Properties

Label 1050.2.j
Level 1050
Weight 2
Character orbit j
Rep. character \(\chi_{1050}(407,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 72
Newform subspaces 6
Sturm bound 480
Trace bound 14

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Defining parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.j (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(480\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(11\), \(17\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1050, [\chi])\).

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

Trace form

\( 72q - 8q^{3} + O(q^{10}) \) \( 72q - 8q^{3} + 8q^{12} - 72q^{16} + 8q^{18} + 16q^{21} - 8q^{22} - 8q^{27} + 32q^{31} + 32q^{33} - 16q^{36} + 40q^{37} - 16q^{43} - 32q^{46} + 8q^{48} + 88q^{51} - 32q^{57} + 16q^{58} - 8q^{63} - 88q^{66} - 8q^{72} + 48q^{73} + 8q^{78} - 96q^{81} - 64q^{82} + 80q^{87} - 8q^{88} - 96q^{91} - 56q^{93} - 16q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1050, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1050.2.j.a \(8\) \(8.384\) 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{7}q^{2}+(-\beta _{4}+\beta _{7})q^{3}-\beta _{2}q^{4}+\cdots\)
1050.2.j.b \(8\) \(8.384\) 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{7}q^{2}+(\beta _{1}-\beta _{6}-\beta _{7})q^{3}-\beta _{2}q^{4}+\cdots\)
1050.2.j.c \(12\) \(8.384\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-4\) \(0\) \(0\) \(q-\beta _{1}q^{2}-\beta _{9}q^{3}+\beta _{7}q^{4}-\beta _{5}q^{6}+\cdots\)
1050.2.j.d \(12\) \(8.384\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-4\) \(0\) \(0\) \(q+\beta _{2}q^{2}+\beta _{4}q^{3}-\beta _{7}q^{4}-\beta _{5}q^{6}+\cdots\)
1050.2.j.e \(16\) \(8.384\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{11}q^{2}+(\beta _{11}-\beta _{13})q^{3}+\beta _{5}q^{4}+\cdots\)
1050.2.j.f \(16\) \(8.384\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{11}q^{2}+(-\beta _{1}+\beta _{11})q^{3}+\beta _{5}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))
$3$ (\( ( 1 + 4 T^{2} + 9 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \))(\( 1 + 4 T + 8 T^{2} + 4 T^{3} - 7 T^{4} - 24 T^{5} - 32 T^{6} - 72 T^{7} - 63 T^{8} + 108 T^{9} + 648 T^{10} + 972 T^{11} + 729 T^{12} \))(\( 1 + 4 T + 8 T^{2} + 20 T^{3} + 57 T^{4} + 104 T^{5} + 160 T^{6} + 312 T^{7} + 513 T^{8} + 540 T^{9} + 648 T^{10} + 972 T^{11} + 729 T^{12} \))(\( 1 + 8 T^{2} + 33 T^{4} + 104 T^{6} + 304 T^{8} + 936 T^{10} + 2673 T^{12} + 5832 T^{14} + 6561 T^{16} \))(\( 1 - 8 T^{2} + 33 T^{4} - 104 T^{6} + 304 T^{8} - 936 T^{10} + 2673 T^{12} - 5832 T^{14} + 6561 T^{16} \))
$5$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$7$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))
$11$ (\( ( 1 - 32 T^{2} + 478 T^{4} - 3872 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 32 T^{2} + 478 T^{4} - 3872 T^{6} + 14641 T^{8} )^{2} \))(\( 1 - 56 T^{2} + 1358 T^{4} - 19128 T^{6} + 195023 T^{8} - 1950544 T^{10} + 21368868 T^{12} - 236015824 T^{14} + 2855331743 T^{16} - 33886418808 T^{18} + 291099360398 T^{20} - 1452495777656 T^{22} + 3138428376721 T^{24} \))(\( 1 - 56 T^{2} + 1358 T^{4} - 19128 T^{6} + 195023 T^{8} - 1950544 T^{10} + 21368868 T^{12} - 236015824 T^{14} + 2855331743 T^{16} - 33886418808 T^{18} + 291099360398 T^{20} - 1452495777656 T^{22} + 3138428376721 T^{24} \))(\( ( 1 - 34 T^{2} + 777 T^{4} - 12410 T^{6} + 157028 T^{8} - 1501610 T^{10} + 11376057 T^{12} - 60233074 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 - 34 T^{2} + 777 T^{4} - 12410 T^{6} + 157028 T^{8} - 1501610 T^{10} + 11376057 T^{12} - 60233074 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( 1 - 28 T^{4} + 52198 T^{8} - 799708 T^{12} + 815730721 T^{16} \))(\( 1 - 28 T^{4} + 52198 T^{8} - 799708 T^{12} + 815730721 T^{16} \))(\( 1 + 32 T^{3} - 14 T^{4} - 272 T^{5} + 512 T^{6} + 5520 T^{7} - 4301 T^{8} - 109488 T^{9} + 220800 T^{10} - 406064 T^{11} - 6852572 T^{12} - 5278832 T^{13} + 37315200 T^{14} - 240545136 T^{15} - 122840861 T^{16} + 2049537360 T^{17} + 2471326208 T^{18} - 17067596624 T^{19} - 11420230094 T^{20} + 339343979936 T^{21} + 23298085122481 T^{24} \))(\( 1 + 32 T^{3} - 14 T^{4} - 272 T^{5} + 512 T^{6} + 5520 T^{7} - 4301 T^{8} - 109488 T^{9} + 220800 T^{10} - 406064 T^{11} - 6852572 T^{12} - 5278832 T^{13} + 37315200 T^{14} - 240545136 T^{15} - 122840861 T^{16} + 2049537360 T^{17} + 2471326208 T^{18} - 17067596624 T^{19} - 11420230094 T^{20} + 339343979936 T^{21} + 23298085122481 T^{24} \))(\( 1 - 110 T^{4} + 16209 T^{8} + 2201330 T^{12} - 281263996 T^{16} + 62872186130 T^{20} + 13222179256689 T^{24} - 2562789363472910 T^{28} + 665416609183179841 T^{32} \))(\( 1 - 110 T^{4} + 16209 T^{8} + 2201330 T^{12} - 281263996 T^{16} + 62872186130 T^{20} + 13222179256689 T^{24} - 2562789363472910 T^{28} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 - 16 T^{2} + 289 T^{4} )^{2}( 1 + 16 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 16 T^{2} + 289 T^{4} )^{2}( 1 + 16 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 28 T + 392 T^{2} + 3716 T^{3} + 26626 T^{4} + 151924 T^{5} + 720808 T^{6} + 3009708 T^{7} + 12126655 T^{8} + 52482936 T^{9} + 248165456 T^{10} + 1175330632 T^{11} + 5125539580 T^{12} + 19980620744 T^{13} + 71719816784 T^{14} + 257848664568 T^{15} + 1012830352255 T^{16} + 4273354971756 T^{17} + 17398552835752 T^{18} + 62340292556852 T^{19} + 185736517624066 T^{20} + 440672549062852 T^{21} + 790269608976008 T^{22} + 959613096613724 T^{23} + 582622237229761 T^{24} \))(\( 1 - 28 T + 392 T^{2} - 3716 T^{3} + 26626 T^{4} - 151924 T^{5} + 720808 T^{6} - 3009708 T^{7} + 12126655 T^{8} - 52482936 T^{9} + 248165456 T^{10} - 1175330632 T^{11} + 5125539580 T^{12} - 19980620744 T^{13} + 71719816784 T^{14} - 257848664568 T^{15} + 1012830352255 T^{16} - 4273354971756 T^{17} + 17398552835752 T^{18} - 62340292556852 T^{19} + 185736517624066 T^{20} - 440672549062852 T^{21} + 790269608976008 T^{22} - 959613096613724 T^{23} + 582622237229761 T^{24} \))(\( 1 + 316 T^{4} + 44362 T^{8} - 6374288 T^{12} - 2550901421 T^{16} - 532386908048 T^{20} + 309458551597642 T^{24} + 184108626964604476 T^{28} + 48661191875666868481 T^{32} \))(\( 1 + 316 T^{4} + 44362 T^{8} - 6374288 T^{12} - 2550901421 T^{16} - 532386908048 T^{20} + 309458551597642 T^{24} + 184108626964604476 T^{28} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 - 48 T^{2} + 1118 T^{4} - 17328 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 48 T^{2} + 1118 T^{4} - 17328 T^{6} + 130321 T^{8} )^{2} \))(\( 1 - 104 T^{2} + 6018 T^{4} - 249384 T^{6} + 7954675 T^{8} - 203686352 T^{10} + 4270353988 T^{12} - 73530773072 T^{14} + 1036661200675 T^{16} - 11732489987304 T^{18} + 102207082380738 T^{20} - 637630890811304 T^{22} + 2213314919066161 T^{24} \))(\( 1 - 104 T^{2} + 6018 T^{4} - 249384 T^{6} + 7954675 T^{8} - 203686352 T^{10} + 4270353988 T^{12} - 73530773072 T^{14} + 1036661200675 T^{16} - 11732489987304 T^{18} + 102207082380738 T^{20} - 637630890811304 T^{22} + 2213314919066161 T^{24} \))(\( ( 1 - 6 T^{2} + 577 T^{4} - 17262 T^{6} + 120564 T^{8} - 6231582 T^{10} + 75195217 T^{12} - 282275286 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 6 T^{2} + 577 T^{4} - 17262 T^{6} + 120564 T^{8} - 6231582 T^{10} + 75195217 T^{12} - 282275286 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 508 T^{4} + 4678 T^{8} - 142159228 T^{12} + 78310985281 T^{16} \))(\( 1 - 508 T^{4} + 4678 T^{8} - 142159228 T^{12} + 78310985281 T^{16} \))(\( 1 - 24 T + 288 T^{2} - 2648 T^{3} + 22078 T^{4} - 163240 T^{5} + 1065248 T^{6} - 6483944 T^{7} + 37525599 T^{8} - 203370768 T^{9} + 1048113088 T^{10} - 5277774096 T^{11} + 25830292740 T^{12} - 121388804208 T^{13} + 554451823552 T^{14} - 2474412134256 T^{15} + 10501201149759 T^{16} - 41732887576792 T^{17} + 157694934685472 T^{18} - 555803705968280 T^{19} + 1728949933033918 T^{20} - 4769452247554024 T^{21} + 11930835229530912 T^{22} - 22867434189934248 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 24 T + 288 T^{2} + 2648 T^{3} + 22078 T^{4} + 163240 T^{5} + 1065248 T^{6} + 6483944 T^{7} + 37525599 T^{8} + 203370768 T^{9} + 1048113088 T^{10} + 5277774096 T^{11} + 25830292740 T^{12} + 121388804208 T^{13} + 554451823552 T^{14} + 2474412134256 T^{15} + 10501201149759 T^{16} + 41732887576792 T^{17} + 157694934685472 T^{18} + 555803705968280 T^{19} + 1728949933033918 T^{20} + 4769452247554024 T^{21} + 11930835229530912 T^{22} + 22867434189934248 T^{23} + 21914624432020321 T^{24} \))(\( 1 - 446 T^{4} + 750657 T^{8} - 288991678 T^{12} + 298801637060 T^{16} - 80871720163198 T^{20} + 58784689278079617 T^{24} - 9773922496681063166 T^{28} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 446 T^{4} + 750657 T^{8} - 288991678 T^{12} + 298801637060 T^{16} - 80871720163198 T^{20} + 58784689278079617 T^{24} - 9773922496681063166 T^{28} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 - 6 T + 62 T^{2} - 174 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 + 6 T + 62 T^{2} + 174 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 - 4 T + 114 T^{2} - 548 T^{3} + 6631 T^{4} - 29256 T^{5} + 241692 T^{6} - 848424 T^{7} + 5576671 T^{8} - 13365172 T^{9} + 80630034 T^{10} - 82044596 T^{11} + 594823321 T^{12} )^{2} \))(\( ( 1 + 4 T + 114 T^{2} + 548 T^{3} + 6631 T^{4} + 29256 T^{5} + 241692 T^{6} + 848424 T^{7} + 5576671 T^{8} + 13365172 T^{9} + 80630034 T^{10} + 82044596 T^{11} + 594823321 T^{12} )^{2} \))(\( ( 1 + 37 T^{2} + 96 T^{3} + 1140 T^{4} + 2784 T^{5} + 31117 T^{6} + 707281 T^{8} )^{4} \))(\( ( 1 + 37 T^{2} - 96 T^{3} + 1140 T^{4} - 2784 T^{5} + 31117 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 6 T + 66 T^{2} - 186 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 - 6 T + 66 T^{2} - 186 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 + 4 T + 28 T^{2} + 76 T^{3} + 807 T^{4} + 1288 T^{5} + 32184 T^{6} + 39928 T^{7} + 775527 T^{8} + 2264116 T^{9} + 25858588 T^{10} + 114516604 T^{11} + 887503681 T^{12} )^{2} \))(\( ( 1 + 4 T + 28 T^{2} + 76 T^{3} + 807 T^{4} + 1288 T^{5} + 32184 T^{6} + 39928 T^{7} + 775527 T^{8} + 2264116 T^{9} + 25858588 T^{10} + 114516604 T^{11} + 887503681 T^{12} )^{2} \))(\( ( 1 + 57 T^{2} + 24 T^{3} + 2588 T^{4} + 744 T^{5} + 54777 T^{6} + 923521 T^{8} )^{4} \))(\( ( 1 + 57 T^{2} + 24 T^{3} + 2588 T^{4} + 744 T^{5} + 54777 T^{6} + 923521 T^{8} )^{4} \))
$37$ (\( 1 - 604 T^{4} - 32474 T^{8} - 1131993244 T^{12} + 3512479453921 T^{16} \))(\( 1 - 604 T^{4} - 32474 T^{8} - 1131993244 T^{12} + 3512479453921 T^{16} \))(\( 1 - 20 T + 200 T^{2} - 1052 T^{3} + 3634 T^{4} - 30620 T^{5} + 438952 T^{6} - 3655156 T^{7} + 16552975 T^{8} - 43779560 T^{9} + 283860944 T^{10} - 4234271992 T^{11} + 34611552220 T^{12} - 156668063704 T^{13} + 388605632336 T^{14} - 2217566052680 T^{15} + 31022940178975 T^{16} - 253462980492292 T^{17} + 1126230738683368 T^{18} - 2906814077812460 T^{19} + 12764350335548914 T^{20} - 136719750264421004 T^{21} + 961716874483569800 T^{22} - 3558352435589208260 T^{23} + 6582952005840035281 T^{24} \))(\( 1 - 20 T + 200 T^{2} - 1052 T^{3} + 3634 T^{4} - 30620 T^{5} + 438952 T^{6} - 3655156 T^{7} + 16552975 T^{8} - 43779560 T^{9} + 283860944 T^{10} - 4234271992 T^{11} + 34611552220 T^{12} - 156668063704 T^{13} + 388605632336 T^{14} - 2217566052680 T^{15} + 31022940178975 T^{16} - 253462980492292 T^{17} + 1126230738683368 T^{18} - 2906814077812460 T^{19} + 12764350335548914 T^{20} - 136719750264421004 T^{21} + 961716874483569800 T^{22} - 3558352435589208260 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 2632 T^{4} + 4493148 T^{8} + 6676855928 T^{12} + 9052768472966 T^{16} + 12513502982876408 T^{20} + 15782090033426233308 T^{24} + \)\(17\!\cdots\!92\)\( T^{28} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 + 2632 T^{4} + 4493148 T^{8} + 6676855928 T^{12} + 9052768472966 T^{16} + 12513502982876408 T^{20} + 15782090033426233308 T^{24} + \)\(17\!\cdots\!92\)\( T^{28} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( ( 1 - 56 T^{2} + 3166 T^{4} - 94136 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 3166 T^{4} - 94136 T^{6} + 2825761 T^{8} )^{2} \))(\( 1 - 212 T^{2} + 22434 T^{4} - 1643716 T^{6} + 97594575 T^{8} - 5014736296 T^{10} + 222514140252 T^{12} - 8429771713576 T^{14} + 275778943846575 T^{16} - 7807822342599556 T^{18} + 179133812590100514 T^{20} - 2845603773752309012 T^{22} + 22563490300366186081 T^{24} \))(\( 1 - 212 T^{2} + 22434 T^{4} - 1643716 T^{6} + 97594575 T^{8} - 5014736296 T^{10} + 222514140252 T^{12} - 8429771713576 T^{14} + 275778943846575 T^{16} - 7807822342599556 T^{18} + 179133812590100514 T^{20} - 2845603773752309012 T^{22} + 22563490300366186081 T^{24} \))(\( ( 1 - 256 T^{2} + 30642 T^{4} - 2250752 T^{6} + 111180659 T^{8} - 3783514112 T^{10} + 86586968562 T^{12} - 1216026685696 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 256 T^{2} + 30642 T^{4} - 2250752 T^{6} + 111180659 T^{8} - 3783514112 T^{10} + 86586968562 T^{12} - 1216026685696 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( 1 + 5444 T^{4} + 13734886 T^{8} + 18611952644 T^{12} + 11688200277601 T^{16} \))(\( 1 + 5444 T^{4} + 13734886 T^{8} + 18611952644 T^{12} + 11688200277601 T^{16} \))(\( 1 + 8 T + 32 T^{2} + 184 T^{3} + 5014 T^{4} + 25096 T^{5} + 57248 T^{6} - 105352 T^{7} + 1869663 T^{8} - 20409392 T^{9} - 214625984 T^{10} - 2505871696 T^{11} - 14493786700 T^{12} - 107752482928 T^{13} - 396843444416 T^{14} - 1622689529744 T^{15} + 6392005734063 T^{16} - 15487633486936 T^{17} + 361885391829152 T^{18} + 6821559864341272 T^{19} + 58604636191891414 T^{20} + 92477040596379112 T^{21} + 691567434025095968 T^{22} + 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \))(\( 1 + 8 T + 32 T^{2} + 184 T^{3} + 5014 T^{4} + 25096 T^{5} + 57248 T^{6} - 105352 T^{7} + 1869663 T^{8} - 20409392 T^{9} - 214625984 T^{10} - 2505871696 T^{11} - 14493786700 T^{12} - 107752482928 T^{13} - 396843444416 T^{14} - 1622689529744 T^{15} + 6392005734063 T^{16} - 15487633486936 T^{17} + 361885391829152 T^{18} + 6821559864341272 T^{19} + 58604636191891414 T^{20} + 92477040596379112 T^{21} + 691567434025095968 T^{22} + 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \))(\( 1 - 10046 T^{4} + 45003681 T^{8} - 124711937854 T^{12} + 257102855207972 T^{16} - 426365297847193054 T^{20} + \)\(52\!\cdots\!81\)\( T^{24} - \)\(40\!\cdots\!46\)\( T^{28} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 - 10046 T^{4} + 45003681 T^{8} - 124711937854 T^{12} + 257102855207972 T^{16} - 426365297847193054 T^{20} + \)\(52\!\cdots\!81\)\( T^{24} - \)\(40\!\cdots\!46\)\( T^{28} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 5788 T^{4} + 16086598 T^{8} - 28243593628 T^{12} + 23811286661761 T^{16} \))(\( 1 - 5788 T^{4} + 16086598 T^{8} - 28243593628 T^{12} + 23811286661761 T^{16} \))(\( 1 + 16 T + 128 T^{2} + 48 T^{3} - 3930 T^{4} + 2832 T^{5} + 549504 T^{6} + 4835248 T^{7} + 9727055 T^{8} - 102211168 T^{9} - 484788992 T^{10} + 7463204576 T^{11} + 87433873108 T^{12} + 350770615072 T^{13} - 1070898883328 T^{14} - 10611870095264 T^{15} + 47464925469455 T^{16} + 1108939986406736 T^{17} + 5923221940146816 T^{18} + 1434756677151216 T^{19} - 93578356580720730 T^{20} + 53718262708932816 T^{21} + 6732688926186246272 T^{22} + 39554547441344196848 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 16 T + 128 T^{2} - 48 T^{3} - 3930 T^{4} - 2832 T^{5} + 549504 T^{6} - 4835248 T^{7} + 9727055 T^{8} + 102211168 T^{9} - 484788992 T^{10} - 7463204576 T^{11} + 87433873108 T^{12} - 350770615072 T^{13} - 1070898883328 T^{14} + 10611870095264 T^{15} + 47464925469455 T^{16} - 1108939986406736 T^{17} + 5923221940146816 T^{18} - 1434756677151216 T^{19} - 93578356580720730 T^{20} - 53718262708932816 T^{21} + 6732688926186246272 T^{22} - 39554547441344196848 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 3016 T^{4} + 12169116 T^{8} + 33492373112 T^{12} + 77269163408966 T^{16} + 163432096719537272 T^{20} + \)\(28\!\cdots\!76\)\( T^{24} + \)\(35\!\cdots\!56\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 3016 T^{4} + 12169116 T^{8} + 33492373112 T^{12} + 77269163408966 T^{16} + 163432096719537272 T^{20} + \)\(28\!\cdots\!76\)\( T^{24} + \)\(35\!\cdots\!56\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 7708 T^{4} + 28156198 T^{8} - 60819827548 T^{12} + 62259690411361 T^{16} \))(\( 1 - 7708 T^{4} + 28156198 T^{8} - 60819827548 T^{12} + 62259690411361 T^{16} \))(\( 1 - 24 T + 288 T^{2} - 2968 T^{3} + 27318 T^{4} - 189144 T^{5} + 1076384 T^{6} - 4855704 T^{7} - 9792929 T^{8} + 439179408 T^{9} - 4825144512 T^{10} + 45292675216 T^{11} - 372958615948 T^{12} + 2400511786448 T^{13} - 13553830934208 T^{14} + 65383712724816 T^{15} - 77270920208849 T^{16} - 2030633528142072 T^{17} + 23857363689477536 T^{18} - 222189563833329528 T^{19} + 1700810222657559798 T^{20} - 9793698340468730744 T^{21} + 50367591465267758112 T^{22} - \)\(22\!\cdots\!28\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 24 T + 288 T^{2} + 2968 T^{3} + 27318 T^{4} + 189144 T^{5} + 1076384 T^{6} + 4855704 T^{7} - 9792929 T^{8} - 439179408 T^{9} - 4825144512 T^{10} - 45292675216 T^{11} - 372958615948 T^{12} - 2400511786448 T^{13} - 13553830934208 T^{14} - 65383712724816 T^{15} - 77270920208849 T^{16} + 2030633528142072 T^{17} + 23857363689477536 T^{18} + 222189563833329528 T^{19} + 1700810222657559798 T^{20} + 9793698340468730744 T^{21} + 50367591465267758112 T^{22} + \)\(22\!\cdots\!28\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 15326 T^{4} + 116331393 T^{8} - 563051617822 T^{12} + 1882603650749540 T^{16} - 4442748092443752382 T^{20} + \)\(72\!\cdots\!73\)\( T^{24} - \)\(75\!\cdots\!66\)\( T^{28} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 - 15326 T^{4} + 116331393 T^{8} - 563051617822 T^{12} + 1882603650749540 T^{16} - 4442748092443752382 T^{20} + \)\(72\!\cdots\!73\)\( T^{24} - \)\(75\!\cdots\!66\)\( T^{28} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( ( 1 - 14 T + 122 T^{2} - 826 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 14 T + 122 T^{2} + 826 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 - 16 T + 270 T^{2} - 3160 T^{3} + 33193 T^{4} - 284568 T^{5} + 2445376 T^{6} - 16789512 T^{7} + 115544833 T^{8} - 648997640 T^{9} + 3271687470 T^{10} - 11438788784 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 16 T + 270 T^{2} + 3160 T^{3} + 33193 T^{4} + 284568 T^{5} + 2445376 T^{6} + 16789512 T^{7} + 115544833 T^{8} + 648997640 T^{9} + 3271687470 T^{10} + 11438788784 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 10 T + 233 T^{2} + 1522 T^{3} + 20008 T^{4} + 89798 T^{5} + 811073 T^{6} + 2053790 T^{7} + 12117361 T^{8} )^{4} \))(\( ( 1 - 10 T + 233 T^{2} - 1522 T^{3} + 20008 T^{4} - 89798 T^{5} + 811073 T^{6} - 2053790 T^{7} + 12117361 T^{8} )^{4} \))
$61$ (\( ( 1 + 8 T + 58 T^{2} + 488 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 8 T + 58 T^{2} + 488 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 178 T^{2} + 664 T^{3} + 12681 T^{4} + 121240 T^{5} + 672800 T^{6} + 7395640 T^{7} + 47186001 T^{8} + 150715384 T^{9} + 2464559698 T^{10} + 51520374361 T^{12} )^{2} \))(\( ( 1 + 178 T^{2} + 664 T^{3} + 12681 T^{4} + 121240 T^{5} + 672800 T^{6} + 7395640 T^{7} + 47186001 T^{8} + 150715384 T^{9} + 2464559698 T^{10} + 51520374361 T^{12} )^{2} \))(\( ( 1 - 8 T + 129 T^{2} - 772 T^{3} + 7388 T^{4} - 47092 T^{5} + 480009 T^{6} - 1815848 T^{7} + 13845841 T^{8} )^{4} \))(\( ( 1 - 8 T + 129 T^{2} - 772 T^{3} + 7388 T^{4} - 47092 T^{5} + 480009 T^{6} - 1815848 T^{7} + 13845841 T^{8} )^{4} \))
$67$ (\( 1 - 12988 T^{4} + 81854758 T^{8} - 261722759548 T^{12} + 406067677556641 T^{16} \))(\( 1 - 12988 T^{4} + 81854758 T^{8} - 261722759548 T^{12} + 406067677556641 T^{16} \))(\( 1 - 64 T^{3} + 2390 T^{4} + 9024 T^{5} + 2048 T^{6} + 508672 T^{7} + 20286015 T^{8} - 4116224 T^{9} + 3266560 T^{10} + 440818304 T^{11} + 142523500468 T^{12} + 29534826368 T^{13} + 14663587840 T^{14} - 1238007878912 T^{15} + 408785942872815 T^{16} + 686770838427904 T^{17} + 185258766682112 T^{18} + 54691861526434752 T^{19} + 970501749360371990 T^{20} - 1741218201362876608 T^{21} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 64 T^{3} + 2390 T^{4} + 9024 T^{5} + 2048 T^{6} + 508672 T^{7} + 20286015 T^{8} - 4116224 T^{9} + 3266560 T^{10} + 440818304 T^{11} + 142523500468 T^{12} + 29534826368 T^{13} + 14663587840 T^{14} - 1238007878912 T^{15} + 408785942872815 T^{16} + 686770838427904 T^{17} + 185258766682112 T^{18} + 54691861526434752 T^{19} + 970501749360371990 T^{20} - 1741218201362876608 T^{21} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 10738 T^{4} + 95935473 T^{8} + 575204178770 T^{12} + 2915472960881732 T^{16} + 11591009006099901170 T^{20} + \)\(38\!\cdots\!93\)\( T^{24} + \)\(87\!\cdots\!18\)\( T^{28} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 + 10738 T^{4} + 95935473 T^{8} + 575204178770 T^{12} + 2915472960881732 T^{16} + 11591009006099901170 T^{20} + \)\(38\!\cdots\!93\)\( T^{24} + \)\(87\!\cdots\!18\)\( T^{28} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 + 84 T^{2} + 8966 T^{4} + 423444 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 84 T^{2} + 8966 T^{4} + 423444 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 440 T^{2} + 91446 T^{4} - 12419864 T^{6} + 1305779695 T^{8} - 116298170672 T^{10} + 8924106004404 T^{12} - 586259078357552 T^{14} + 33182057065617295 T^{16} - 1590988104660206744 T^{18} + 59051577418299860406 T^{20} - \)\(14\!\cdots\!40\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 440 T^{2} + 91446 T^{4} - 12419864 T^{6} + 1305779695 T^{8} - 116298170672 T^{10} + 8924106004404 T^{12} - 586259078357552 T^{14} + 33182057065617295 T^{16} - 1590988104660206744 T^{18} + 59051577418299860406 T^{20} - \)\(14\!\cdots\!40\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 132 T^{2} + 11366 T^{4} - 665412 T^{6} + 25411681 T^{8} )^{4} \))(\( ( 1 - 132 T^{2} + 11366 T^{4} - 665412 T^{6} + 25411681 T^{8} )^{4} \))
$73$ (\( 1 - 1148 T^{4} + 2279238 T^{8} - 32601180668 T^{12} + 806460091894081 T^{16} \))(\( 1 - 1148 T^{4} + 2279238 T^{8} - 32601180668 T^{12} + 806460091894081 T^{16} \))(\( 1 - 24 T + 288 T^{2} - 2936 T^{3} + 13126 T^{4} + 142696 T^{5} - 2894944 T^{6} + 37628616 T^{7} - 323587345 T^{8} + 985106832 T^{9} + 7253083968 T^{10} - 194887191344 T^{11} + 2401164656404 T^{12} - 14226764968112 T^{13} + 38651684465472 T^{14} + 383223304464144 T^{15} - 9189311407860145 T^{16} + 78006814913505288 T^{17} - 438104110389982816 T^{18} + 1576419579081065512 T^{19} + 10585595166201707206 T^{20} - \)\(17\!\cdots\!68\)\( T^{21} + \)\(12\!\cdots\!12\)\( T^{22} - \)\(75\!\cdots\!48\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 - 24 T + 288 T^{2} - 2936 T^{3} + 13126 T^{4} + 142696 T^{5} - 2894944 T^{6} + 37628616 T^{7} - 323587345 T^{8} + 985106832 T^{9} + 7253083968 T^{10} - 194887191344 T^{11} + 2401164656404 T^{12} - 14226764968112 T^{13} + 38651684465472 T^{14} + 383223304464144 T^{15} - 9189311407860145 T^{16} + 78006814913505288 T^{17} - 438104110389982816 T^{18} + 1576419579081065512 T^{19} + 10585595166201707206 T^{20} - \)\(17\!\cdots\!68\)\( T^{21} + \)\(12\!\cdots\!12\)\( T^{22} - \)\(75\!\cdots\!48\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 23186 T^{4} + 265691377 T^{8} + 2044672338098 T^{12} + 12134752575351268 T^{16} + 58065097823340485618 T^{20} + \)\(21\!\cdots\!37\)\( T^{24} + \)\(53\!\cdots\!06\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 + 23186 T^{4} + 265691377 T^{8} + 2044672338098 T^{12} + 12134752575351268 T^{16} + 58065097823340485618 T^{20} + \)\(21\!\cdots\!37\)\( T^{24} + \)\(53\!\cdots\!06\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( ( 1 - 124 T^{2} + 11206 T^{4} - 773884 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 124 T^{2} + 11206 T^{4} - 773884 T^{6} + 38950081 T^{8} )^{2} \))(\( 1 - 508 T^{2} + 135562 T^{4} - 24649964 T^{6} + 3381685343 T^{8} - 366753014104 T^{10} + 32137748903980 T^{12} - 2288905561023064 T^{14} + 131716918026362783 T^{16} - 5992097027444251244 T^{18} + \)\(20\!\cdots\!82\)\( T^{20} - \)\(48\!\cdots\!08\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 - 508 T^{2} + 135562 T^{4} - 24649964 T^{6} + 3381685343 T^{8} - 366753014104 T^{10} + 32137748903980 T^{12} - 2288905561023064 T^{14} + 131716918026362783 T^{16} - 5992097027444251244 T^{18} + \)\(20\!\cdots\!82\)\( T^{20} - \)\(48\!\cdots\!08\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 464 T^{2} + 97500 T^{4} - 12633136 T^{6} + 1158681542 T^{8} - 78843401776 T^{10} + 3797632897500 T^{12} - 112792579361744 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 464 T^{2} + 97500 T^{4} - 12633136 T^{6} + 1158681542 T^{8} - 78843401776 T^{10} + 3797632897500 T^{12} - 112792579361744 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 + 10436 T^{4} + 85152166 T^{8} + 495275037956 T^{12} + 2252292232139041 T^{16} \))(\( 1 + 10436 T^{4} + 85152166 T^{8} + 495275037956 T^{12} + 2252292232139041 T^{16} \))(\( 1 - 24 T + 288 T^{2} - 3784 T^{3} + 70066 T^{4} - 929320 T^{5} + 9284000 T^{6} - 107957496 T^{7} + 1400664563 T^{8} - 14071243520 T^{9} + 124154702400 T^{10} - 1299362373632 T^{11} + 13371946678180 T^{12} - 107847077011456 T^{13} + 855301744833600 T^{14} - 8045754118570240 T^{15} + 66473188444178723 T^{16} - 425248964460509928 T^{17} + 3035314426357796000 T^{18} - 25218074905680163640 T^{19} + \)\(15\!\cdots\!06\)\( T^{20} - \)\(70\!\cdots\!52\)\( T^{21} + \)\(44\!\cdots\!12\)\( T^{22} - \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 24 T + 288 T^{2} + 3784 T^{3} + 70066 T^{4} + 929320 T^{5} + 9284000 T^{6} + 107957496 T^{7} + 1400664563 T^{8} + 14071243520 T^{9} + 124154702400 T^{10} + 1299362373632 T^{11} + 13371946678180 T^{12} + 107847077011456 T^{13} + 855301744833600 T^{14} + 8045754118570240 T^{15} + 66473188444178723 T^{16} + 425248964460509928 T^{17} + 3035314426357796000 T^{18} + 25218074905680163640 T^{19} + \)\(15\!\cdots\!06\)\( T^{20} + \)\(70\!\cdots\!52\)\( T^{21} + \)\(44\!\cdots\!12\)\( T^{22} + \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 - 19412 T^{4} + 133452090 T^{8} - 480056569264 T^{12} + 2140269989799107 T^{16} - 22782678762289645744 T^{20} + \)\(30\!\cdots\!90\)\( T^{24} - \)\(20\!\cdots\!32\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 19412 T^{4} + 133452090 T^{8} - 480056569264 T^{12} + 2140269989799107 T^{16} - 22782678762289645744 T^{20} + \)\(30\!\cdots\!90\)\( T^{24} - \)\(20\!\cdots\!32\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 - 14 T + 222 T^{2} - 1246 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 + 14 T + 222 T^{2} + 1246 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 - 24 T + 358 T^{2} - 5272 T^{3} + 62767 T^{4} - 696208 T^{5} + 7360436 T^{6} - 61962512 T^{7} + 497177407 T^{8} - 3716596568 T^{9} + 22461722278 T^{10} - 134017426776 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 24 T + 358 T^{2} + 5272 T^{3} + 62767 T^{4} + 696208 T^{5} + 7360436 T^{6} + 61962512 T^{7} + 497177407 T^{8} + 3716596568 T^{9} + 22461722278 T^{10} + 134017426776 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 8 T + 193 T^{2} - 1204 T^{3} + 18084 T^{4} - 107156 T^{5} + 1528753 T^{6} - 5639752 T^{7} + 62742241 T^{8} )^{4} \))(\( ( 1 + 8 T + 193 T^{2} + 1204 T^{3} + 18084 T^{4} + 107156 T^{5} + 1528753 T^{6} + 5639752 T^{7} + 62742241 T^{8} )^{4} \))
$97$ (\( 1 - 21436 T^{4} + 227134086 T^{8} - 1897713667516 T^{12} + 7837433594376961 T^{16} \))(\( 1 - 21436 T^{4} + 227134086 T^{8} - 1897713667516 T^{12} + 7837433594376961 T^{16} \))(\( 1 + 8 T + 32 T^{2} - 1240 T^{3} + 7270 T^{4} + 77896 T^{5} + 1159328 T^{6} - 23481432 T^{7} - 27667953 T^{8} - 164181296 T^{9} + 23294478656 T^{10} - 157379895024 T^{11} - 641510857644 T^{12} - 15265849817328 T^{13} + 219177749674304 T^{14} - 149843835964208 T^{15} - 2449423985831793 T^{16} - 201643046305608024 T^{17} + 965687768530327712 T^{18} + 6293863167707090248 T^{19} + 56978142231120506470 T^{20} - \)\(94\!\cdots\!80\)\( T^{21} + \)\(23\!\cdots\!68\)\( T^{22} + \)\(57\!\cdots\!24\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 8 T + 32 T^{2} - 1240 T^{3} + 7270 T^{4} + 77896 T^{5} + 1159328 T^{6} - 23481432 T^{7} - 27667953 T^{8} - 164181296 T^{9} + 23294478656 T^{10} - 157379895024 T^{11} - 641510857644 T^{12} - 15265849817328 T^{13} + 219177749674304 T^{14} - 149843835964208 T^{15} - 2449423985831793 T^{16} - 201643046305608024 T^{17} + 965687768530327712 T^{18} + 6293863167707090248 T^{19} + 56978142231120506470 T^{20} - \)\(94\!\cdots\!80\)\( T^{21} + \)\(23\!\cdots\!68\)\( T^{22} + \)\(57\!\cdots\!24\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 2440 T^{4} + 62893852 T^{8} + 1375626588856 T^{12} + 4265661590968390 T^{16} + \)\(12\!\cdots\!36\)\( T^{20} + \)\(49\!\cdots\!72\)\( T^{24} + \)\(16\!\cdots\!40\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 2440 T^{4} + 62893852 T^{8} + 1375626588856 T^{12} + 4265661590968390 T^{16} + \)\(12\!\cdots\!36\)\( T^{20} + \)\(49\!\cdots\!72\)\( T^{24} + \)\(16\!\cdots\!40\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))
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