 74088 T^{6} + 269496 T^{7} + 5102928 T^{8} + 9971352 T^{9} + 101426472 T^{10} + 127645560 T^{11} + 2700666001 T^{12} + 184732301448 T^{14} + 3512479453921 T^{16} \))(\( 1 + 80 T^{2} + 3214 T^{4} + 35840 T^{6} - 485165 T^{8} + 49064960 T^{10} + 6023553454 T^{12} + 205258112720 T^{14} + 3512479453921 T^{16} \))(\( 1 + 72 T^{2} + 1441 T^{4} - 2520 T^{5} + 74088 T^{6} - 269496 T^{7} + 5102928 T^{8} - 9971352 T^{9} + 101426472 T^{10} - 127645560 T^{11} + 2700666001 T^{12} + 184732301448 T^{14} + 3512479453921 T^{16} \))(\( 1 + 72 T^{2} + 1441 T^{4} - 2520 T^{5} + 74088 T^{6} - 269496 T^{7} + 5102928 T^{8} - 9971352 T^{9} + 101426472 T^{10} - 127645560 T^{11} + 2700666001 T^{12} + 184732301448 T^{14} + 3512479453921 T^{16} \))(\( 1 + 80 T^{2} + 3214 T^{4} + 35840 T^{6} - 485165 T^{8} + 49064960 T^{10} + 6023553454 T^{12} + 205258112720 T^{14} + 3512479453921 T^{16} \))(\( 1 + 72 T^{2} + 1441 T^{4} + 2520 T^{5} + 74088 T^{6} + 269496 T^{7} + 5102928 T^{8} + 9971352 T^{9} + 101426472 T^{10} + 127645560 T^{11} + 2700666001 T^{12} + 184732301448 T^{14} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 16 T + 232 T^{2} + 2000 T^{3} + 15591 T^{4} + 82000 T^{5} + 389992 T^{6} + 1102736 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 20 T^{2} - 1146 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 82000 T^{5} + 389992 T^{6} - 1102736 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 16 T + 232 T^{2} + 2000 T^{3} + 15591 T^{4} + 82000 T^{5} + 389992 T^{6} + 1102736 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 20 T^{2} - 1146 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 82000 T^{5} + 389992 T^{6} - 1102736 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 296 T^{2} + 39900 T^{4} - 3207640 T^{6} + 168538406 T^{8} - 5930926360 T^{10} + 136410159900 T^{12} - 1871123462504 T^{14} + 11688200277601 T^{16} \))(\( ( 1 - 104 T^{2} + 5250 T^{4} - 192296 T^{6} + 3418801 T^{8} )^{2} \))(\( 1 - 296 T^{2} + 39900 T^{4} - 3207640 T^{6} + 168538406 T^{8} - 5930926360 T^{10} + 136410159900 T^{12} - 1871123462504 T^{14} + 11688200277601 T^{16} \))(\( 1 - 296 T^{2} + 39900 T^{4} - 3207640 T^{6} + 168538406 T^{8} - 5930926360 T^{10} + 136410159900 T^{12} - 1871123462504 T^{14} + 11688200277601 T^{16} \))(\( ( 1 - 104 T^{2} + 5250 T^{4} - 192296 T^{6} + 3418801 T^{8} )^{2} \))(\( 1 - 296 T^{2} + 39900 T^{4} - 3207640 T^{6} + 168538406 T^{8} - 5930926360 T^{10} + 136410159900 T^{12} - 1871123462504 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 14001 T^{4} + 79344 T^{5} + 532582 T^{6} + 2135988 T^{7} + 16324772 T^{8} + 100391436 T^{9} + 1176473638 T^{10} + 8237732112 T^{11} + 68320413681 T^{12} + 368786771256 T^{13} + 1961817189878 T^{14} + 6079477445556 T^{15} + 23811286661761 T^{16} \))(\( ( 1 + 12 T + 148 T^{2} + 1200 T^{3} + 10047 T^{4} + 56400 T^{5} + 326932 T^{6} + 1245876 T^{7} + 4879681 T^{8} )^{2} \))(\( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 14001 T^{4} + 79344 T^{5} + 532582 T^{6} + 2135988 T^{7} + 16324772 T^{8} + 100391436 T^{9} + 1176473638 T^{10} + 8237732112 T^{11} + 68320413681 T^{12} + 368786771256 T^{13} + 1961817189878 T^{14} + 6079477445556 T^{15} + 23811286661761 T^{16} \))(\( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 14001 T^{4} - 79344 T^{5} + 532582 T^{6} - 2135988 T^{7} + 16324772 T^{8} - 100391436 T^{9} + 1176473638 T^{10} - 8237732112 T^{11} + 68320413681 T^{12} - 368786771256 T^{13} + 1961817189878 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16} \))(\( ( 1 - 12 T + 148 T^{2} - 1200 T^{3} + 10047 T^{4} - 56400 T^{5} + 326932 T^{6} - 1245876 T^{7} + 4879681 T^{8} )^{2} \))(\( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 14001 T^{4} - 79344 T^{5} + 532582 T^{6} - 2135988 T^{7} + 16324772 T^{8} - 100391436 T^{9} + 1176473638 T^{10} - 8237732112 T^{11} + 68320413681 T^{12} - 368786771256 T^{13} + 1961817189878 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 4 T - 66 T^{2} - 984 T^{3} - 631 T^{4} + 50712 T^{5} + 363998 T^{6} - 1081556 T^{7} - 22439580 T^{8} - 57322468 T^{9} + 1022470382 T^{10} + 7549850424 T^{11} - 4978893511 T^{12} - 411504365112 T^{13} - 1462847834514 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} \))(\( ( 1 - 6 T - 61 T^{2} + 54 T^{3} + 4692 T^{4} + 2862 T^{5} - 171349 T^{6} - 893262 T^{7} + 7890481 T^{8} )^{2} \))(\( 1 + 4 T - 66 T^{2} - 984 T^{3} - 631 T^{4} + 50712 T^{5} + 363998 T^{6} - 1081556 T^{7} - 22439580 T^{8} - 57322468 T^{9} + 1022470382 T^{10} + 7549850424 T^{11} - 4978893511 T^{12} - 411504365112 T^{13} - 1462847834514 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} \))(\( 1 - 4 T - 66 T^{2} + 984 T^{3} - 631 T^{4} - 50712 T^{5} + 363998 T^{6} + 1081556 T^{7} - 22439580 T^{8} + 57322468 T^{9} + 1022470382 T^{10} - 7549850424 T^{11} - 4978893511 T^{12} + 411504365112 T^{13} - 1462847834514 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16} \))(\( ( 1 + 6 T - 61 T^{2} - 54 T^{3} + 4692 T^{4} - 2862 T^{5} - 171349 T^{6} + 893262 T^{7} + 7890481 T^{8} )^{2} \))(\( 1 - 4 T - 66 T^{2} + 984 T^{3} - 631 T^{4} - 50712 T^{5} + 363998 T^{6} + 1081556 T^{7} - 22439580 T^{8} + 57322468 T^{9} + 1022470382 T^{10} - 7549850424 T^{11} - 4978893511 T^{12} + 411504365112 T^{13} - 1462847834514 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 24 T + 224 T^{2} + 1392 T^{3} + 9358 T^{4} + 16056 T^{5} - 613888 T^{6} - 6806952 T^{7} - 48734957 T^{8} - 401610168 T^{9} - 2136944128 T^{10} + 3297565224 T^{11} + 113394264238 T^{12} + 995174624208 T^{13} + 9448439535584 T^{14} + 59727635635656 T^{15} + 146830437604321 T^{16} \))(\( 1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 104631898 T^{10} - 57278765447 T^{12} - 1602860278358 T^{14} + 146830437604321 T^{16} \))(\( 1 - 24 T + 224 T^{2} - 1392 T^{3} + 9358 T^{4} - 16056 T^{5} - 613888 T^{6} + 6806952 T^{7} - 48734957 T^{8} + 401610168 T^{9} - 2136944128 T^{10} - 3297565224 T^{11} + 113394264238 T^{12} - 995174624208 T^{13} + 9448439535584 T^{14} - 59727635635656 T^{15} + 146830437604321 T^{16} \))(\( 1 + 24 T + 224 T^{2} + 1392 T^{3} + 9358 T^{4} + 16056 T^{5} - 613888 T^{6} - 6806952 T^{7} - 48734957 T^{8} - 401610168 T^{9} - 2136944128 T^{10} + 3297565224 T^{11} + 113394264238 T^{12} + 995174624208 T^{13} + 9448439535584 T^{14} + 59727635635656 T^{15} + 146830437604321 T^{16} \))(\( 1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 104631898 T^{10} - 57278765447 T^{12} - 1602860278358 T^{14} + 146830437604321 T^{16} \))(\( 1 - 24 T + 224 T^{2} - 1392 T^{3} + 9358 T^{4} - 16056 T^{5} - 613888 T^{6} + 6806952 T^{7} - 48734957 T^{8} + 401610168 T^{9} - 2136944128 T^{10} - 3297565224 T^{11} + 113394264238 T^{12} - 995174624208 T^{13} + 9448439535584 T^{14} - 59727635635656 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16} \))(\( ( 1 - 12 T + 176 T^{2} - 1536 T^{3} + 15591 T^{4} - 93696 T^{5} + 654896 T^{6} - 2723772 T^{7} + 13845841 T^{8} )^{2} \))(\( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16} \))(\( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16} \))(\( ( 1 - 12 T + 176 T^{2} - 1536 T^{3} + 15591 T^{4} - 93696 T^{5} + 654896 T^{6} - 2723772 T^{7} + 13845841 T^{8} )^{2} \))(\( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16} \))
$67$ (\( 1 - 48 T + 1108 T^{2} - 16320 T^{3} + 166858 T^{4} - 1140048 T^{5} + 3573520 T^{6} + 22726896 T^{7} - 369529661 T^{8} + 1522702032 T^{9} + 16041531280 T^{10} - 342884256624 T^{11} + 3362375747818 T^{12} - 22034041746240 T^{13} + 100227887443252 T^{14} - 290914157055504 T^{15} + 406067677556641 T^{16} \))(\( ( 1 + 34 T^{2} - 3333 T^{4} + 152626 T^{6} + 20151121 T^{8} )^{2} \))(\( 1 + 48 T + 1108 T^{2} + 16320 T^{3} + 166858 T^{4} + 1140048 T^{5} + 3573520 T^{6} - 22726896 T^{7} - 369529661 T^{8} - 1522702032 T^{9} + 16041531280 T^{10} + 342884256624 T^{11} + 3362375747818 T^{12} + 22034041746240 T^{13} + 100227887443252 T^{14} + 290914157055504 T^{15} + 406067677556641 T^{16} \))(\( 1 + 48 T + 1108 T^{2} + 16320 T^{3} + 166858 T^{4} + 1140048 T^{5} + 3573520 T^{6} - 22726896 T^{7} - 369529661 T^{8} - 1522702032 T^{9} + 16041531280 T^{10} + 342884256624 T^{11} + 3362375747818 T^{12} + 22034041746240 T^{13} + 100227887443252 T^{14} + 290914157055504 T^{15} + 406067677556641 T^{16} \))(\( ( 1 + 34 T^{2} - 3333 T^{4} + 152626 T^{6} + 20151121 T^{8} )^{2} \))(\( 1 - 48 T + 1108 T^{2} - 16320 T^{3} + 166858 T^{4} - 1140048 T^{5} + 3573520 T^{6} + 22726896 T^{7} - 369529661 T^{8} + 1522702032 T^{9} + 16041531280 T^{10} - 342884256624 T^{11} + 3362375747818 T^{12} - 22034041746240 T^{13} + 100227887443252 T^{14} - 290914157055504 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16} \))(\( ( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16} \))(\( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16} \))(\( ( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16} \))
$73$ (\( ( 1 + 12 T - 6 T^{2} + 48 T^{3} + 6659 T^{4} + 3504 T^{5} - 31974 T^{6} + 4668204 T^{7} + 28398241 T^{8} )^{2} \))(\( 1 - 220 T^{2} + 26794 T^{4} - 2408560 T^{6} + 180497395 T^{8} - 12835216240 T^{10} + 760902469354 T^{12} - 33293529783580 T^{14} + 806460091894081 T^{16} \))(\( ( 1 - 12 T - 6 T^{2} - 48 T^{3} + 6659 T^{4} - 3504 T^{5} - 31974 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 12 T - 6 T^{2} - 48 T^{3} + 6659 T^{4} - 3504 T^{5} - 31974 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \))(\( 1 - 220 T^{2} + 26794 T^{4} - 2408560 T^{6} + 180497395 T^{8} - 12835216240 T^{10} + 760902469354 T^{12} - 33293529783580 T^{14} + 806460091894081 T^{16} \))(\( ( 1 + 12 T - 6 T^{2} + 48 T^{3} + 6659 T^{4} + 3504 T^{5} - 31974 T^{6} + 4668204 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 - 24 T + 252 T^{2} - 1584 T^{3} + 5050 T^{4} - 17832 T^{5} - 542160 T^{6} + 18801768 T^{7} - 226869549 T^{8} + 1485339672 T^{9} - 3383620560 T^{10} - 8791871448 T^{11} + 196697909050 T^{12} - 4874057336016 T^{13} + 61258038791292 T^{14} - 460893815667816 T^{15} + 1517108809906561 T^{16} \))(\( ( 1 + 14 T + 7 T^{2} + 434 T^{3} + 13996 T^{4} + 34286 T^{5} + 43687 T^{6} + 6902546 T^{7} + 38950081 T^{8} )^{2} \))(\( 1 - 24 T + 252 T^{2} - 1584 T^{3} + 5050 T^{4} - 17832 T^{5} - 542160 T^{6} + 18801768 T^{7} - 226869549 T^{8} + 1485339672 T^{9} - 3383620560 T^{10} - 8791871448 T^{11} + 196697909050 T^{12} - 4874057336016 T^{13} + 61258038791292 T^{14} - 460893815667816 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 24 T + 252 T^{2} - 1584 T^{3} + 5050 T^{4} - 17832 T^{5} - 542160 T^{6} + 18801768 T^{7} - 226869549 T^{8} + 1485339672 T^{9} - 3383620560 T^{10} - 8791871448 T^{11} + 196697909050 T^{12} - 4874057336016 T^{13} + 61258038791292 T^{14} - 460893815667816 T^{15} + 1517108809906561 T^{16} \))(\( ( 1 + 14 T + 7 T^{2} + 434 T^{3} + 13996 T^{4} + 34286 T^{5} + 43687 T^{6} + 6902546 T^{7} + 38950081 T^{8} )^{2} \))(\( 1 - 24 T + 252 T^{2} - 1584 T^{3} + 5050 T^{4} - 17832 T^{5} - 542160 T^{6} + 18801768 T^{7} - 226869549 T^{8} + 1485339672 T^{9} - 3383620560 T^{10} - 8791871448 T^{11} + 196697909050 T^{12} - 4874057336016 T^{13} + 61258038791292 T^{14} - 460893815667816 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 232 T^{2} + 39708 T^{4} - 4492952 T^{6} + 427617638 T^{8} - 30951946328 T^{10} + 1884475010268 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16} \))(\( ( 1 - 278 T^{2} + 32811 T^{4} - 1915142 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 232 T^{2} + 39708 T^{4} - 4492952 T^{6} + 427617638 T^{8} - 30951946328 T^{10} + 1884475010268 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16} \))(\( 1 - 232 T^{2} + 39708 T^{4} - 4492952 T^{6} + 427617638 T^{8} - 30951946328 T^{10} + 1884475010268 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16} \))(\( ( 1 - 278 T^{2} + 32811 T^{4} - 1915142 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 232 T^{2} + 39708 T^{4} - 4492952 T^{6} + 427617638 T^{8} - 30951946328 T^{10} + 1884475010268 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 16 T - 48 T^{2} - 1824 T^{3} + 2078 T^{4} + 135504 T^{5} + 38912 T^{6} - 7970384 T^{7} - 72309597 T^{8} - 709364176 T^{9} + 308221952 T^{10} + 95526119376 T^{11} + 130378376798 T^{12} - 10185324434976 T^{13} - 23855101966128 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16} \))(\( ( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 - 16 T - 48 T^{2} + 1824 T^{3} + 2078 T^{4} - 135504 T^{5} + 38912 T^{6} + 7970384 T^{7} - 72309597 T^{8} + 709364176 T^{9} + 308221952 T^{10} - 95526119376 T^{11} + 130378376798 T^{12} + 10185324434976 T^{13} - 23855101966128 T^{14} - 707701358328464 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 16 T - 48 T^{2} - 1824 T^{3} + 2078 T^{4} + 135504 T^{5} + 38912 T^{6} - 7970384 T^{7} - 72309597 T^{8} - 709364176 T^{9} + 308221952 T^{10} + 95526119376 T^{11} + 130378376798 T^{12} - 10185324434976 T^{13} - 23855101966128 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16} \))(\( ( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 - 16 T - 48 T^{2} + 1824 T^{3} + 2078 T^{4} - 135504 T^{5} + 38912 T^{6} + 7970384 T^{7} - 72309597 T^{8} + 709364176 T^{9} + 308221952 T^{10} - 95526119376 T^{11} + 130378376798 T^{12} + 10185324434976 T^{13} - 23855101966128 T^{14} - 707701358328464 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 + 24 T + 276 T^{2} + 1608 T^{3} + 10022 T^{4} + 155976 T^{5} + 2596884 T^{6} + 21904152 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 190 T^{2} + 20643 T^{4} + 1787710 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 24 T + 276 T^{2} - 1608 T^{3} + 10022 T^{4} - 155976 T^{5} + 2596884 T^{6} - 21904152 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 24 T + 276 T^{2} - 1608 T^{3} + 10022 T^{4} - 155976 T^{5} + 2596884 T^{6} - 21904152 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 190 T^{2} + 20643 T^{4} + 1787710 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 24 T + 276 T^{2} + 1608 T^{3} + 10022 T^{4} + 155976 T^{5} + 2596884 T^{6} + 21904152 T^{7} + 88529281 T^{8} )^{2} \))
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