Properties

Label 225.2.m
Level 225
Weight 2
Character orbit m
Rep. character \(\chi_{225}(19,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 48
Newform subspaces 3
Sturm bound 60
Trace bound 1

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Defining parameters

Level: \( N \) = \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 225.m (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 3 \)
Sturm bound: \(60\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(225, [\chi])\).

Total New Old
Modular forms 136 56 80
Cusp forms 104 48 56
Eisenstein series 32 8 24

Trace form

\( 48q + 5q^{2} + 9q^{4} + 20q^{8} + O(q^{10}) \) \( 48q + 5q^{2} + 9q^{4} + 20q^{8} - 15q^{10} + 10q^{11} - 5q^{13} - q^{14} - 31q^{16} + 7q^{19} - 5q^{20} + 40q^{22} + 15q^{23} - 20q^{25} - 18q^{26} + 45q^{28} - 11q^{29} - 21q^{31} - 13q^{34} - 25q^{35} - 45q^{38} - 60q^{40} + 18q^{41} - 24q^{44} - 3q^{46} - 40q^{47} - 58q^{49} - 5q^{50} - 110q^{52} - 10q^{55} - 10q^{56} - 100q^{58} - 12q^{59} - 21q^{61} + 40q^{62} + 14q^{64} + 65q^{65} + 60q^{67} + 90q^{70} + 2q^{71} - 25q^{73} + 64q^{74} + 36q^{76} + 30q^{77} + 31q^{79} - 20q^{80} + 35q^{83} + 55q^{85} + 45q^{86} + 120q^{88} + 7q^{89} + 2q^{91} - 40q^{92} - 15q^{94} + 25q^{95} + 80q^{97} - 50q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
225.2.m.a \(8\) \(1.797\) 8.0.58140625.2 None \(5\) \(0\) \(0\) \(0\) \(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
225.2.m.b \(16\) \(1.797\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{3}+\beta _{4}-\beta _{10}+\beta _{15})q^{2}+\cdots\)
225.2.m.c \(24\) \(1.797\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(225, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(225, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 5 T + 15 T^{2} - 30 T^{3} + 41 T^{4} - 30 T^{5} - 20 T^{6} + 110 T^{7} - 199 T^{8} + 220 T^{9} - 80 T^{10} - 240 T^{11} + 656 T^{12} - 960 T^{13} + 960 T^{14} - 640 T^{15} + 256 T^{16} \))(\( 1 + 3 T^{2} - 10 T^{3} + 6 T^{4} - 30 T^{5} + 61 T^{6} - 80 T^{7} + 165 T^{8} - 310 T^{9} + 503 T^{10} - 690 T^{11} + 1290 T^{12} - 2060 T^{13} + 2585 T^{14} - 4150 T^{15} + 6461 T^{16} - 8300 T^{17} + 10340 T^{18} - 16480 T^{19} + 20640 T^{20} - 22080 T^{21} + 32192 T^{22} - 39680 T^{23} + 42240 T^{24} - 40960 T^{25} + 62464 T^{26} - 61440 T^{27} + 24576 T^{28} - 81920 T^{29} + 49152 T^{30} + 65536 T^{32} \))
$3$ 1
$5$ (\( 1 + 5 T^{2} + 20 T^{3} + 5 T^{4} + 100 T^{5} + 125 T^{6} + 625 T^{8} \))(\( 1 + 5 T^{2} - 10 T^{3} + 10 T^{4} + 50 T^{5} + 25 T^{6} + 450 T^{7} - 725 T^{8} + 2250 T^{9} + 625 T^{10} + 6250 T^{11} + 6250 T^{12} - 31250 T^{13} + 78125 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - 35 T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 345205 T^{10} + 1467011 T^{12} - 4117715 T^{14} + 5764801 T^{16} \))(\( 1 - 56 T^{2} + 1628 T^{4} - 32234 T^{6} + 485518 T^{8} - 5899406 T^{10} + 59922825 T^{12} - 520141390 T^{14} + 3906488461 T^{16} - 25486928110 T^{18} + 143874702825 T^{20} - 694059216494 T^{22} + 2798914651918 T^{24} - 9105307176266 T^{26} + 22533615563228 T^{28} - 37980492079544 T^{30} + 33232930569601 T^{32} \))
$11$ (\( ( 1 - 2 T - 7 T^{2} + 36 T^{3} + 5 T^{4} + 396 T^{5} - 847 T^{6} - 2662 T^{7} + 14641 T^{8} )^{2} \))(\( 1 - 6 T + 4 T^{2} + 68 T^{3} - 237 T^{4} + 38 T^{5} + 2400 T^{6} - 9328 T^{7} - 1188 T^{8} + 114992 T^{9} - 345452 T^{10} + 52314 T^{11} + 3321817 T^{12} - 16648114 T^{13} + 30762856 T^{14} + 86386670 T^{15} - 607501509 T^{16} + 950253370 T^{17} + 3722305576 T^{18} - 22158639734 T^{19} + 48634722697 T^{20} + 8425222014 T^{21} - 611989290572 T^{22} + 2240868767632 T^{23} - 254658350628 T^{24} - 21994936061648 T^{25} + 62249819042400 T^{26} + 10841843483218 T^{27} - 743807525282877 T^{28} + 2347544425787308 T^{29} + 1518999334332964 T^{30} - 25063489016493906 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 + 5 T + 30 T^{2} + 60 T^{3} + 346 T^{4} + 655 T^{5} + 6335 T^{6} + 13320 T^{7} + 88856 T^{8} + 173160 T^{9} + 1070615 T^{10} + 1439035 T^{11} + 9882106 T^{12} + 22277580 T^{13} + 144804270 T^{14} + 313742585 T^{15} + 815730721 T^{16} \))(\( 1 + 22 T^{2} + 80 T^{3} + 81 T^{4} + 1760 T^{5} + 1994 T^{6} + 7400 T^{7} + 54580 T^{8} + 208560 T^{9} + 320132 T^{10} + 4629240 T^{11} + 13798055 T^{12} + 28576760 T^{13} + 253427140 T^{14} + 524829280 T^{15} + 2023398311 T^{16} + 6822780640 T^{17} + 42829186660 T^{18} + 62783141720 T^{19} + 394086248855 T^{20} + 1718804407320 T^{21} + 1545216018788 T^{22} + 13086830705520 T^{23} + 44522582752180 T^{24} + 78473295360200 T^{25} + 274889832746906 T^{26} + 3154202293505120 T^{27} + 1887144894920961 T^{28} + 24230008527380240 T^{29} + 86622280485384358 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 10 T + 90 T^{2} - 720 T^{3} + 4451 T^{4} - 26310 T^{5} + 136780 T^{6} - 638720 T^{7} + 2802941 T^{8} - 10858240 T^{9} + 39529420 T^{10} - 129261030 T^{11} + 371751971 T^{12} - 1022297040 T^{13} + 2172381210 T^{14} - 4103386730 T^{15} + 6975757441 T^{16} \))(\( 1 + 10 T + 55 T^{2} + 550 T^{3} + 4137 T^{4} + 21370 T^{5} + 140590 T^{6} + 875240 T^{7} + 4267373 T^{8} + 23180400 T^{9} + 125011430 T^{10} + 567252720 T^{11} + 2721539524 T^{12} + 13138878600 T^{13} + 55061701425 T^{14} + 238439860100 T^{15} + 1051104835255 T^{16} + 4053477621700 T^{17} + 15912831711825 T^{18} + 64551310561800 T^{19} + 227305702584004 T^{20} + 805417745261040 T^{21} + 3017472017413670 T^{22} + 9511814575609200 T^{23} + 29768158958272493 T^{24} + 103792853025234280 T^{25} + 283428582464124910 T^{26} + 732390424094117210 T^{27} + 2410308195419521257 T^{28} + 5447517918098265350 T^{29} + 9260780460767051095 T^{30} + 28624230515098157930 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 5 T - 8 T^{2} + 40 T^{3} + 878 T^{4} + 1705 T^{5} - 1861 T^{6} + 31550 T^{7} + 293380 T^{8} + 599450 T^{9} - 671821 T^{10} + 11694595 T^{11} + 114421838 T^{12} + 99043960 T^{13} - 376367048 T^{14} + 4469358695 T^{15} + 16983563041 T^{16} \))(\( 1 + 2 T - 53 T^{2} - 216 T^{3} + 1273 T^{4} + 10232 T^{5} - 10629 T^{6} - 233030 T^{7} - 154021 T^{8} + 2536598 T^{9} + 3525662 T^{10} + 321666 T^{11} + 122263380 T^{12} - 353312856 T^{13} - 6187122082 T^{14} + 3769508888 T^{15} + 148112464363 T^{16} + 71620668872 T^{17} - 2233551071602 T^{18} - 2423372879304 T^{19} + 15933485944980 T^{20} + 796476860934 T^{21} + 165867874898222 T^{22} + 2267393265403922 T^{23} - 2615825363137861 T^{24} - 75195914213440370 T^{25} - 65167103254166829 T^{26} + 1191928329046576808 T^{27} + 2817549891971222953 T^{28} - 9083444427847524744 T^{29} - 42347354346492858413 T^{30} + 30362254059749596598 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 5 T + 45 T^{2} - 100 T^{3} - 104 T^{4} - 7645 T^{5} + 18890 T^{6} - 8420 T^{7} + 1238691 T^{8} - 193660 T^{9} + 9992810 T^{10} - 93016715 T^{11} - 29103464 T^{12} - 643634300 T^{13} + 6661615005 T^{14} + 17024127235 T^{15} + 78310985281 T^{16} \))(\( 1 - 20 T + 181 T^{2} - 1150 T^{3} + 7948 T^{4} - 59890 T^{5} + 379519 T^{6} - 2059710 T^{7} + 11607418 T^{8} - 70214570 T^{9} + 395238326 T^{10} - 1984546720 T^{11} + 9942722840 T^{12} - 53914737500 T^{13} + 283746609760 T^{14} - 1328416871210 T^{15} + 6099237415331 T^{16} - 30553588037830 T^{17} + 150101956563040 T^{18} - 655980611162500 T^{19} + 2782381502268440 T^{20} - 12773223389444960 T^{21} + 58509456956281814 T^{22} - 239068354686162790 T^{23} + 908988340148414458 T^{24} - 3709852148341955730 T^{25} + 15722148109292854831 T^{26} - 57063776401465088030 T^{27} + \)\(17\!\cdots\!08\)\( T^{28} - \)\(57\!\cdots\!50\)\( T^{29} + \)\(20\!\cdots\!29\)\( T^{30} - \)\(53\!\cdots\!40\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 5 T - 28 T^{2} - 140 T^{3} + 1268 T^{4} + 5045 T^{5} + 48219 T^{6} - 207000 T^{7} - 1738520 T^{8} - 6003000 T^{9} + 40552179 T^{10} + 123042505 T^{11} + 896832308 T^{12} - 2871560860 T^{13} - 16655052988 T^{14} - 86249381545 T^{15} + 500246412961 T^{16} \))(\( 1 + 16 T + 101 T^{2} + 378 T^{3} + 2782 T^{4} + 28638 T^{5} + 156384 T^{6} + 372476 T^{7} + 1621718 T^{8} + 24643166 T^{9} + 158941381 T^{10} + 390275636 T^{11} + 1282706252 T^{12} + 24547929228 T^{13} + 196193283170 T^{14} + 670123125368 T^{15} + 1692236078907 T^{16} + 19433570635672 T^{17} + 164998551145970 T^{18} + 598699445941692 T^{19} + 907233760620812 T^{20} + 8005001721065764 T^{21} + 94542040090746301 T^{22} + 425091565362154294 T^{23} + 811258612334286998 T^{24} + 5403563704507781644 T^{25} + 65791879972418633184 T^{26} + \)\(34\!\cdots\!02\)\( T^{27} + \)\(98\!\cdots\!62\)\( T^{28} + \)\(38\!\cdots\!42\)\( T^{29} + \)\(30\!\cdots\!81\)\( T^{30} + \)\(13\!\cdots\!84\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 9 T + 55 T^{2} + 390 T^{3} + 2980 T^{4} + 20297 T^{5} + 114748 T^{6} + 589990 T^{7} + 3582095 T^{8} + 18289690 T^{9} + 110272828 T^{10} + 604667927 T^{11} + 2752092580 T^{12} + 11165368890 T^{13} + 48812702455 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \))(\( 1 - 6 T - 27 T^{2} + 222 T^{3} + 1398 T^{4} - 1096 T^{5} - 82911 T^{6} + 83080 T^{7} + 2495194 T^{8} + 3301786 T^{9} - 44430232 T^{10} - 311988762 T^{11} + 1830869280 T^{12} + 7523309618 T^{13} - 27963150078 T^{14} + 31095224934 T^{15} - 66108612217 T^{16} + 963951972954 T^{17} - 26872587224958 T^{18} + 224126916829838 T^{19} + 1690846228334880 T^{20} - 8931973377601062 T^{21} - 39431994447683992 T^{22} + 90840764095102246 T^{23} + 2128128599276558554 T^{24} + 2196603809108546680 T^{25} - 67956200901865191711 T^{26} - 27847690678459694776 T^{27} + \)\(11\!\cdots\!78\)\( T^{28} + \)\(54\!\cdots\!02\)\( T^{29} - \)\(20\!\cdots\!67\)\( T^{30} - \)\(14\!\cdots\!06\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 - 30 T + 480 T^{2} - 5675 T^{3} + 56171 T^{4} - 489630 T^{5} + 3826535 T^{6} - 26904445 T^{7} + 171416106 T^{8} - 995464465 T^{9} + 5238526415 T^{10} - 24801228390 T^{11} + 105273497531 T^{12} - 393526955975 T^{13} + 1231548676320 T^{14} - 2847956313990 T^{15} + 3512479453921 T^{16} \))(\( 1 + 10 T + 159 T^{2} + 1490 T^{3} + 15403 T^{4} + 128130 T^{5} + 1158066 T^{6} + 8990890 T^{7} + 72816793 T^{8} + 529727610 T^{9} + 3982453044 T^{10} + 27036193580 T^{11} + 190987118220 T^{12} + 1239202673080 T^{13} + 8238752278565 T^{14} + 50853391327250 T^{15} + 322012573505931 T^{16} + 1881575479108250 T^{17} + 11278851869355485 T^{18} + 62769332999521240 T^{19} + 357940608470313420 T^{20} + 1874796645055196060 T^{21} + 10217884947593238996 T^{22} + 50288036386477742130 T^{23} + \)\(25\!\cdots\!53\)\( T^{24} + \)\(11\!\cdots\!30\)\( T^{25} + \)\(55\!\cdots\!34\)\( T^{26} + \)\(22\!\cdots\!90\)\( T^{27} + \)\(10\!\cdots\!43\)\( T^{28} + \)\(36\!\cdots\!30\)\( T^{29} + \)\(14\!\cdots\!51\)\( T^{30} + \)\(33\!\cdots\!30\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 - 4 T - 30 T^{2} + 240 T^{3} + 195 T^{4} - 18372 T^{5} + 77388 T^{6} + 230240 T^{7} - 1438395 T^{8} + 9439840 T^{9} + 130089228 T^{10} - 1266216612 T^{11} + 551023395 T^{12} + 27805488240 T^{13} - 142503127230 T^{14} - 779017095524 T^{15} + 7984925229121 T^{16} \))(\( 1 - 14 T - 82 T^{2} + 2028 T^{3} - 747 T^{4} - 120014 T^{5} + 311054 T^{6} + 3602970 T^{7} - 14348876 T^{8} - 63645696 T^{9} + 522280968 T^{10} + 1039813212 T^{11} - 35365817705 T^{12} - 21504097848 T^{13} + 2388690604652 T^{14} + 220861336106 T^{15} - 115364883721937 T^{16} + 9055314780346 T^{17} + 4015388906420012 T^{18} - 1482083927782008 T^{19} - 99935348403898505 T^{20} + 120468808491927612 T^{21} + 2480889041090385288 T^{22} - 12395271310130866176 T^{23} - \)\(11\!\cdots\!96\)\( T^{24} + \)\(11\!\cdots\!70\)\( T^{25} + \)\(41\!\cdots\!54\)\( T^{26} - \)\(66\!\cdots\!74\)\( T^{27} - \)\(16\!\cdots\!07\)\( T^{28} + \)\(18\!\cdots\!88\)\( T^{29} - \)\(31\!\cdots\!02\)\( T^{30} - \)\(21\!\cdots\!14\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - 215 T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 2918101045 T^{10} + 78328149711 T^{12} - 1359093055535 T^{14} + 11688200277601 T^{16} \))(\( 1 - 300 T^{2} + 44842 T^{4} - 4521820 T^{6} + 352848163 T^{8} - 23099919100 T^{10} + 1325953845604 T^{12} - 67791756187600 T^{14} + 3089570546914805 T^{16} - 125346957190872400 T^{18} + 4533172333304800804 T^{20} - \)\(14\!\cdots\!00\)\( T^{22} + \)\(41\!\cdots\!63\)\( T^{24} - \)\(97\!\cdots\!80\)\( T^{26} + \)\(17\!\cdots\!42\)\( T^{28} - \)\(22\!\cdots\!00\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 110 T^{2} + 90 T^{3} + 4101 T^{4} + 9900 T^{5} - 3830 T^{6} + 910260 T^{7} - 5610889 T^{8} + 42782220 T^{9} - 8460470 T^{10} + 1027847700 T^{11} + 20011571781 T^{12} + 20641050630 T^{13} + 1185713686190 T^{14} + 23811286661761 T^{16} \))(\( 1 + 40 T + 865 T^{2} + 12910 T^{3} + 142622 T^{4} + 1167030 T^{5} + 6143280 T^{6} - 637840 T^{7} - 479197322 T^{8} - 6905707270 T^{9} - 64790306935 T^{10} - 444387466120 T^{11} - 1988154991916 T^{12} - 288127717900 T^{13} + 103937589685250 T^{14} + 1288860429553200 T^{15} + 10336092465289955 T^{16} + 60576440189000400 T^{17} + 229598135614717250 T^{18} - 29914284055531700 T^{19} - 9701562139107658796 T^{20} - \)\(10\!\cdots\!40\)\( T^{21} - \)\(69\!\cdots\!15\)\( T^{22} - \)\(34\!\cdots\!10\)\( T^{23} - \)\(11\!\cdots\!42\)\( T^{24} - \)\(71\!\cdots\!80\)\( T^{25} + \)\(32\!\cdots\!20\)\( T^{26} + \)\(28\!\cdots\!90\)\( T^{27} + \)\(16\!\cdots\!02\)\( T^{28} + \)\(70\!\cdots\!70\)\( T^{29} + \)\(22\!\cdots\!85\)\( T^{30} + \)\(48\!\cdots\!20\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 10 T + 100 T^{2} - 1625 T^{3} + 11531 T^{4} - 95310 T^{5} + 857875 T^{6} - 5635035 T^{7} + 42472426 T^{8} - 298656855 T^{9} + 2409770875 T^{10} - 14189466870 T^{11} + 90985136411 T^{12} - 679567676125 T^{13} + 2216436112900 T^{14} - 11747111398370 T^{15} + 62259690411361 T^{16} \))(\( 1 + 10 T + 31 T^{2} - 750 T^{3} - 4567 T^{4} + 7470 T^{5} + 369374 T^{6} + 601780 T^{7} - 13186187 T^{8} - 114344200 T^{9} + 595715966 T^{10} + 8185408800 T^{11} + 16758234740 T^{12} - 506558110520 T^{13} - 2917035559495 T^{14} + 4163213158300 T^{15} + 189869197228511 T^{16} + 220650297389900 T^{17} - 8193952886621455 T^{18} - 75414851819886040 T^{19} + 132230532809509940 T^{20} + 3423101068522538400 T^{21} + 13203663800735085614 T^{22} - \)\(13\!\cdots\!00\)\( T^{23} - \)\(82\!\cdots\!07\)\( T^{24} + \)\(19\!\cdots\!40\)\( T^{25} + \)\(64\!\cdots\!26\)\( T^{26} + \)\(69\!\cdots\!90\)\( T^{27} - \)\(22\!\cdots\!47\)\( T^{28} - \)\(19\!\cdots\!50\)\( T^{29} + \)\(42\!\cdots\!39\)\( T^{30} + \)\(73\!\cdots\!70\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 118 T^{2} - 900 T^{3} + 3393 T^{4} + 96300 T^{5} + 615034 T^{6} - 2943900 T^{7} - 65890945 T^{8} - 173690100 T^{9} + 2140933354 T^{10} + 19777997700 T^{11} + 41114205873 T^{12} - 643431869100 T^{13} - 4977302969638 T^{14} + 146830437604321 T^{16} \))(\( 1 + 12 T - 48 T^{2} - 796 T^{3} + 7633 T^{4} + 76112 T^{5} - 428429 T^{6} - 2657110 T^{7} + 51753489 T^{8} + 263375518 T^{9} - 2799769853 T^{10} - 8680064174 T^{11} + 262778488535 T^{12} + 989243918004 T^{13} - 11934572611572 T^{14} - 2210541870272 T^{15} + 962870787479413 T^{16} - 130421970346048 T^{17} - 41544247260882132 T^{18} + 203169926635743516 T^{19} + 3184181808612956135 T^{20} - 6205588794871964026 T^{21} - \)\(11\!\cdots\!73\)\( T^{22} + \)\(65\!\cdots\!42\)\( T^{23} + \)\(75\!\cdots\!69\)\( T^{24} - \)\(23\!\cdots\!90\)\( T^{25} - \)\(21\!\cdots\!29\)\( T^{26} + \)\(22\!\cdots\!08\)\( T^{27} + \)\(13\!\cdots\!73\)\( T^{28} - \)\(83\!\cdots\!84\)\( T^{29} - \)\(29\!\cdots\!28\)\( T^{30} + \)\(43\!\cdots\!88\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 9 T - 165 T^{2} - 1800 T^{3} + 7560 T^{4} + 154437 T^{5} + 526138 T^{6} - 4559670 T^{7} - 69838275 T^{8} - 278139870 T^{9} + 1957759498 T^{10} + 35054264697 T^{11} + 104674557960 T^{12} - 1520273341800 T^{13} - 8500861769565 T^{14} + 28284685524189 T^{15} + 191707312997281 T^{16} \))(\( 1 - 66 T^{2} + 280 T^{3} + 2565 T^{4} + 73000 T^{5} - 42550 T^{6} - 4179060 T^{7} + 27062520 T^{8} + 116843800 T^{9} + 1866464452 T^{10} + 5426819300 T^{11} - 75968338617 T^{12} + 773051497020 T^{13} + 3102841052040 T^{14} + 12380110098160 T^{15} + 378207085764575 T^{16} + 755186715987760 T^{17} + 11545671554640840 T^{18} + 175468001845096620 T^{19} - 1051845537525141897 T^{20} + 4583471506975409300 T^{21} + 96160947298538715172 T^{22} + \)\(36\!\cdots\!00\)\( T^{23} + \)\(51\!\cdots\!20\)\( T^{24} - \)\(48\!\cdots\!60\)\( T^{25} - \)\(30\!\cdots\!50\)\( T^{26} + \)\(31\!\cdots\!00\)\( T^{27} + \)\(68\!\cdots\!65\)\( T^{28} + \)\(45\!\cdots\!80\)\( T^{29} - \)\(65\!\cdots\!06\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 - 20 T + 250 T^{2} - 2600 T^{3} + 26091 T^{4} - 241820 T^{5} + 2034700 T^{6} - 15361680 T^{7} + 117317461 T^{8} - 1029232560 T^{9} + 9133768300 T^{10} - 72730508660 T^{11} + 525762898011 T^{12} - 3510325278200 T^{13} + 22614595542250 T^{14} - 121214232106460 T^{15} + 406067677556641 T^{16} \))(\( 1 + 40 T + 1030 T^{2} + 19080 T^{3} + 285687 T^{4} + 3555880 T^{5} + 38106080 T^{6} + 351599640 T^{7} + 2795567883 T^{8} + 18168145600 T^{9} + 82372777780 T^{10} - 40068357040 T^{11} - 6615976940891 T^{12} - 103670166025040 T^{13} - 1174625422928650 T^{14} - 11401205121382320 T^{15} - 98095935267474960 T^{16} - 763880743132615440 T^{17} - 5272893523526709850 T^{18} - 31180150144189105520 T^{19} - \)\(13\!\cdots\!11\)\( T^{20} - 54097294835944203280 T^{21} + \)\(74\!\cdots\!20\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{23} + \)\(11\!\cdots\!03\)\( T^{24} + \)\(95\!\cdots\!80\)\( T^{25} + \)\(69\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!40\)\( T^{27} + \)\(23\!\cdots\!07\)\( T^{28} + \)\(10\!\cdots\!60\)\( T^{29} + \)\(37\!\cdots\!70\)\( T^{30} + \)\(98\!\cdots\!20\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 + 6 T + 910 T^{3} + 4875 T^{4} - 34402 T^{5} + 474398 T^{6} + 3835500 T^{7} - 25760305 T^{8} + 272320500 T^{9} + 2391440318 T^{10} - 12312854222 T^{11} + 123881944875 T^{12} + 1641848709410 T^{13} + 54570720950346 T^{15} + 645753531245761 T^{16} \))(\( 1 - 8 T - 249 T^{2} + 2580 T^{3} + 22325 T^{4} - 321084 T^{5} - 587178 T^{6} + 14941106 T^{7} - 19432655 T^{8} + 485658130 T^{9} - 1289469490 T^{10} - 87893386040 T^{11} + 259786638060 T^{12} + 3694736827880 T^{13} + 7661490804445 T^{14} - 51486950465450 T^{15} - 2151535697455665 T^{16} - 3655573483046950 T^{17} + 38621575145207245 T^{18} + 1322386952803358680 T^{19} + 6601615174443178860 T^{20} - \)\(15\!\cdots\!40\)\( T^{21} - \)\(16\!\cdots\!90\)\( T^{22} + \)\(44\!\cdots\!30\)\( T^{23} - \)\(12\!\cdots\!55\)\( T^{24} + \)\(68\!\cdots\!86\)\( T^{25} - \)\(19\!\cdots\!78\)\( T^{26} - \)\(74\!\cdots\!64\)\( T^{27} + \)\(36\!\cdots\!25\)\( T^{28} + \)\(30\!\cdots\!80\)\( T^{29} - \)\(20\!\cdots\!69\)\( T^{30} - \)\(46\!\cdots\!08\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 15 T + 195 T^{2} - 1340 T^{3} + 15786 T^{4} - 132465 T^{5} + 1900340 T^{6} - 14658400 T^{7} + 154250461 T^{8} - 1070063200 T^{9} + 10126911860 T^{10} - 51531136905 T^{11} + 448294632426 T^{12} - 2777915934620 T^{13} + 29510174126355 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} \))(\( 1 + 20 T + 426 T^{2} + 5180 T^{3} + 61943 T^{4} + 512020 T^{5} + 4294444 T^{6} + 28526480 T^{7} + 237139103 T^{8} + 1934704520 T^{9} + 18247662296 T^{10} + 153434416460 T^{11} + 1126098548485 T^{12} + 8592560378500 T^{13} + 56602836572850 T^{14} + 546633756136020 T^{15} + 3853646456873416 T^{16} + 39904264197929460 T^{17} + 301636516096717650 T^{18} + 3342652060762934500 T^{19} + 31979217969627214885 T^{20} + \)\(31\!\cdots\!80\)\( T^{21} + \)\(27\!\cdots\!44\)\( T^{22} + \)\(21\!\cdots\!40\)\( T^{23} + \)\(19\!\cdots\!43\)\( T^{24} + \)\(16\!\cdots\!40\)\( T^{25} + \)\(18\!\cdots\!56\)\( T^{26} + \)\(16\!\cdots\!40\)\( T^{27} + \)\(14\!\cdots\!03\)\( T^{28} + \)\(86\!\cdots\!40\)\( T^{29} + \)\(51\!\cdots\!34\)\( T^{30} + \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 15 T - 58 T^{2} + 2180 T^{3} - 7802 T^{4} - 148865 T^{5} + 1559169 T^{6} + 5584500 T^{7} - 169067020 T^{8} + 441175500 T^{9} + 9730773729 T^{10} - 73396250735 T^{11} - 303888531962 T^{12} + 6707982949820 T^{13} - 14099072420218 T^{14} - 288058634792385 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 20 T - 26 T^{2} - 3900 T^{3} - 29155 T^{4} + 347300 T^{5} + 6752970 T^{6} + 1986700 T^{7} - 796980480 T^{8} - 5624999100 T^{9} + 44951710692 T^{10} + 864380570440 T^{11} + 1361154199023 T^{12} - 70535599976700 T^{13} - 564113457968440 T^{14} + 2384852381457900 T^{15} + 60762066594476935 T^{16} + 188403338135174100 T^{17} - 3520632091181034040 T^{18} - 34776801676912191300 T^{19} + 53017066305435970863 T^{20} + \)\(26\!\cdots\!60\)\( T^{21} + \)\(10\!\cdots\!32\)\( T^{22} - \)\(10\!\cdots\!00\)\( T^{23} - \)\(12\!\cdots\!80\)\( T^{24} + \)\(23\!\cdots\!00\)\( T^{25} + \)\(63\!\cdots\!70\)\( T^{26} + \)\(25\!\cdots\!00\)\( T^{27} - \)\(17\!\cdots\!55\)\( T^{28} - \)\(18\!\cdots\!00\)\( T^{29} - \)\(95\!\cdots\!06\)\( T^{30} + \)\(58\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 45 T + 1115 T^{2} - 19890 T^{3} + 284376 T^{4} - 3450645 T^{5} + 37197280 T^{6} - 367888080 T^{7} + 3431240591 T^{8} - 30534710640 T^{9} + 256252061920 T^{10} - 1973033952615 T^{11} + 13496007492696 T^{12} - 78347518389270 T^{13} + 364538516306435 T^{14} - 1221122294533215 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 10 T + 202 T^{2} - 1560 T^{3} - 11179 T^{4} - 496950 T^{5} + 1018034 T^{6} - 15276030 T^{7} + 550764400 T^{8} - 1724892540 T^{9} + 40801859152 T^{10} - 455708099560 T^{11} + 3046921596515 T^{12} - 45917600363320 T^{13} + 368604392570920 T^{14} - 3224407333551250 T^{15} + 38409379558310191 T^{16} - 267625808684753750 T^{17} + 2539315660421067880 T^{18} - 26255086958941652840 T^{19} + \)\(14\!\cdots\!15\)\( T^{20} - \)\(17\!\cdots\!80\)\( T^{21} + \)\(13\!\cdots\!88\)\( T^{22} - \)\(46\!\cdots\!80\)\( T^{23} + \)\(12\!\cdots\!00\)\( T^{24} - \)\(28\!\cdots\!90\)\( T^{25} + \)\(15\!\cdots\!66\)\( T^{26} - \)\(63\!\cdots\!50\)\( T^{27} - \)\(11\!\cdots\!19\)\( T^{28} - \)\(13\!\cdots\!80\)\( T^{29} + \)\(14\!\cdots\!58\)\( T^{30} + \)\(61\!\cdots\!70\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 25 T + 342 T^{2} - 5000 T^{3} + 78868 T^{4} - 928525 T^{5} + 9098049 T^{6} - 101226750 T^{7} + 1076434080 T^{8} - 9009180750 T^{9} + 72065646129 T^{10} - 654581340725 T^{11} + 4948355063188 T^{12} - 27920297245000 T^{13} + 169967601508662 T^{14} - 1105783372388225 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 18 T - 288 T^{2} - 7264 T^{3} + 14783 T^{4} + 1100918 T^{5} + 4764016 T^{6} - 40139780 T^{7} - 746571076 T^{8} - 9051008048 T^{9} + 13821315872 T^{10} + 1302515170094 T^{11} + 4134501893785 T^{12} - 64272919989494 T^{13} - 170401937947852 T^{14} + 989839747535702 T^{15} - 10684112304900357 T^{16} + 88095737530677478 T^{17} - 1349753750484935692 T^{18} - 45310416132073595686 T^{19} + \)\(25\!\cdots\!85\)\( T^{20} + \)\(72\!\cdots\!06\)\( T^{21} + \)\(68\!\cdots\!92\)\( T^{22} - \)\(40\!\cdots\!92\)\( T^{23} - \)\(29\!\cdots\!56\)\( T^{24} - \)\(14\!\cdots\!20\)\( T^{25} + \)\(14\!\cdots\!16\)\( T^{26} + \)\(30\!\cdots\!02\)\( T^{27} + \)\(36\!\cdots\!43\)\( T^{28} - \)\(15\!\cdots\!16\)\( T^{29} - \)\(56\!\cdots\!08\)\( T^{30} + \)\(31\!\cdots\!82\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 60 T + 1830 T^{2} + 35660 T^{3} + 467711 T^{4} + 3770460 T^{5} + 6583460 T^{6} - 292271720 T^{7} - 4451764059 T^{8} - 28350356840 T^{9} + 61943775140 T^{10} + 3441197039580 T^{11} + 41406118545791 T^{12} + 306224553564620 T^{13} + 1524338769020070 T^{14} + 4847897068686780 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 40 T + 1170 T^{2} - 28840 T^{3} + 597037 T^{4} - 11124280 T^{5} + 188742510 T^{6} - 2956583740 T^{7} + 43370532748 T^{8} - 598437364480 T^{9} + 7830452653220 T^{10} - 97502185336460 T^{11} + 1158651974029219 T^{12} - 13184566116562660 T^{13} + 143770204308646900 T^{14} - 1504531362263433880 T^{15} + 15122407401217954415 T^{16} - \)\(14\!\cdots\!60\)\( T^{17} + \)\(13\!\cdots\!00\)\( T^{18} - \)\(12\!\cdots\!80\)\( T^{19} + \)\(10\!\cdots\!39\)\( T^{20} - \)\(83\!\cdots\!20\)\( T^{21} + \)\(65\!\cdots\!80\)\( T^{22} - \)\(48\!\cdots\!40\)\( T^{23} + \)\(33\!\cdots\!28\)\( T^{24} - \)\(22\!\cdots\!80\)\( T^{25} + \)\(13\!\cdots\!90\)\( T^{26} - \)\(79\!\cdots\!40\)\( T^{27} + \)\(41\!\cdots\!17\)\( T^{28} - \)\(19\!\cdots\!80\)\( T^{29} + \)\(76\!\cdots\!30\)\( T^{30} - \)\(25\!\cdots\!20\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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