Properties

Label 1575.2.m
Level 1575
Weight 2
Character orbit m
Rep. character \(\chi_{1575}(1268,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 72
Newform subspaces 6
Sturm bound 480
Trace bound 8

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Defining parameters

Level: \( N \) = \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1575.m (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(480\)
Trace bound: \(8\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1575, [\chi])\).

Total New Old
Modular forms 528 72 456
Cusp forms 432 72 360
Eisenstein series 96 0 96

Trace form

\( 72q + O(q^{10}) \) \( 72q + 8q^{13} - 40q^{16} + 16q^{22} - 8q^{37} - 80q^{43} - 64q^{46} - 72q^{52} + 56q^{58} + 128q^{61} + 96q^{67} + 40q^{73} - 64q^{76} - 40q^{82} - 24q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1575, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1575.2.m.a \(8\) \(12.576\) \(\Q(\zeta_{24})\) None \(-4\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{2}q^{2}+(-1+\zeta_{24}^{2}-\zeta_{24}^{6}+\cdots)q^{4}+\cdots\)
1575.2.m.b \(8\) \(12.576\) \(\Q(\zeta_{24})\) None \(4\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{6}q^{2}+(-1+\zeta_{24}^{2}-\zeta_{24}^{6}+\cdots)q^{4}+\cdots\)
1575.2.m.c \(12\) \(12.576\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}+(-\beta _{1}+\beta _{3}+\beta _{4}-2\beta _{10}+\cdots)q^{4}+\cdots\)
1575.2.m.d \(12\) \(12.576\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{2}+(\beta _{1}-\beta _{3}-\beta _{4}+2\beta _{10})q^{4}+\cdots\)
1575.2.m.e \(16\) \(12.576\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-4\) \(0\) \(0\) \(0\) \(q+(-1-\beta _{2}-\beta _{5}-\beta _{12})q^{2}+(\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1575.2.m.f \(16\) \(12.576\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(4\) \(0\) \(0\) \(0\) \(q+(1+\beta _{2}+\beta _{5}+\beta _{12})q^{2}+(\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1575, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1575, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 2 T^{3} + T^{4} - 4 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8} )^{2} \))(\( 1 + 8 T^{3} - 5 T^{4} + 32 T^{6} - 32 T^{7} - 5 T^{8} + 80 T^{9} - 96 T^{10} - 72 T^{11} + 193 T^{12} - 144 T^{13} - 384 T^{14} + 640 T^{15} - 80 T^{16} - 1024 T^{17} + 2048 T^{18} - 1280 T^{20} + 4096 T^{21} + 4096 T^{24} \))(\( 1 - 8 T^{3} - 5 T^{4} + 32 T^{6} + 32 T^{7} - 5 T^{8} - 80 T^{9} - 96 T^{10} + 72 T^{11} + 193 T^{12} + 144 T^{13} - 384 T^{14} - 640 T^{15} - 80 T^{16} + 1024 T^{17} + 2048 T^{18} - 1280 T^{20} - 4096 T^{21} + 4096 T^{24} \))(\( ( 1 + 2 T + 2 T^{2} - T^{4} + 4 T^{5} + 10 T^{6} + 6 T^{7} + T^{8} + 12 T^{9} + 40 T^{10} + 32 T^{11} - 16 T^{12} + 128 T^{14} + 256 T^{15} + 256 T^{16} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - T^{4} - 4 T^{5} + 10 T^{6} - 6 T^{7} + T^{8} - 12 T^{9} + 40 T^{10} - 32 T^{11} - 16 T^{12} + 128 T^{14} - 256 T^{15} + 256 T^{16} )^{2} \))
$3$ 1
$5$ 1
$7$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))
$11$ (\( ( 1 - 16 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 16 T^{2} + 121 T^{4} )^{4} \))(\( 1 - 52 T^{2} + 1458 T^{4} - 30084 T^{6} + 494911 T^{8} - 6805160 T^{10} + 80542716 T^{12} - 823424360 T^{14} + 7245991951 T^{16} - 53295641124 T^{18} + 312535248498 T^{20} - 1348746079252 T^{22} + 3138428376721 T^{24} \))(\( 1 - 52 T^{2} + 1458 T^{4} - 30084 T^{6} + 494911 T^{8} - 6805160 T^{10} + 80542716 T^{12} - 823424360 T^{14} + 7245991951 T^{16} - 53295641124 T^{18} + 312535248498 T^{20} - 1348746079252 T^{22} + 3138428376721 T^{24} \))(\( ( 1 - 40 T^{2} + 790 T^{4} - 12352 T^{6} + 157771 T^{8} - 1494592 T^{10} + 11566390 T^{12} - 70862440 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 - 40 T^{2} + 790 T^{4} - 12352 T^{6} + 157771 T^{8} - 1494592 T^{10} + 11566390 T^{12} - 70862440 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( 1 - 92 T^{4} - 17562 T^{8} - 2627612 T^{12} + 815730721 T^{16} \))(\( 1 - 92 T^{4} - 17562 T^{8} - 2627612 T^{12} + 815730721 T^{16} \))(\( 1 - 4 T + 8 T^{2} - 108 T^{3} + 298 T^{4} - 204 T^{5} + 4264 T^{6} - 17924 T^{7} + 42687 T^{8} - 200904 T^{9} + 1148048 T^{10} - 3595224 T^{11} + 7337420 T^{12} - 46737912 T^{13} + 194020112 T^{14} - 441386088 T^{15} + 1219183407 T^{16} - 6655055732 T^{17} + 20581513576 T^{18} - 12800697468 T^{19} + 243087754858 T^{20} - 1145285932284 T^{21} + 1102867934792 T^{22} - 7168641576148 T^{23} + 23298085122481 T^{24} \))(\( 1 - 4 T + 8 T^{2} - 108 T^{3} + 298 T^{4} - 204 T^{5} + 4264 T^{6} - 17924 T^{7} + 42687 T^{8} - 200904 T^{9} + 1148048 T^{10} - 3595224 T^{11} + 7337420 T^{12} - 46737912 T^{13} + 194020112 T^{14} - 441386088 T^{15} + 1219183407 T^{16} - 6655055732 T^{17} + 20581513576 T^{18} - 12800697468 T^{19} + 243087754858 T^{20} - 1145285932284 T^{21} + 1102867934792 T^{22} - 7168641576148 T^{23} + 23298085122481 T^{24} \))(\( 1 + 8 T^{4} + 604 T^{8} - 1145416 T^{12} - 1263574010 T^{16} - 32714226376 T^{20} + 492701355484 T^{24} + 186384680979848 T^{28} + 665416609183179841 T^{32} \))(\( 1 + 8 T^{4} + 604 T^{8} - 1145416 T^{12} - 1263574010 T^{16} - 32714226376 T^{20} + 492701355484 T^{24} + 186384680979848 T^{28} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 + 4 T + 8 T^{2} + 68 T^{3} + 289 T^{4} )^{4} \))(\( ( 1 - 4 T + 8 T^{2} - 68 T^{3} + 289 T^{4} )^{4} \))(\( 1 + 8 T + 32 T^{2} + 232 T^{3} + 1238 T^{4} + 3432 T^{5} + 14752 T^{6} + 67976 T^{7} + 254479 T^{8} + 1447664 T^{9} + 6834752 T^{10} + 33133872 T^{11} + 165039860 T^{12} + 563275824 T^{13} + 1975243328 T^{14} + 7112373232 T^{15} + 21254340559 T^{16} + 96516199432 T^{17} + 356077417888 T^{18} + 1408282325736 T^{19} + 8635987711958 T^{20} + 27512387347304 T^{21} + 64511804814368 T^{22} + 274175170461064 T^{23} + 582622237229761 T^{24} \))(\( 1 - 8 T + 32 T^{2} - 232 T^{3} + 1238 T^{4} - 3432 T^{5} + 14752 T^{6} - 67976 T^{7} + 254479 T^{8} - 1447664 T^{9} + 6834752 T^{10} - 33133872 T^{11} + 165039860 T^{12} - 563275824 T^{13} + 1975243328 T^{14} - 7112373232 T^{15} + 21254340559 T^{16} - 96516199432 T^{17} + 356077417888 T^{18} - 1408282325736 T^{19} + 8635987711958 T^{20} - 27512387347304 T^{21} + 64511804814368 T^{22} - 274175170461064 T^{23} + 582622237229761 T^{24} \))(\( ( 1 - 4 T + 8 T^{2} - 36 T^{3} - 412 T^{4} + 1492 T^{5} - 2024 T^{6} + 1332 T^{7} + 90694 T^{8} + 22644 T^{9} - 584936 T^{10} + 7330196 T^{11} - 34410652 T^{12} - 51114852 T^{13} + 193100552 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} + 36 T^{3} - 412 T^{4} - 1492 T^{5} - 2024 T^{6} - 1332 T^{7} + 90694 T^{8} - 22644 T^{9} - 584936 T^{10} - 7330196 T^{11} - 34410652 T^{12} + 51114852 T^{13} + 193100552 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 34 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 34 T^{2} + 361 T^{4} )^{4} \))(\( 1 - 84 T^{2} + 4642 T^{4} - 179332 T^{6} + 5560223 T^{8} - 138383432 T^{10} + 2894445596 T^{12} - 49956418952 T^{14} + 724613821583 T^{16} - 8436831931492 T^{18} + 78837699636322 T^{20} - 515009565655284 T^{22} + 2213314919066161 T^{24} \))(\( 1 - 84 T^{2} + 4642 T^{4} - 179332 T^{6} + 5560223 T^{8} - 138383432 T^{10} + 2894445596 T^{12} - 49956418952 T^{14} + 724613821583 T^{16} - 8436831931492 T^{18} + 78837699636322 T^{20} - 515009565655284 T^{22} + 2213314919066161 T^{24} \))(\( ( 1 - 16 T^{2} + 796 T^{4} - 20848 T^{6} + 311782 T^{8} - 7526128 T^{10} + 103735516 T^{12} - 752734096 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 16 T^{2} + 796 T^{4} - 20848 T^{6} + 311782 T^{8} - 7526128 T^{10} + 103735516 T^{12} - 752734096 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 - 12 T + 72 T^{2} - 204 T^{3} + 542 T^{4} - 4692 T^{5} + 38088 T^{6} - 146004 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 12 T + 72 T^{2} + 204 T^{3} + 542 T^{4} + 4692 T^{5} + 38088 T^{6} + 146004 T^{7} + 279841 T^{8} )^{2} \))(\( 1 + 8 T + 32 T^{2} + 184 T^{3} + 518 T^{4} - 1752 T^{5} - 13664 T^{6} - 145640 T^{7} - 745937 T^{8} + 382288 T^{9} + 8743232 T^{10} + 77776560 T^{11} + 600339476 T^{12} + 1788860880 T^{13} + 4625169728 T^{14} + 4651298096 T^{15} - 208743756017 T^{16} - 937388994520 T^{17} - 2022762387296 T^{18} - 5965254183144 T^{19} + 40565090375558 T^{20} + 331412089709192 T^{21} + 1325648358836768 T^{22} + 7622478063311416 T^{23} + 21914624432020321 T^{24} \))(\( 1 - 8 T + 32 T^{2} - 184 T^{3} + 518 T^{4} + 1752 T^{5} - 13664 T^{6} + 145640 T^{7} - 745937 T^{8} - 382288 T^{9} + 8743232 T^{10} - 77776560 T^{11} + 600339476 T^{12} - 1788860880 T^{13} + 4625169728 T^{14} - 4651298096 T^{15} - 208743756017 T^{16} + 937388994520 T^{17} - 2022762387296 T^{18} + 5965254183144 T^{19} + 40565090375558 T^{20} - 331412089709192 T^{21} + 1325648358836768 T^{22} - 7622478063311416 T^{23} + 21914624432020321 T^{24} \))(\( ( 1 - 24 T + 288 T^{2} - 2568 T^{3} + 19882 T^{4} - 135216 T^{5} + 816480 T^{6} - 4528368 T^{7} + 22934091 T^{8} - 104152464 T^{9} + 431917920 T^{10} - 1645173072 T^{11} + 5563798762 T^{12} - 16528528824 T^{13} + 42634336032 T^{14} - 81715810728 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 + 24 T + 288 T^{2} + 2568 T^{3} + 19882 T^{4} + 135216 T^{5} + 816480 T^{6} + 4528368 T^{7} + 22934091 T^{8} + 104152464 T^{9} + 431917920 T^{10} + 1645173072 T^{11} + 5563798762 T^{12} + 16528528824 T^{13} + 42634336032 T^{14} + 81715810728 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 + 64 T^{2} + 2514 T^{4} + 53824 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + 64 T^{2} + 2514 T^{4} + 53824 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + 16 T + 216 T^{2} + 1904 T^{3} + 15483 T^{4} + 98368 T^{5} + 589488 T^{6} + 2852672 T^{7} + 13021203 T^{8} + 46436656 T^{9} + 152772696 T^{10} + 328178384 T^{11} + 594823321 T^{12} )^{2} \))(\( ( 1 - 16 T + 216 T^{2} - 1904 T^{3} + 15483 T^{4} - 98368 T^{5} + 589488 T^{6} - 2852672 T^{7} + 13021203 T^{8} - 46436656 T^{9} + 152772696 T^{10} - 328178384 T^{11} + 594823321 T^{12} )^{2} \))(\( ( 1 + 152 T^{2} + 11062 T^{4} + 514880 T^{6} + 17268331 T^{8} + 433014080 T^{10} + 7823942422 T^{12} + 90413144792 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 152 T^{2} + 11062 T^{4} + 514880 T^{6} + 17268331 T^{8} + 433014080 T^{10} + 7823942422 T^{12} + 90413144792 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 - 4 T + 18 T^{2} - 124 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 - 4 T + 18 T^{2} - 124 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 + 114 T^{2} - 168 T^{3} + 6583 T^{4} - 10856 T^{5} + 253628 T^{6} - 336536 T^{7} + 6326263 T^{8} - 5004888 T^{9} + 105281394 T^{10} + 887503681 T^{12} )^{2} \))(\( ( 1 + 114 T^{2} - 168 T^{3} + 6583 T^{4} - 10856 T^{5} + 253628 T^{6} - 336536 T^{7} + 6326263 T^{8} - 5004888 T^{9} + 105281394 T^{10} + 887503681 T^{12} )^{2} \))(\( ( 1 + 4 T + 52 T^{2} + 292 T^{3} + 1510 T^{4} + 9052 T^{5} + 49972 T^{6} + 119164 T^{7} + 923521 T^{8} )^{4} \))(\( ( 1 + 4 T + 52 T^{2} + 292 T^{3} + 1510 T^{4} + 9052 T^{5} + 49972 T^{6} + 119164 T^{7} + 923521 T^{8} )^{4} \))
$37$ (\( 1 + 292 T^{4} - 2730714 T^{8} + 547255012 T^{12} + 3512479453921 T^{16} \))(\( 1 + 292 T^{4} - 2730714 T^{8} + 547255012 T^{12} + 3512479453921 T^{16} \))(\( 1 + 4 T + 8 T^{2} + 28 T^{3} + 122 T^{4} + 124 T^{5} - 88 T^{6} - 102780 T^{7} - 827937 T^{8} - 2005016 T^{9} - 4255984 T^{10} - 201457960 T^{11} - 3018818132 T^{12} - 7453944520 T^{13} - 5826442096 T^{14} - 101560075448 T^{15} - 1551687235857 T^{16} - 7127171900460 T^{17} - 225783923992 T^{18} + 11771552764492 T^{19} + 428522493378362 T^{20} + 3638928714262156 T^{21} + 38468674979342792 T^{22} + 711670487117841652 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 4 T + 8 T^{2} + 28 T^{3} + 122 T^{4} + 124 T^{5} - 88 T^{6} - 102780 T^{7} - 827937 T^{8} - 2005016 T^{9} - 4255984 T^{10} - 201457960 T^{11} - 3018818132 T^{12} - 7453944520 T^{13} - 5826442096 T^{14} - 101560075448 T^{15} - 1551687235857 T^{16} - 7127171900460 T^{17} - 225783923992 T^{18} + 11771552764492 T^{19} + 428522493378362 T^{20} + 3638928714262156 T^{21} + 38468674979342792 T^{22} + 711670487117841652 T^{23} + 6582952005840035281 T^{24} \))(\( 1 - 1828 T^{4} + 580138 T^{8} - 1379443216 T^{12} + 4448356754803 T^{16} - 2585298677141776 T^{20} + 2037722805438821098 T^{24} - \)\(12\!\cdots\!68\)\( T^{28} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - 1828 T^{4} + 580138 T^{8} - 1379443216 T^{12} + 4448356754803 T^{16} - 2585298677141776 T^{20} + 2037722805438821098 T^{24} - \)\(12\!\cdots\!68\)\( T^{28} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( ( 1 - 100 T^{2} + 5094 T^{4} - 168100 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 100 T^{2} + 5094 T^{4} - 168100 T^{6} + 2825761 T^{8} )^{2} \))(\( 1 - 232 T^{2} + 30646 T^{4} - 2820520 T^{6} + 198407727 T^{8} - 11090030832 T^{10} + 503431026356 T^{12} - 18642341828592 T^{14} + 560652817055247 T^{16} - 13397764013825320 T^{18} + 244706018571642166 T^{20} - 3114056959955357032 T^{22} + 22563490300366186081 T^{24} \))(\( 1 - 232 T^{2} + 30646 T^{4} - 2820520 T^{6} + 198407727 T^{8} - 11090030832 T^{10} + 503431026356 T^{12} - 18642341828592 T^{14} + 560652817055247 T^{16} - 13397764013825320 T^{18} + 244706018571642166 T^{20} - 3114056959955357032 T^{22} + 22563490300366186081 T^{24} \))(\( ( 1 - 152 T^{2} + 8572 T^{4} - 197672 T^{6} + 2694598 T^{8} - 332286632 T^{10} + 24222423292 T^{12} - 722015844632 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 152 T^{2} + 8572 T^{4} - 197672 T^{6} + 2694598 T^{8} - 332286632 T^{10} + 24222423292 T^{12} - 722015844632 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + 1202 T^{4} + 3418801 T^{8} )^{2} \))(\( ( 1 + 1202 T^{4} + 3418801 T^{8} )^{2} \))(\( 1 + 40 T + 800 T^{2} + 11608 T^{3} + 144630 T^{4} + 1609192 T^{5} + 16036512 T^{6} + 146487000 T^{7} + 1250458847 T^{8} + 9986195856 T^{9} + 74890568512 T^{10} + 532487979120 T^{11} + 3591291202036 T^{12} + 22896983102160 T^{13} + 138472661178688 T^{14} + 793972473922992 T^{15} + 4275069956582447 T^{16} + 21534825789741000 T^{17} + 101372614391645088 T^{18} + 437408334444495544 T^{19} + 1690464406149432630 T^{20} + 5834095039362873544 T^{21} + 17289185850627399200 T^{22} + 37171749578848908280 T^{23} + 39959630797262576401 T^{24} \))(\( 1 + 40 T + 800 T^{2} + 11608 T^{3} + 144630 T^{4} + 1609192 T^{5} + 16036512 T^{6} + 146487000 T^{7} + 1250458847 T^{8} + 9986195856 T^{9} + 74890568512 T^{10} + 532487979120 T^{11} + 3591291202036 T^{12} + 22896983102160 T^{13} + 138472661178688 T^{14} + 793972473922992 T^{15} + 4275069956582447 T^{16} + 21534825789741000 T^{17} + 101372614391645088 T^{18} + 437408334444495544 T^{19} + 1690464406149432630 T^{20} + 5834095039362873544 T^{21} + 17289185850627399200 T^{22} + 37171749578848908280 T^{23} + 39959630797262576401 T^{24} \))(\( 1 + 476 T^{4} + 663082 T^{8} + 7749842672 T^{12} + 7079769593971 T^{16} + 26495169876876272 T^{20} + 7750235216472226282 T^{24} + \)\(19\!\cdots\!76\)\( T^{28} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 + 476 T^{4} + 663082 T^{8} + 7749842672 T^{12} + 7079769593971 T^{16} + 26495169876876272 T^{20} + 7750235216472226282 T^{24} + \)\(19\!\cdots\!76\)\( T^{28} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( ( 1 + 16 T + 128 T^{2} + 880 T^{3} + 5986 T^{4} + 41360 T^{5} + 282752 T^{6} + 1661168 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 16 T + 128 T^{2} - 880 T^{3} + 5986 T^{4} - 41360 T^{5} + 282752 T^{6} - 1661168 T^{7} + 4879681 T^{8} )^{2} \))(\( 1 + 24 T + 288 T^{2} + 2920 T^{3} + 30374 T^{4} + 283512 T^{5} + 2319776 T^{6} + 18683912 T^{7} + 148359759 T^{8} + 1083670000 T^{9} + 7566143296 T^{10} + 53830728976 T^{11} + 378456487636 T^{12} + 2530044261872 T^{13} + 16713610540864 T^{14} + 112509870410000 T^{15} + 723948297156879 T^{16} + 4285061928427384 T^{17} + 25005365019046304 T^{18} + 143633734128706056 T^{19} + 723244021064328614 T^{20} + 3267860981460079640 T^{21} + 15148550083919054112 T^{22} + 59331821162016295272 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 24 T + 288 T^{2} - 2920 T^{3} + 30374 T^{4} - 283512 T^{5} + 2319776 T^{6} - 18683912 T^{7} + 148359759 T^{8} - 1083670000 T^{9} + 7566143296 T^{10} - 53830728976 T^{11} + 378456487636 T^{12} - 2530044261872 T^{13} + 16713610540864 T^{14} - 112509870410000 T^{15} + 723948297156879 T^{16} - 4285061928427384 T^{17} + 25005365019046304 T^{18} - 143633734128706056 T^{19} + 723244021064328614 T^{20} - 3267860981460079640 T^{21} + 15148550083919054112 T^{22} - 59331821162016295272 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 20 T + 200 T^{2} - 1260 T^{3} + 6308 T^{4} - 44860 T^{5} + 429400 T^{6} - 3480900 T^{7} + 25034950 T^{8} - 163602300 T^{9} + 948544600 T^{10} - 4657499780 T^{11} + 30781027748 T^{12} - 288974708820 T^{13} + 2155843065800 T^{14} - 10132462409260 T^{15} + 23811286661761 T^{16} )^{2} \))(\( ( 1 + 20 T + 200 T^{2} + 1260 T^{3} + 6308 T^{4} + 44860 T^{5} + 429400 T^{6} + 3480900 T^{7} + 25034950 T^{8} + 163602300 T^{9} + 948544600 T^{10} + 4657499780 T^{11} + 30781027748 T^{12} + 288974708820 T^{13} + 2155843065800 T^{14} + 10132462409260 T^{15} + 23811286661761 T^{16} )^{2} \))
$53$ (\( ( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 66992 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 16 T + 128 T^{2} + 1264 T^{3} + 11806 T^{4} + 66992 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} )^{2} \))(\( 1 - 40 T + 800 T^{2} - 11880 T^{3} + 147222 T^{4} - 1477736 T^{5} + 11899040 T^{6} - 74456936 T^{7} + 249326879 T^{8} + 1557249200 T^{9} - 37151695552 T^{10} + 419908594096 T^{11} - 3501062916044 T^{12} + 22255155487088 T^{13} - 104359112805568 T^{14} + 231838589148400 T^{15} + 1967309001538799 T^{16} - 31137555057789448 T^{17} + 263734619648416160 T^{18} - 1735912940938169032 T^{19} + 9165996141741389142 T^{20} - 39201191470609340040 T^{21} + \)\(13\!\cdots\!00\)\( T^{22} - \)\(37\!\cdots\!80\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 40 T + 800 T^{2} + 11880 T^{3} + 147222 T^{4} + 1477736 T^{5} + 11899040 T^{6} + 74456936 T^{7} + 249326879 T^{8} - 1557249200 T^{9} - 37151695552 T^{10} - 419908594096 T^{11} - 3501062916044 T^{12} - 22255155487088 T^{13} - 104359112805568 T^{14} - 231838589148400 T^{15} + 1967309001538799 T^{16} + 31137555057789448 T^{17} + 263734619648416160 T^{18} + 1735912940938169032 T^{19} + 9165996141741389142 T^{20} + 39201191470609340040 T^{21} + \)\(13\!\cdots\!00\)\( T^{22} + \)\(37\!\cdots\!80\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 16 T + 128 T^{2} - 1104 T^{3} + 9500 T^{4} - 64112 T^{5} + 419200 T^{6} - 3139632 T^{7} + 23623462 T^{8} - 166400496 T^{9} + 1177532800 T^{10} - 9544802224 T^{11} + 74959569500 T^{12} - 461687824272 T^{13} + 2837038224512 T^{14} - 18795378237392 T^{15} + 62259690411361 T^{16} )^{2} \))(\( ( 1 + 16 T + 128 T^{2} + 1104 T^{3} + 9500 T^{4} + 64112 T^{5} + 419200 T^{6} + 3139632 T^{7} + 23623462 T^{8} + 166400496 T^{9} + 1177532800 T^{10} + 9544802224 T^{11} + 74959569500 T^{12} + 461687824272 T^{13} + 2837038224512 T^{14} + 18795378237392 T^{15} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 + 28 T^{2} + 4086 T^{4} + 97468 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 4086 T^{4} + 97468 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 40 T + 986 T^{2} + 16824 T^{3} + 222455 T^{4} + 2325552 T^{5} + 19830604 T^{6} + 137207568 T^{7} + 774365855 T^{8} + 3455296296 T^{9} + 11947717946 T^{10} + 28596971960 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 - 40 T + 986 T^{2} - 16824 T^{3} + 222455 T^{4} - 2325552 T^{5} + 19830604 T^{6} - 137207568 T^{7} + 774365855 T^{8} - 3455296296 T^{9} + 11947717946 T^{10} - 28596971960 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 104 T^{2} + 6988 T^{4} + 285656 T^{6} + 18409798 T^{8} + 994368536 T^{10} + 84676118668 T^{12} + 4386775498664 T^{14} + 146830437604321 T^{16} )^{2} \))(\( ( 1 + 104 T^{2} + 6988 T^{4} + 285656 T^{6} + 18409798 T^{8} + 994368536 T^{10} + 84676118668 T^{12} + 4386775498664 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 + 4 T + 78 T^{2} + 244 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 4 T + 78 T^{2} + 244 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 16 T + 254 T^{2} + 2800 T^{3} + 31591 T^{4} + 270400 T^{5} + 2316548 T^{6} + 16494400 T^{7} + 117550111 T^{8} + 635546800 T^{9} + 3516843614 T^{10} + 13513540816 T^{11} + 51520374361 T^{12} )^{2} \))(\( ( 1 + 16 T + 254 T^{2} + 2800 T^{3} + 31591 T^{4} + 270400 T^{5} + 2316548 T^{6} + 16494400 T^{7} + 117550111 T^{8} + 635546800 T^{9} + 3516843614 T^{10} + 13513540816 T^{11} + 51520374361 T^{12} )^{2} \))(\( ( 1 - 28 T + 508 T^{2} - 6052 T^{3} + 55366 T^{4} - 369172 T^{5} + 1890268 T^{6} - 6355468 T^{7} + 13845841 T^{8} )^{4} \))(\( ( 1 - 28 T + 508 T^{2} - 6052 T^{3} + 55366 T^{4} - 369172 T^{5} + 1890268 T^{6} - 6355468 T^{7} + 13845841 T^{8} )^{4} \))
$67$ (\( 1 - 2972 T^{4} + 21854310 T^{8} - 59889131612 T^{12} + 406067677556641 T^{16} \))(\( 1 - 2972 T^{4} + 21854310 T^{8} - 59889131612 T^{12} + 406067677556641 T^{16} \))(\( 1 - 48 T + 1152 T^{2} - 18992 T^{3} + 248566 T^{4} - 2816976 T^{5} + 29214848 T^{6} - 287145808 T^{7} + 2747657919 T^{8} - 25947593440 T^{9} + 239514732288 T^{10} - 2122341535712 T^{11} + 17856126120308 T^{12} - 142196882892704 T^{13} + 1075181633240832 T^{14} - 7804076045794720 T^{15} + 55368387192377199 T^{16} - 387682764750601456 T^{17} + 2642727885393245312 T^{18} - 17072879135116363248 T^{19} + \)\(10\!\cdots\!06\)\( T^{20} - \)\(51\!\cdots\!24\)\( T^{21} + \)\(20\!\cdots\!48\)\( T^{22} - \)\(58\!\cdots\!84\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 48 T + 1152 T^{2} - 18992 T^{3} + 248566 T^{4} - 2816976 T^{5} + 29214848 T^{6} - 287145808 T^{7} + 2747657919 T^{8} - 25947593440 T^{9} + 239514732288 T^{10} - 2122341535712 T^{11} + 17856126120308 T^{12} - 142196882892704 T^{13} + 1075181633240832 T^{14} - 7804076045794720 T^{15} + 55368387192377199 T^{16} - 387682764750601456 T^{17} + 2642727885393245312 T^{18} - 17072879135116363248 T^{19} + \)\(10\!\cdots\!06\)\( T^{20} - \)\(51\!\cdots\!24\)\( T^{21} + \)\(20\!\cdots\!48\)\( T^{22} - \)\(58\!\cdots\!84\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 7684 T^{4} + 38988106 T^{8} - 68000407504 T^{12} + 258909628702099 T^{16} - 1370284439662411984 T^{20} + \)\(15\!\cdots\!46\)\( T^{24} - \)\(62\!\cdots\!24\)\( T^{28} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 - 7684 T^{4} + 38988106 T^{8} - 68000407504 T^{12} + 258909628702099 T^{16} - 1370284439662411984 T^{20} + \)\(15\!\cdots\!46\)\( T^{24} - \)\(62\!\cdots\!24\)\( T^{28} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 - 160 T^{2} + 14754 T^{4} - 806560 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 160 T^{2} + 14754 T^{4} - 806560 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 404 T^{2} + 75490 T^{4} - 8307268 T^{6} + 571593487 T^{8} - 25299911144 T^{10} + 1141503138780 T^{12} - 127536852076904 T^{14} + 14525151353321647 T^{16} - 1064163389407837828 T^{18} + 48747934073742497890 T^{20} - \)\(13\!\cdots\!04\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 404 T^{2} + 75490 T^{4} - 8307268 T^{6} + 571593487 T^{8} - 25299911144 T^{10} + 1141503138780 T^{12} - 127536852076904 T^{14} + 14525151353321647 T^{16} - 1064163389407837828 T^{18} + 48747934073742497890 T^{20} - \)\(13\!\cdots\!04\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 104 T^{2} + 10246 T^{4} - 920192 T^{6} + 79259707 T^{8} - 4638687872 T^{10} + 260368083526 T^{12} - 13322429527784 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 104 T^{2} + 10246 T^{4} - 920192 T^{6} + 79259707 T^{8} - 4638687872 T^{10} + 260368083526 T^{12} - 13322429527784 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( 1 + 12868 T^{4} + 85213638 T^{8} + 365428565188 T^{12} + 806460091894081 T^{16} \))(\( 1 + 12868 T^{4} + 85213638 T^{8} + 365428565188 T^{12} + 806460091894081 T^{16} \))(\( 1 - 20 T + 200 T^{2} - 1452 T^{3} + 7482 T^{4} - 35804 T^{5} + 273832 T^{6} - 2000484 T^{7} + 27883951 T^{8} - 188107816 T^{9} - 317778928 T^{10} + 34324729064 T^{11} - 473910414484 T^{12} + 2505705221672 T^{13} - 1693443907312 T^{14} - 73177138256872 T^{15} + 791855160530191 T^{16} - 4147146552651012 T^{17} + 41440153853169448 T^{18} - 395541056577748988 T^{19} + 6033934407551514042 T^{20} - 85481543900405009676 T^{21} + \)\(85\!\cdots\!00\)\( T^{22} - \)\(62\!\cdots\!40\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 - 20 T + 200 T^{2} - 1452 T^{3} + 7482 T^{4} - 35804 T^{5} + 273832 T^{6} - 2000484 T^{7} + 27883951 T^{8} - 188107816 T^{9} - 317778928 T^{10} + 34324729064 T^{11} - 473910414484 T^{12} + 2505705221672 T^{13} - 1693443907312 T^{14} - 73177138256872 T^{15} + 791855160530191 T^{16} - 4147146552651012 T^{17} + 41440153853169448 T^{18} - 395541056577748988 T^{19} + 6033934407551514042 T^{20} - 85481543900405009676 T^{21} + \)\(85\!\cdots\!00\)\( T^{22} - \)\(62\!\cdots\!40\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 - 10744 T^{4} + 21741724 T^{8} + 248072229176 T^{12} - 2082801346307642 T^{16} + 7044814949547279416 T^{20} + \)\(17\!\cdots\!44\)\( T^{24} - \)\(24\!\cdots\!24\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 10744 T^{4} + 21741724 T^{8} + 248072229176 T^{12} - 2082801346307642 T^{16} + 7044814949547279416 T^{20} + \)\(17\!\cdots\!44\)\( T^{24} - \)\(24\!\cdots\!24\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( ( 1 - 142 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{4} \))(\( 1 - 372 T^{2} + 73090 T^{4} - 9878564 T^{6} + 1031952175 T^{8} - 90900376040 T^{10} + 7353042693660 T^{12} - 567309246865640 T^{14} + 40194620804376175 T^{16} - 2401354986961351844 T^{18} + \)\(11\!\cdots\!90\)\( T^{20} - \)\(35\!\cdots\!72\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 - 372 T^{2} + 73090 T^{4} - 9878564 T^{6} + 1031952175 T^{8} - 90900376040 T^{10} + 7353042693660 T^{12} - 567309246865640 T^{14} + 40194620804376175 T^{16} - 2401354986961351844 T^{18} + \)\(11\!\cdots\!90\)\( T^{20} - \)\(35\!\cdots\!72\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 388 T^{2} + 72202 T^{4} - 8684944 T^{6} + 776480659 T^{8} - 54202735504 T^{10} + 2812273748362 T^{12} - 94317932742148 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 388 T^{2} + 72202 T^{4} - 8684944 T^{6} + 776480659 T^{8} - 54202735504 T^{10} + 2812273748362 T^{12} - 94317932742148 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 - 32 T + 512 T^{2} - 5984 T^{3} + 59122 T^{4} - 496672 T^{5} + 3527168 T^{6} - 18297184 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 32 T + 512 T^{2} + 5984 T^{3} + 59122 T^{4} + 496672 T^{5} + 3527168 T^{6} + 18297184 T^{7} + 47458321 T^{8} )^{2} \))(\( 1 + 24 T + 288 T^{2} + 2760 T^{3} + 26518 T^{4} + 273208 T^{5} + 2728608 T^{6} + 24159272 T^{7} + 240213375 T^{8} + 2803489520 T^{9} + 29745965888 T^{10} + 283235701200 T^{11} + 2619914616372 T^{12} + 23508563199600 T^{13} + 204919959002432 T^{14} + 1602998862172240 T^{15} + 11400123459243375 T^{16} + 95164354313291896 T^{17} + 892092118297640352 T^{18} + 7413786218774013416 T^{19} + 59726285411863089238 T^{20} + \)\(51\!\cdots\!80\)\( T^{21} + \)\(44\!\cdots\!12\)\( T^{22} + \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 - 24 T + 288 T^{2} - 2760 T^{3} + 26518 T^{4} - 273208 T^{5} + 2728608 T^{6} - 24159272 T^{7} + 240213375 T^{8} - 2803489520 T^{9} + 29745965888 T^{10} - 283235701200 T^{11} + 2619914616372 T^{12} - 23508563199600 T^{13} + 204919959002432 T^{14} - 1602998862172240 T^{15} + 11400123459243375 T^{16} - 95164354313291896 T^{17} + 892092118297640352 T^{18} - 7413786218774013416 T^{19} + 59726285411863089238 T^{20} - \)\(51\!\cdots\!80\)\( T^{21} + \)\(44\!\cdots\!12\)\( T^{22} - \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( ( 1 - 8 T + 32 T^{2} - 312 T^{3} - 12412 T^{4} + 81992 T^{5} - 210080 T^{6} - 402504 T^{7} + 81059878 T^{8} - 33407832 T^{9} - 1447241120 T^{10} + 46881959704 T^{11} - 589052680252 T^{12} - 1228980680616 T^{13} + 10462091947808 T^{14} - 217088407917016 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} + 312 T^{3} - 12412 T^{4} - 81992 T^{5} - 210080 T^{6} + 402504 T^{7} + 81059878 T^{8} + 33407832 T^{9} - 1447241120 T^{10} - 46881959704 T^{11} - 589052680252 T^{12} + 1228980680616 T^{13} + 10462091947808 T^{14} + 217088407917016 T^{15} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( ( 1 - 22 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 - 22 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 28 T + 684 T^{2} + 11188 T^{3} + 161827 T^{4} + 1876496 T^{5} + 19427016 T^{6} + 167008144 T^{7} + 1281831667 T^{8} + 7887193172 T^{9} + 42915692844 T^{10} + 156353664572 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 28 T + 684 T^{2} - 11188 T^{3} + 161827 T^{4} - 1876496 T^{5} + 19427016 T^{6} - 167008144 T^{7} + 1281831667 T^{8} - 7887193172 T^{9} + 42915692844 T^{10} - 156353664572 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 488 T^{2} + 119308 T^{4} + 18415256 T^{6} + 1956713638 T^{8} + 145867242776 T^{10} + 7485651289228 T^{12} + 242526869988968 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 488 T^{2} + 119308 T^{4} + 18415256 T^{6} + 1956713638 T^{8} + 145867242776 T^{10} + 7485651289228 T^{12} + 242526869988968 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 - 3836 T^{4} - 51582714 T^{8} - 339598321916 T^{12} + 7837433594376961 T^{16} \))(\( 1 - 3836 T^{4} - 51582714 T^{8} - 339598321916 T^{12} + 7837433594376961 T^{16} \))(\( 1 + 12 T + 72 T^{2} + 884 T^{3} + 15450 T^{4} + 102628 T^{5} + 509864 T^{6} + 1529948 T^{7} - 62272049 T^{8} - 410787880 T^{9} - 1704538608 T^{10} - 69640900504 T^{11} - 1520682376276 T^{12} - 6755167348888 T^{13} - 16038003762672 T^{14} - 374915006803240 T^{15} - 5512899724366769 T^{16} + 13138184051516636 T^{17} + 424702438321119656 T^{18} + 8292166339419780964 T^{19} + \)\(12\!\cdots\!50\)\( T^{20} + \)\(67\!\cdots\!28\)\( T^{21} + \)\(53\!\cdots\!28\)\( T^{22} + \)\(85\!\cdots\!36\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 12 T + 72 T^{2} + 884 T^{3} + 15450 T^{4} + 102628 T^{5} + 509864 T^{6} + 1529948 T^{7} - 62272049 T^{8} - 410787880 T^{9} - 1704538608 T^{10} - 69640900504 T^{11} - 1520682376276 T^{12} - 6755167348888 T^{13} - 16038003762672 T^{14} - 374915006803240 T^{15} - 5512899724366769 T^{16} + 13138184051516636 T^{17} + 424702438321119656 T^{18} + 8292166339419780964 T^{19} + \)\(12\!\cdots\!50\)\( T^{20} + \)\(67\!\cdots\!28\)\( T^{21} + \)\(53\!\cdots\!28\)\( T^{22} + \)\(85\!\cdots\!36\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 14984 T^{4} + 94954780 T^{8} - 251532176968 T^{12} - 11937369278985146 T^{16} - 22267962775341800008 T^{20} + \)\(74\!\cdots\!80\)\( T^{24} + \)\(10\!\cdots\!44\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 14984 T^{4} + 94954780 T^{8} - 251532176968 T^{12} - 11937369278985146 T^{16} - 22267962775341800008 T^{20} + \)\(74\!\cdots\!80\)\( T^{24} + \)\(10\!\cdots\!44\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))
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