Properties

Label 75.2.i
Level 75
Weight 2
Character orbit i
Rep. character \(\chi_{75}(4,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 16
Newform subspaces 1
Sturm bound 20
Trace bound 0

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 75.i (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 1 \)
Sturm bound: \(20\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(75, [\chi])\).

Total New Old
Modular forms 48 16 32
Cusp forms 32 16 16
Eisenstein series 16 0 16

Trace form

\( 16q + 2q^{4} + 2q^{6} - 30q^{8} + 4q^{9} + O(q^{10}) \) \( 16q + 2q^{4} + 2q^{6} - 30q^{8} + 4q^{9} - 6q^{11} - 12q^{14} - 10q^{16} + 10q^{17} - 2q^{19} + 20q^{20} + 4q^{21} - 30q^{22} - 20q^{23} + 24q^{24} - 10q^{25} + 12q^{26} + 30q^{28} + 16q^{29} - 20q^{30} + 6q^{31} + 10q^{33} - 36q^{34} + 10q^{35} - 2q^{36} - 10q^{37} + 30q^{38} - 8q^{39} + 10q^{40} - 14q^{41} - 10q^{42} + 26q^{44} + 16q^{46} + 40q^{47} + 20q^{50} - 32q^{51} + 40q^{52} + 10q^{53} - 2q^{54} + 10q^{55} + 10q^{58} + 12q^{59} - 30q^{60} - 10q^{62} - 10q^{63} + 8q^{64} - 70q^{65} + 16q^{66} - 40q^{67} - 12q^{69} + 30q^{70} - 8q^{71} - 30q^{72} - 20q^{73} - 52q^{74} - 32q^{76} - 40q^{77} - 20q^{79} - 4q^{81} + 10q^{83} + 12q^{84} - 20q^{85} - 36q^{86} + 40q^{87} - 40q^{88} + 18q^{89} + 30q^{90} + 26q^{91} + 10q^{92} - 38q^{94} - 40q^{95} - 26q^{96} + 40q^{97} + 60q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
75.2.i.a \(16\) \(0.599\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(-1-\beta _{2}-\beta _{4}+\beta _{6}-\beta _{7}-\beta _{8}+\cdots)q^{2}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + 10 T^{3} + 6 T^{4} + 30 T^{5} + 61 T^{6} + 80 T^{7} + 165 T^{8} + 310 T^{9} + 503 T^{10} + 690 T^{11} + 1290 T^{12} + 2060 T^{13} + 2585 T^{14} + 4150 T^{15} + 6461 T^{16} + 8300 T^{17} + 10340 T^{18} + 16480 T^{19} + 20640 T^{20} + 22080 T^{21} + 32192 T^{22} + 39680 T^{23} + 42240 T^{24} + 40960 T^{25} + 62464 T^{26} + 61440 T^{27} + 24576 T^{28} + 81920 T^{29} + 49152 T^{30} + 65536 T^{32} \)
$3$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$5$ \( 1 + 5 T^{2} + 10 T^{3} + 10 T^{4} - 50 T^{5} + 25 T^{6} - 450 T^{7} - 725 T^{8} - 2250 T^{9} + 625 T^{10} - 6250 T^{11} + 6250 T^{12} + 31250 T^{13} + 78125 T^{14} + 390625 T^{16} \)
$7$ \( 1 - 56 T^{2} + 1628 T^{4} - 32234 T^{6} + 485518 T^{8} - 5899406 T^{10} + 59922825 T^{12} - 520141390 T^{14} + 3906488461 T^{16} - 25486928110 T^{18} + 143874702825 T^{20} - 694059216494 T^{22} + 2798914651918 T^{24} - 9105307176266 T^{26} + 22533615563228 T^{28} - 37980492079544 T^{30} + 33232930569601 T^{32} \)
$11$ \( 1 + 6 T + 4 T^{2} - 68 T^{3} - 237 T^{4} - 38 T^{5} + 2400 T^{6} + 9328 T^{7} - 1188 T^{8} - 114992 T^{9} - 345452 T^{10} - 52314 T^{11} + 3321817 T^{12} + 16648114 T^{13} + 30762856 T^{14} - 86386670 T^{15} - 607501509 T^{16} - 950253370 T^{17} + 3722305576 T^{18} + 22158639734 T^{19} + 48634722697 T^{20} - 8425222014 T^{21} - 611989290572 T^{22} - 2240868767632 T^{23} - 254658350628 T^{24} + 21994936061648 T^{25} + 62249819042400 T^{26} - 10841843483218 T^{27} - 743807525282877 T^{28} - 2347544425787308 T^{29} + 1518999334332964 T^{30} + 25063489016493906 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 + 22 T^{2} + 80 T^{3} + 81 T^{4} + 1760 T^{5} + 1994 T^{6} + 7400 T^{7} + 54580 T^{8} + 208560 T^{9} + 320132 T^{10} + 4629240 T^{11} + 13798055 T^{12} + 28576760 T^{13} + 253427140 T^{14} + 524829280 T^{15} + 2023398311 T^{16} + 6822780640 T^{17} + 42829186660 T^{18} + 62783141720 T^{19} + 394086248855 T^{20} + 1718804407320 T^{21} + 1545216018788 T^{22} + 13086830705520 T^{23} + 44522582752180 T^{24} + 78473295360200 T^{25} + 274889832746906 T^{26} + 3154202293505120 T^{27} + 1887144894920961 T^{28} + 24230008527380240 T^{29} + 86622280485384358 T^{30} + 665416609183179841 T^{32} \)
$17$ \( 1 - 10 T + 55 T^{2} - 550 T^{3} + 4137 T^{4} - 21370 T^{5} + 140590 T^{6} - 875240 T^{7} + 4267373 T^{8} - 23180400 T^{9} + 125011430 T^{10} - 567252720 T^{11} + 2721539524 T^{12} - 13138878600 T^{13} + 55061701425 T^{14} - 238439860100 T^{15} + 1051104835255 T^{16} - 4053477621700 T^{17} + 15912831711825 T^{18} - 64551310561800 T^{19} + 227305702584004 T^{20} - 805417745261040 T^{21} + 3017472017413670 T^{22} - 9511814575609200 T^{23} + 29768158958272493 T^{24} - 103792853025234280 T^{25} + 283428582464124910 T^{26} - 732390424094117210 T^{27} + 2410308195419521257 T^{28} - 5447517918098265350 T^{29} + 9260780460767051095 T^{30} - 28624230515098157930 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 + 2 T - 53 T^{2} - 216 T^{3} + 1273 T^{4} + 10232 T^{5} - 10629 T^{6} - 233030 T^{7} - 154021 T^{8} + 2536598 T^{9} + 3525662 T^{10} + 321666 T^{11} + 122263380 T^{12} - 353312856 T^{13} - 6187122082 T^{14} + 3769508888 T^{15} + 148112464363 T^{16} + 71620668872 T^{17} - 2233551071602 T^{18} - 2423372879304 T^{19} + 15933485944980 T^{20} + 796476860934 T^{21} + 165867874898222 T^{22} + 2267393265403922 T^{23} - 2615825363137861 T^{24} - 75195914213440370 T^{25} - 65167103254166829 T^{26} + 1191928329046576808 T^{27} + 2817549891971222953 T^{28} - 9083444427847524744 T^{29} - 42347354346492858413 T^{30} + 30362254059749596598 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 + 20 T + 181 T^{2} + 1150 T^{3} + 7948 T^{4} + 59890 T^{5} + 379519 T^{6} + 2059710 T^{7} + 11607418 T^{8} + 70214570 T^{9} + 395238326 T^{10} + 1984546720 T^{11} + 9942722840 T^{12} + 53914737500 T^{13} + 283746609760 T^{14} + 1328416871210 T^{15} + 6099237415331 T^{16} + 30553588037830 T^{17} + 150101956563040 T^{18} + 655980611162500 T^{19} + 2782381502268440 T^{20} + 12773223389444960 T^{21} + 58509456956281814 T^{22} + 239068354686162790 T^{23} + 908988340148414458 T^{24} + 3709852148341955730 T^{25} + 15722148109292854831 T^{26} + 57063776401465088030 T^{27} + \)\(17\!\cdots\!08\)\( T^{28} + \)\(57\!\cdots\!50\)\( T^{29} + \)\(20\!\cdots\!29\)\( T^{30} + \)\(53\!\cdots\!40\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 - 16 T + 101 T^{2} - 378 T^{3} + 2782 T^{4} - 28638 T^{5} + 156384 T^{6} - 372476 T^{7} + 1621718 T^{8} - 24643166 T^{9} + 158941381 T^{10} - 390275636 T^{11} + 1282706252 T^{12} - 24547929228 T^{13} + 196193283170 T^{14} - 670123125368 T^{15} + 1692236078907 T^{16} - 19433570635672 T^{17} + 164998551145970 T^{18} - 598699445941692 T^{19} + 907233760620812 T^{20} - 8005001721065764 T^{21} + 94542040090746301 T^{22} - 425091565362154294 T^{23} + 811258612334286998 T^{24} - 5403563704507781644 T^{25} + 65791879972418633184 T^{26} - \)\(34\!\cdots\!02\)\( T^{27} + \)\(98\!\cdots\!62\)\( T^{28} - \)\(38\!\cdots\!42\)\( T^{29} + \)\(30\!\cdots\!81\)\( T^{30} - \)\(13\!\cdots\!84\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 - 6 T - 27 T^{2} + 222 T^{3} + 1398 T^{4} - 1096 T^{5} - 82911 T^{6} + 83080 T^{7} + 2495194 T^{8} + 3301786 T^{9} - 44430232 T^{10} - 311988762 T^{11} + 1830869280 T^{12} + 7523309618 T^{13} - 27963150078 T^{14} + 31095224934 T^{15} - 66108612217 T^{16} + 963951972954 T^{17} - 26872587224958 T^{18} + 224126916829838 T^{19} + 1690846228334880 T^{20} - 8931973377601062 T^{21} - 39431994447683992 T^{22} + 90840764095102246 T^{23} + 2128128599276558554 T^{24} + 2196603809108546680 T^{25} - 67956200901865191711 T^{26} - 27847690678459694776 T^{27} + \)\(11\!\cdots\!78\)\( T^{28} + \)\(54\!\cdots\!02\)\( T^{29} - \)\(20\!\cdots\!67\)\( T^{30} - \)\(14\!\cdots\!06\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 + 10 T + 159 T^{2} + 1490 T^{3} + 15403 T^{4} + 128130 T^{5} + 1158066 T^{6} + 8990890 T^{7} + 72816793 T^{8} + 529727610 T^{9} + 3982453044 T^{10} + 27036193580 T^{11} + 190987118220 T^{12} + 1239202673080 T^{13} + 8238752278565 T^{14} + 50853391327250 T^{15} + 322012573505931 T^{16} + 1881575479108250 T^{17} + 11278851869355485 T^{18} + 62769332999521240 T^{19} + 357940608470313420 T^{20} + 1874796645055196060 T^{21} + 10217884947593238996 T^{22} + 50288036386477742130 T^{23} + \)\(25\!\cdots\!53\)\( T^{24} + \)\(11\!\cdots\!30\)\( T^{25} + \)\(55\!\cdots\!34\)\( T^{26} + \)\(22\!\cdots\!90\)\( T^{27} + \)\(10\!\cdots\!43\)\( T^{28} + \)\(36\!\cdots\!30\)\( T^{29} + \)\(14\!\cdots\!51\)\( T^{30} + \)\(33\!\cdots\!30\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 + 14 T - 82 T^{2} - 2028 T^{3} - 747 T^{4} + 120014 T^{5} + 311054 T^{6} - 3602970 T^{7} - 14348876 T^{8} + 63645696 T^{9} + 522280968 T^{10} - 1039813212 T^{11} - 35365817705 T^{12} + 21504097848 T^{13} + 2388690604652 T^{14} - 220861336106 T^{15} - 115364883721937 T^{16} - 9055314780346 T^{17} + 4015388906420012 T^{18} + 1482083927782008 T^{19} - 99935348403898505 T^{20} - 120468808491927612 T^{21} + 2480889041090385288 T^{22} + 12395271310130866176 T^{23} - \)\(11\!\cdots\!96\)\( T^{24} - \)\(11\!\cdots\!70\)\( T^{25} + \)\(41\!\cdots\!54\)\( T^{26} + \)\(66\!\cdots\!74\)\( T^{27} - \)\(16\!\cdots\!07\)\( T^{28} - \)\(18\!\cdots\!88\)\( T^{29} - \)\(31\!\cdots\!02\)\( T^{30} + \)\(21\!\cdots\!14\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 - 300 T^{2} + 44842 T^{4} - 4521820 T^{6} + 352848163 T^{8} - 23099919100 T^{10} + 1325953845604 T^{12} - 67791756187600 T^{14} + 3089570546914805 T^{16} - 125346957190872400 T^{18} + 4533172333304800804 T^{20} - \)\(14\!\cdots\!00\)\( T^{22} + \)\(41\!\cdots\!63\)\( T^{24} - \)\(97\!\cdots\!80\)\( T^{26} + \)\(17\!\cdots\!42\)\( T^{28} - \)\(22\!\cdots\!00\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 40 T + 865 T^{2} - 12910 T^{3} + 142622 T^{4} - 1167030 T^{5} + 6143280 T^{6} + 637840 T^{7} - 479197322 T^{8} + 6905707270 T^{9} - 64790306935 T^{10} + 444387466120 T^{11} - 1988154991916 T^{12} + 288127717900 T^{13} + 103937589685250 T^{14} - 1288860429553200 T^{15} + 10336092465289955 T^{16} - 60576440189000400 T^{17} + 229598135614717250 T^{18} + 29914284055531700 T^{19} - 9701562139107658796 T^{20} + \)\(10\!\cdots\!40\)\( T^{21} - \)\(69\!\cdots\!15\)\( T^{22} + \)\(34\!\cdots\!10\)\( T^{23} - \)\(11\!\cdots\!42\)\( T^{24} + \)\(71\!\cdots\!80\)\( T^{25} + \)\(32\!\cdots\!20\)\( T^{26} - \)\(28\!\cdots\!90\)\( T^{27} + \)\(16\!\cdots\!02\)\( T^{28} - \)\(70\!\cdots\!70\)\( T^{29} + \)\(22\!\cdots\!85\)\( T^{30} - \)\(48\!\cdots\!20\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 - 10 T + 31 T^{2} + 750 T^{3} - 4567 T^{4} - 7470 T^{5} + 369374 T^{6} - 601780 T^{7} - 13186187 T^{8} + 114344200 T^{9} + 595715966 T^{10} - 8185408800 T^{11} + 16758234740 T^{12} + 506558110520 T^{13} - 2917035559495 T^{14} - 4163213158300 T^{15} + 189869197228511 T^{16} - 220650297389900 T^{17} - 8193952886621455 T^{18} + 75414851819886040 T^{19} + 132230532809509940 T^{20} - 3423101068522538400 T^{21} + 13203663800735085614 T^{22} + \)\(13\!\cdots\!00\)\( T^{23} - \)\(82\!\cdots\!07\)\( T^{24} - \)\(19\!\cdots\!40\)\( T^{25} + \)\(64\!\cdots\!26\)\( T^{26} - \)\(69\!\cdots\!90\)\( T^{27} - \)\(22\!\cdots\!47\)\( T^{28} + \)\(19\!\cdots\!50\)\( T^{29} + \)\(42\!\cdots\!39\)\( T^{30} - \)\(73\!\cdots\!70\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 12 T - 48 T^{2} + 796 T^{3} + 7633 T^{4} - 76112 T^{5} - 428429 T^{6} + 2657110 T^{7} + 51753489 T^{8} - 263375518 T^{9} - 2799769853 T^{10} + 8680064174 T^{11} + 262778488535 T^{12} - 989243918004 T^{13} - 11934572611572 T^{14} + 2210541870272 T^{15} + 962870787479413 T^{16} + 130421970346048 T^{17} - 41544247260882132 T^{18} - 203169926635743516 T^{19} + 3184181808612956135 T^{20} + 6205588794871964026 T^{21} - \)\(11\!\cdots\!73\)\( T^{22} - \)\(65\!\cdots\!42\)\( T^{23} + \)\(75\!\cdots\!69\)\( T^{24} + \)\(23\!\cdots\!90\)\( T^{25} - \)\(21\!\cdots\!29\)\( T^{26} - \)\(22\!\cdots\!08\)\( T^{27} + \)\(13\!\cdots\!73\)\( T^{28} + \)\(83\!\cdots\!84\)\( T^{29} - \)\(29\!\cdots\!28\)\( T^{30} - \)\(43\!\cdots\!88\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 - 66 T^{2} + 280 T^{3} + 2565 T^{4} + 73000 T^{5} - 42550 T^{6} - 4179060 T^{7} + 27062520 T^{8} + 116843800 T^{9} + 1866464452 T^{10} + 5426819300 T^{11} - 75968338617 T^{12} + 773051497020 T^{13} + 3102841052040 T^{14} + 12380110098160 T^{15} + 378207085764575 T^{16} + 755186715987760 T^{17} + 11545671554640840 T^{18} + 175468001845096620 T^{19} - 1051845537525141897 T^{20} + 4583471506975409300 T^{21} + 96160947298538715172 T^{22} + \)\(36\!\cdots\!00\)\( T^{23} + \)\(51\!\cdots\!20\)\( T^{24} - \)\(48\!\cdots\!60\)\( T^{25} - \)\(30\!\cdots\!50\)\( T^{26} + \)\(31\!\cdots\!00\)\( T^{27} + \)\(68\!\cdots\!65\)\( T^{28} + \)\(45\!\cdots\!80\)\( T^{29} - \)\(65\!\cdots\!06\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 + 40 T + 1030 T^{2} + 19080 T^{3} + 285687 T^{4} + 3555880 T^{5} + 38106080 T^{6} + 351599640 T^{7} + 2795567883 T^{8} + 18168145600 T^{9} + 82372777780 T^{10} - 40068357040 T^{11} - 6615976940891 T^{12} - 103670166025040 T^{13} - 1174625422928650 T^{14} - 11401205121382320 T^{15} - 98095935267474960 T^{16} - 763880743132615440 T^{17} - 5272893523526709850 T^{18} - 31180150144189105520 T^{19} - \)\(13\!\cdots\!11\)\( T^{20} - 54097294835944203280 T^{21} + \)\(74\!\cdots\!20\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{23} + \)\(11\!\cdots\!03\)\( T^{24} + \)\(95\!\cdots\!80\)\( T^{25} + \)\(69\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!40\)\( T^{27} + \)\(23\!\cdots\!07\)\( T^{28} + \)\(10\!\cdots\!60\)\( T^{29} + \)\(37\!\cdots\!70\)\( T^{30} + \)\(98\!\cdots\!20\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 + 8 T - 249 T^{2} - 2580 T^{3} + 22325 T^{4} + 321084 T^{5} - 587178 T^{6} - 14941106 T^{7} - 19432655 T^{8} - 485658130 T^{9} - 1289469490 T^{10} + 87893386040 T^{11} + 259786638060 T^{12} - 3694736827880 T^{13} + 7661490804445 T^{14} + 51486950465450 T^{15} - 2151535697455665 T^{16} + 3655573483046950 T^{17} + 38621575145207245 T^{18} - 1322386952803358680 T^{19} + 6601615174443178860 T^{20} + \)\(15\!\cdots\!40\)\( T^{21} - \)\(16\!\cdots\!90\)\( T^{22} - \)\(44\!\cdots\!30\)\( T^{23} - \)\(12\!\cdots\!55\)\( T^{24} - \)\(68\!\cdots\!86\)\( T^{25} - \)\(19\!\cdots\!78\)\( T^{26} + \)\(74\!\cdots\!64\)\( T^{27} + \)\(36\!\cdots\!25\)\( T^{28} - \)\(30\!\cdots\!80\)\( T^{29} - \)\(20\!\cdots\!69\)\( T^{30} + \)\(46\!\cdots\!08\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 + 20 T + 426 T^{2} + 5180 T^{3} + 61943 T^{4} + 512020 T^{5} + 4294444 T^{6} + 28526480 T^{7} + 237139103 T^{8} + 1934704520 T^{9} + 18247662296 T^{10} + 153434416460 T^{11} + 1126098548485 T^{12} + 8592560378500 T^{13} + 56602836572850 T^{14} + 546633756136020 T^{15} + 3853646456873416 T^{16} + 39904264197929460 T^{17} + 301636516096717650 T^{18} + 3342652060762934500 T^{19} + 31979217969627214885 T^{20} + \)\(31\!\cdots\!80\)\( T^{21} + \)\(27\!\cdots\!44\)\( T^{22} + \)\(21\!\cdots\!40\)\( T^{23} + \)\(19\!\cdots\!43\)\( T^{24} + \)\(16\!\cdots\!40\)\( T^{25} + \)\(18\!\cdots\!56\)\( T^{26} + \)\(16\!\cdots\!40\)\( T^{27} + \)\(14\!\cdots\!03\)\( T^{28} + \)\(86\!\cdots\!40\)\( T^{29} + \)\(51\!\cdots\!34\)\( T^{30} + \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 + 20 T - 26 T^{2} - 3900 T^{3} - 29155 T^{4} + 347300 T^{5} + 6752970 T^{6} + 1986700 T^{7} - 796980480 T^{8} - 5624999100 T^{9} + 44951710692 T^{10} + 864380570440 T^{11} + 1361154199023 T^{12} - 70535599976700 T^{13} - 564113457968440 T^{14} + 2384852381457900 T^{15} + 60762066594476935 T^{16} + 188403338135174100 T^{17} - 3520632091181034040 T^{18} - 34776801676912191300 T^{19} + 53017066305435970863 T^{20} + \)\(26\!\cdots\!60\)\( T^{21} + \)\(10\!\cdots\!32\)\( T^{22} - \)\(10\!\cdots\!00\)\( T^{23} - \)\(12\!\cdots\!80\)\( T^{24} + \)\(23\!\cdots\!00\)\( T^{25} + \)\(63\!\cdots\!70\)\( T^{26} + \)\(25\!\cdots\!00\)\( T^{27} - \)\(17\!\cdots\!55\)\( T^{28} - \)\(18\!\cdots\!00\)\( T^{29} - \)\(95\!\cdots\!06\)\( T^{30} + \)\(58\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 10 T + 202 T^{2} + 1560 T^{3} - 11179 T^{4} + 496950 T^{5} + 1018034 T^{6} + 15276030 T^{7} + 550764400 T^{8} + 1724892540 T^{9} + 40801859152 T^{10} + 455708099560 T^{11} + 3046921596515 T^{12} + 45917600363320 T^{13} + 368604392570920 T^{14} + 3224407333551250 T^{15} + 38409379558310191 T^{16} + 267625808684753750 T^{17} + 2539315660421067880 T^{18} + 26255086958941652840 T^{19} + \)\(14\!\cdots\!15\)\( T^{20} + \)\(17\!\cdots\!80\)\( T^{21} + \)\(13\!\cdots\!88\)\( T^{22} + \)\(46\!\cdots\!80\)\( T^{23} + \)\(12\!\cdots\!00\)\( T^{24} + \)\(28\!\cdots\!90\)\( T^{25} + \)\(15\!\cdots\!66\)\( T^{26} + \)\(63\!\cdots\!50\)\( T^{27} - \)\(11\!\cdots\!19\)\( T^{28} + \)\(13\!\cdots\!80\)\( T^{29} + \)\(14\!\cdots\!58\)\( T^{30} - \)\(61\!\cdots\!70\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - 18 T - 288 T^{2} + 7264 T^{3} + 14783 T^{4} - 1100918 T^{5} + 4764016 T^{6} + 40139780 T^{7} - 746571076 T^{8} + 9051008048 T^{9} + 13821315872 T^{10} - 1302515170094 T^{11} + 4134501893785 T^{12} + 64272919989494 T^{13} - 170401937947852 T^{14} - 989839747535702 T^{15} - 10684112304900357 T^{16} - 88095737530677478 T^{17} - 1349753750484935692 T^{18} + 45310416132073595686 T^{19} + \)\(25\!\cdots\!85\)\( T^{20} - \)\(72\!\cdots\!06\)\( T^{21} + \)\(68\!\cdots\!92\)\( T^{22} + \)\(40\!\cdots\!92\)\( T^{23} - \)\(29\!\cdots\!56\)\( T^{24} + \)\(14\!\cdots\!20\)\( T^{25} + \)\(14\!\cdots\!16\)\( T^{26} - \)\(30\!\cdots\!02\)\( T^{27} + \)\(36\!\cdots\!43\)\( T^{28} + \)\(15\!\cdots\!16\)\( T^{29} - \)\(56\!\cdots\!08\)\( T^{30} - \)\(31\!\cdots\!82\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 - 40 T + 1170 T^{2} - 28840 T^{3} + 597037 T^{4} - 11124280 T^{5} + 188742510 T^{6} - 2956583740 T^{7} + 43370532748 T^{8} - 598437364480 T^{9} + 7830452653220 T^{10} - 97502185336460 T^{11} + 1158651974029219 T^{12} - 13184566116562660 T^{13} + 143770204308646900 T^{14} - 1504531362263433880 T^{15} + 15122407401217954415 T^{16} - \)\(14\!\cdots\!60\)\( T^{17} + \)\(13\!\cdots\!00\)\( T^{18} - \)\(12\!\cdots\!80\)\( T^{19} + \)\(10\!\cdots\!39\)\( T^{20} - \)\(83\!\cdots\!20\)\( T^{21} + \)\(65\!\cdots\!80\)\( T^{22} - \)\(48\!\cdots\!40\)\( T^{23} + \)\(33\!\cdots\!28\)\( T^{24} - \)\(22\!\cdots\!80\)\( T^{25} + \)\(13\!\cdots\!90\)\( T^{26} - \)\(79\!\cdots\!40\)\( T^{27} + \)\(41\!\cdots\!17\)\( T^{28} - \)\(19\!\cdots\!80\)\( T^{29} + \)\(76\!\cdots\!30\)\( T^{30} - \)\(25\!\cdots\!20\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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