Properties

Label 672.2.bl
Level 672
Weight 2
Character orbit bl
Rep. character \(\chi_{672}(31,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 32
Newform subspaces 2
Sturm bound 256
Trace bound 3

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Defining parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.bl (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(256\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(672, [\chi])\).

Total New Old
Modular forms 288 32 256
Cusp forms 224 32 192
Eisenstein series 64 0 64

Trace form

\( 32q - 16q^{9} + O(q^{10}) \) \( 32q - 16q^{9} + 16q^{21} + 8q^{25} - 24q^{33} + 8q^{37} + 48q^{49} + 16q^{53} - 16q^{57} + 48q^{61} + 16q^{65} + 24q^{73} - 64q^{77} - 16q^{81} + 64q^{85} + 8q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(672, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
672.2.bl.a \(16\) \(5.366\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(-8\) \(0\) \(-4\) \(q+(-1-\beta _{1})q^{3}-\beta _{6}q^{5}+(-\beta _{9}+\beta _{13}+\cdots)q^{7}+\cdots\)
672.2.bl.b \(16\) \(5.366\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(8\) \(0\) \(4\) \(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(1+\beta _{1}-\beta _{4}+\beta _{12}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(672, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T + T^{2} )^{8} \))(\( ( 1 - T + T^{2} )^{8} \))
$5$ (\( 1 + 18 T^{2} + 163 T^{4} - 96 T^{5} + 926 T^{6} - 1152 T^{7} + 3145 T^{8} - 5280 T^{9} + 5460 T^{10} + 2976 T^{11} - 42898 T^{12} + 208512 T^{13} - 629144 T^{14} + 1493856 T^{15} - 4046526 T^{16} + 7469280 T^{17} - 15728600 T^{18} + 26064000 T^{19} - 26811250 T^{20} + 9300000 T^{21} + 85312500 T^{22} - 412500000 T^{23} + 1228515625 T^{24} - 2250000000 T^{25} + 9042968750 T^{26} - 4687500000 T^{27} + 39794921875 T^{28} + 109863281250 T^{30} + 152587890625 T^{32} \))(\( 1 + 18 T^{2} + 163 T^{4} - 96 T^{5} + 926 T^{6} - 1152 T^{7} + 3145 T^{8} - 5280 T^{9} + 5460 T^{10} + 2976 T^{11} - 42898 T^{12} + 208512 T^{13} - 629144 T^{14} + 1493856 T^{15} - 4046526 T^{16} + 7469280 T^{17} - 15728600 T^{18} + 26064000 T^{19} - 26811250 T^{20} + 9300000 T^{21} + 85312500 T^{22} - 412500000 T^{23} + 1228515625 T^{24} - 2250000000 T^{25} + 9042968750 T^{26} - 4687500000 T^{27} + 39794921875 T^{28} + 109863281250 T^{30} + 152587890625 T^{32} \))
$7$ (\( 1 + 4 T - 4 T^{2} - 80 T^{3} - 158 T^{4} + 340 T^{5} + 1792 T^{6} - 180 T^{7} - 9813 T^{8} - 1260 T^{9} + 87808 T^{10} + 116620 T^{11} - 379358 T^{12} - 1344560 T^{13} - 470596 T^{14} + 3294172 T^{15} + 5764801 T^{16} \))(\( 1 - 4 T - 4 T^{2} + 80 T^{3} - 158 T^{4} - 340 T^{5} + 1792 T^{6} + 180 T^{7} - 9813 T^{8} + 1260 T^{9} + 87808 T^{10} - 116620 T^{11} - 379358 T^{12} + 1344560 T^{13} - 470596 T^{14} - 3294172 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 - 12 T + 122 T^{2} - 888 T^{3} + 5699 T^{4} - 31224 T^{5} + 156214 T^{6} - 710532 T^{7} + 3034729 T^{8} - 12231408 T^{9} + 47424804 T^{10} - 178064688 T^{11} + 653004526 T^{12} - 2345920584 T^{13} + 8247983112 T^{14} - 28390858272 T^{15} + 95259200226 T^{16} - 312299440992 T^{17} + 998005956552 T^{18} - 3122420297304 T^{19} + 9560639265166 T^{20} - 28677496067088 T^{21} + 84015933199044 T^{22} - 238355539266768 T^{23} + 650521112578249 T^{24} - 1675397288781612 T^{25} + 4051788846620614 T^{26} - 8908571603157864 T^{27} + 17885903318932979 T^{28} - 30656168383810728 T^{29} + 46329479697155402 T^{30} - 50126978032987812 T^{31} + 45949729863572161 T^{32} \))(\( 1 + 12 T + 122 T^{2} + 888 T^{3} + 5699 T^{4} + 31224 T^{5} + 156214 T^{6} + 710532 T^{7} + 3034729 T^{8} + 12231408 T^{9} + 47424804 T^{10} + 178064688 T^{11} + 653004526 T^{12} + 2345920584 T^{13} + 8247983112 T^{14} + 28390858272 T^{15} + 95259200226 T^{16} + 312299440992 T^{17} + 998005956552 T^{18} + 3122420297304 T^{19} + 9560639265166 T^{20} + 28677496067088 T^{21} + 84015933199044 T^{22} + 238355539266768 T^{23} + 650521112578249 T^{24} + 1675397288781612 T^{25} + 4051788846620614 T^{26} + 8908571603157864 T^{27} + 17885903318932979 T^{28} + 30656168383810728 T^{29} + 46329479697155402 T^{30} + 50126978032987812 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 - 84 T^{2} + 3350 T^{4} - 88392 T^{6} + 1800769 T^{8} - 30564216 T^{10} + 453375110 T^{12} - 6200312460 T^{14} + 81612322276 T^{16} - 1047852805740 T^{18} + 12948846516710 T^{20} - 147527632866744 T^{22} + 1468942594724449 T^{24} - 12185587811516808 T^{26} + 78048585160311350 T^{28} - 330739616398740276 T^{30} + 665416609183179841 T^{32} \))(\( 1 - 84 T^{2} + 3350 T^{4} - 88392 T^{6} + 1800769 T^{8} - 30564216 T^{10} + 453375110 T^{12} - 6200312460 T^{14} + 81612322276 T^{16} - 1047852805740 T^{18} + 12948846516710 T^{20} - 147527632866744 T^{22} + 1468942594724449 T^{24} - 12185587811516808 T^{26} + 78048585160311350 T^{28} - 330739616398740276 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 48 T^{2} + 724 T^{4} - 2880 T^{5} + 5408 T^{6} - 128640 T^{7} + 180202 T^{8} - 1624320 T^{9} + 8804880 T^{10} - 9221184 T^{11} + 226101200 T^{12} - 511068864 T^{13} + 2089948240 T^{14} - 16835139072 T^{15} + 2634028179 T^{16} - 286197364224 T^{17} + 603995041360 T^{18} - 2510881328832 T^{19} + 18884198325200 T^{20} - 13092762650688 T^{21} + 212528398536720 T^{22} - 666521313327360 T^{23} + 1257045442383082 T^{24} - 15255144432574080 T^{25} + 10902495013628192 T^{26} - 98703061365983040 T^{27} + 421818499754346964 T^{28} + 8082135674851244592 T^{30} + 48661191875666868481 T^{32} \))(\( 1 + 48 T^{2} + 724 T^{4} - 2880 T^{5} + 5408 T^{6} - 128640 T^{7} + 180202 T^{8} - 1624320 T^{9} + 8804880 T^{10} - 9221184 T^{11} + 226101200 T^{12} - 511068864 T^{13} + 2089948240 T^{14} - 16835139072 T^{15} + 2634028179 T^{16} - 286197364224 T^{17} + 603995041360 T^{18} - 2510881328832 T^{19} + 18884198325200 T^{20} - 13092762650688 T^{21} + 212528398536720 T^{22} - 666521313327360 T^{23} + 1257045442383082 T^{24} - 15255144432574080 T^{25} + 10902495013628192 T^{26} - 98703061365983040 T^{27} + 421818499754346964 T^{28} + 8082135674851244592 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( 1 - 4 T - 46 T^{2} + 296 T^{3} + 931 T^{4} - 11288 T^{5} - 1970 T^{6} + 262932 T^{7} - 361639 T^{8} - 4327472 T^{9} + 10279876 T^{10} + 57644240 T^{11} - 192706850 T^{12} - 648662264 T^{13} + 3481638296 T^{14} + 4680631584 T^{15} - 67921193790 T^{16} + 88932000096 T^{17} + 1256871424856 T^{18} - 4449174468776 T^{19} - 25113749398850 T^{20} + 142732845019760 T^{21} + 483625822990756 T^{22} - 3868204922113808 T^{23} - 6141918754584199 T^{24} + 84844921752428028 T^{25} - 12078200527867970 T^{26} - 1314942042443096072 T^{27} + 2060596189650595891 T^{28} + 12447683104828089464 T^{29} - 36754307546012669566 T^{30} - 60724508119499193196 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 4 T - 46 T^{2} - 296 T^{3} + 931 T^{4} + 11288 T^{5} - 1970 T^{6} - 262932 T^{7} - 361639 T^{8} + 4327472 T^{9} + 10279876 T^{10} - 57644240 T^{11} - 192706850 T^{12} + 648662264 T^{13} + 3481638296 T^{14} - 4680631584 T^{15} - 67921193790 T^{16} - 88932000096 T^{17} + 1256871424856 T^{18} + 4449174468776 T^{19} - 25113749398850 T^{20} - 142732845019760 T^{21} + 483625822990756 T^{22} + 3868204922113808 T^{23} - 6141918754584199 T^{24} - 84844921752428028 T^{25} - 12078200527867970 T^{26} + 1314942042443096072 T^{27} + 2060596189650595891 T^{28} - 12447683104828089464 T^{29} - 36754307546012669566 T^{30} + 60724508119499193196 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 88 T^{2} + 3860 T^{4} + 5664 T^{5} + 94672 T^{6} + 516288 T^{7} + 1094890 T^{8} + 23434368 T^{9} + 10505704 T^{10} + 632564832 T^{11} + 1223438416 T^{12} + 9864851424 T^{13} + 71879999784 T^{14} + 85610565888 T^{15} + 2190561467667 T^{16} + 1969043015424 T^{17} + 38024519885736 T^{18} + 120025647275808 T^{19} + 342368229771856 T^{20} + 4071404228489376 T^{21} + 1555221231210856 T^{22} + 79789932500762496 T^{23} + 85741914674314090 T^{24} + 929913505281409344 T^{25} + 3921930669618578128 T^{26} + 5396714468824482528 T^{27} + 84590450307598439060 T^{28} + \)\(10\!\cdots\!92\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 88 T^{2} + 3860 T^{4} - 5664 T^{5} + 94672 T^{6} - 516288 T^{7} + 1094890 T^{8} - 23434368 T^{9} + 10505704 T^{10} - 632564832 T^{11} + 1223438416 T^{12} - 9864851424 T^{13} + 71879999784 T^{14} - 85610565888 T^{15} + 2190561467667 T^{16} - 1969043015424 T^{17} + 38024519885736 T^{18} - 120025647275808 T^{19} + 342368229771856 T^{20} - 4071404228489376 T^{21} + 1555221231210856 T^{22} - 79789932500762496 T^{23} + 85741914674314090 T^{24} - 929913505281409344 T^{25} + 3921930669618578128 T^{26} - 5396714468824482528 T^{27} + 84590450307598439060 T^{28} + \)\(10\!\cdots\!92\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 + 126 T^{2} + 96 T^{3} + 8529 T^{4} + 8640 T^{5} + 396966 T^{6} + 391008 T^{7} + 13413812 T^{8} + 11339232 T^{9} + 333848406 T^{10} + 210720960 T^{11} + 6032399649 T^{12} + 1969070304 T^{13} + 74947738446 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 126 T^{2} + 96 T^{3} + 8529 T^{4} + 8640 T^{5} + 396966 T^{6} + 391008 T^{7} + 13413812 T^{8} + 11339232 T^{9} + 333848406 T^{10} + 210720960 T^{11} + 6032399649 T^{12} + 1969070304 T^{13} + 74947738446 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( 1 - 4 T - 92 T^{2} + 1008 T^{3} + 1502 T^{4} - 68580 T^{5} + 270808 T^{6} + 1724724 T^{7} - 17946455 T^{8} + 19528132 T^{9} + 413026600 T^{10} - 1954928252 T^{11} - 1427100162 T^{12} + 30393015032 T^{13} - 45586379748 T^{14} - 34504405332 T^{15} - 431043300748 T^{16} - 1069636565292 T^{17} - 43808510937828 T^{18} + 905438310818312 T^{19} - 1317956968710402 T^{20} - 55967936120674052 T^{21} + 366562627850914600 T^{22} + 537269960024670652 T^{23} - 15306370623338221655 T^{24} + 45601050891441129804 T^{25} + \)\(22\!\cdots\!08\)\( T^{26} - \)\(17\!\cdots\!80\)\( T^{27} + \)\(11\!\cdots\!22\)\( T^{28} + \)\(24\!\cdots\!28\)\( T^{29} - \)\(69\!\cdots\!32\)\( T^{30} - \)\(93\!\cdots\!04\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 4 T - 92 T^{2} - 1008 T^{3} + 1502 T^{4} + 68580 T^{5} + 270808 T^{6} - 1724724 T^{7} - 17946455 T^{8} - 19528132 T^{9} + 413026600 T^{10} + 1954928252 T^{11} - 1427100162 T^{12} - 30393015032 T^{13} - 45586379748 T^{14} + 34504405332 T^{15} - 431043300748 T^{16} + 1069636565292 T^{17} - 43808510937828 T^{18} - 905438310818312 T^{19} - 1317956968710402 T^{20} + 55967936120674052 T^{21} + 366562627850914600 T^{22} - 537269960024670652 T^{23} - 15306370623338221655 T^{24} - 45601050891441129804 T^{25} + \)\(22\!\cdots\!08\)\( T^{26} + \)\(17\!\cdots\!80\)\( T^{27} + \)\(11\!\cdots\!22\)\( T^{28} - \)\(24\!\cdots\!28\)\( T^{29} - \)\(69\!\cdots\!32\)\( T^{30} + \)\(93\!\cdots\!04\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 - 4 T - 158 T^{2} + 744 T^{3} + 12779 T^{4} - 73728 T^{5} - 645026 T^{6} + 5204196 T^{7} + 19119049 T^{8} - 286453064 T^{9} - 55617596 T^{10} + 12416572840 T^{11} - 31278446130 T^{12} - 387805803760 T^{13} + 2209401589944 T^{14} + 5693211840792 T^{15} - 96714100430062 T^{16} + 210648838109304 T^{17} + 3024670776633336 T^{18} - 19643527377855280 T^{19} - 58620843877446930 T^{20} + 861014293104327880 T^{21} - 142699534862292764 T^{22} - 27193527076019385512 T^{23} + 67155266791008841129 T^{24} + \)\(67\!\cdots\!92\)\( T^{25} - \)\(31\!\cdots\!74\)\( T^{26} - \)\(13\!\cdots\!64\)\( T^{27} + \)\(84\!\cdots\!99\)\( T^{28} + \)\(18\!\cdots\!68\)\( T^{29} - \)\(14\!\cdots\!62\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - 4 T - 158 T^{2} + 744 T^{3} + 12779 T^{4} - 73728 T^{5} - 645026 T^{6} + 5204196 T^{7} + 19119049 T^{8} - 286453064 T^{9} - 55617596 T^{10} + 12416572840 T^{11} - 31278446130 T^{12} - 387805803760 T^{13} + 2209401589944 T^{14} + 5693211840792 T^{15} - 96714100430062 T^{16} + 210648838109304 T^{17} + 3024670776633336 T^{18} - 19643527377855280 T^{19} - 58620843877446930 T^{20} + 861014293104327880 T^{21} - 142699534862292764 T^{22} - 27193527076019385512 T^{23} + 67155266791008841129 T^{24} + \)\(67\!\cdots\!92\)\( T^{25} - \)\(31\!\cdots\!74\)\( T^{26} - \)\(13\!\cdots\!64\)\( T^{27} + \)\(84\!\cdots\!99\)\( T^{28} + \)\(18\!\cdots\!68\)\( T^{29} - \)\(14\!\cdots\!62\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 - 336 T^{2} + 56632 T^{4} - 6304752 T^{6} + 518498076 T^{8} - 33643406160 T^{10} + 1813689475976 T^{12} - 85242291537264 T^{14} + 3637250979607238 T^{16} - 143292292074140784 T^{18} + 5125052987323417736 T^{20} - \)\(15\!\cdots\!60\)\( T^{22} + \)\(41\!\cdots\!96\)\( T^{24} - \)\(84\!\cdots\!52\)\( T^{26} + \)\(12\!\cdots\!92\)\( T^{28} - \)\(12\!\cdots\!96\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 336 T^{2} + 56632 T^{4} - 6304752 T^{6} + 518498076 T^{8} - 33643406160 T^{10} + 1813689475976 T^{12} - 85242291537264 T^{14} + 3637250979607238 T^{16} - 143292292074140784 T^{18} + 5125052987323417736 T^{20} - \)\(15\!\cdots\!60\)\( T^{22} + \)\(41\!\cdots\!96\)\( T^{24} - \)\(84\!\cdots\!52\)\( T^{26} + \)\(12\!\cdots\!92\)\( T^{28} - \)\(12\!\cdots\!96\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - 228 T^{2} + 31878 T^{4} - 3192264 T^{6} + 258267313 T^{8} - 17507868792 T^{10} + 1031738237334 T^{12} - 53260073427900 T^{14} + 2436766852473924 T^{16} - 98477875768187100 T^{18} + 3527307717535716534 T^{20} - \)\(11\!\cdots\!08\)\( T^{22} + \)\(30\!\cdots\!13\)\( T^{24} - \)\(68\!\cdots\!36\)\( T^{26} + \)\(12\!\cdots\!78\)\( T^{28} - \)\(16\!\cdots\!72\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 - 228 T^{2} + 31878 T^{4} - 3192264 T^{6} + 258267313 T^{8} - 17507868792 T^{10} + 1031738237334 T^{12} - 53260073427900 T^{14} + 2436766852473924 T^{16} - 98477875768187100 T^{18} + 3527307717535716534 T^{20} - \)\(11\!\cdots\!08\)\( T^{22} + \)\(30\!\cdots\!13\)\( T^{24} - \)\(68\!\cdots\!36\)\( T^{26} + \)\(12\!\cdots\!78\)\( T^{28} - \)\(16\!\cdots\!72\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 8 T - 144 T^{2} + 464 T^{3} + 17172 T^{4} + 11000 T^{5} - 1140704 T^{6} - 6276488 T^{7} + 46712874 T^{8} + 620504992 T^{9} - 149042736 T^{10} - 37377235304 T^{11} - 131884930992 T^{12} + 1479781798792 T^{13} + 11497201556624 T^{14} - 26133075365216 T^{15} - 645304022262061 T^{16} - 1228254542165152 T^{17} + 25397318238582416 T^{18} + 153635385695981816 T^{19} - 643556391947973552 T^{20} - 8572282292436527128 T^{21} - 1606563744567300144 T^{22} + \)\(31\!\cdots\!96\)\( T^{23} + \)\(11\!\cdots\!14\)\( T^{24} - \)\(70\!\cdots\!96\)\( T^{25} - \)\(60\!\cdots\!96\)\( T^{26} + \)\(27\!\cdots\!00\)\( T^{27} + \)\(19\!\cdots\!52\)\( T^{28} + \)\(25\!\cdots\!28\)\( T^{29} - \)\(36\!\cdots\!36\)\( T^{30} - \)\(96\!\cdots\!44\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 8 T - 144 T^{2} - 464 T^{3} + 17172 T^{4} - 11000 T^{5} - 1140704 T^{6} + 6276488 T^{7} + 46712874 T^{8} - 620504992 T^{9} - 149042736 T^{10} + 37377235304 T^{11} - 131884930992 T^{12} - 1479781798792 T^{13} + 11497201556624 T^{14} + 26133075365216 T^{15} - 645304022262061 T^{16} + 1228254542165152 T^{17} + 25397318238582416 T^{18} - 153635385695981816 T^{19} - 643556391947973552 T^{20} + 8572282292436527128 T^{21} - 1606563744567300144 T^{22} - \)\(31\!\cdots\!96\)\( T^{23} + \)\(11\!\cdots\!14\)\( T^{24} + \)\(70\!\cdots\!96\)\( T^{25} - \)\(60\!\cdots\!96\)\( T^{26} - \)\(27\!\cdots\!00\)\( T^{27} + \)\(19\!\cdots\!52\)\( T^{28} - \)\(25\!\cdots\!28\)\( T^{29} - \)\(36\!\cdots\!36\)\( T^{30} + \)\(96\!\cdots\!44\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 8 T - 78 T^{2} + 1184 T^{3} - 2973 T^{4} - 20320 T^{5} + 566974 T^{6} - 5965256 T^{7} + 2148105 T^{8} + 493398784 T^{9} - 3412297068 T^{10} - 591604736 T^{11} + 160029842478 T^{12} - 1529452153232 T^{13} + 7166583780200 T^{14} + 53834495957056 T^{15} - 875021360553214 T^{16} + 2853228285723968 T^{17} + 20130933838581800 T^{18} - 227700248216720464 T^{19} + 1262712431505651918 T^{20} - 247406434232654848 T^{21} - 75631384494579869772 T^{22} + \)\(57\!\cdots\!08\)\( T^{23} + \)\(13\!\cdots\!05\)\( T^{24} - \)\(19\!\cdots\!48\)\( T^{25} + \)\(99\!\cdots\!26\)\( T^{26} - \)\(18\!\cdots\!40\)\( T^{27} - \)\(14\!\cdots\!93\)\( T^{28} + \)\(30\!\cdots\!32\)\( T^{29} - \)\(10\!\cdots\!82\)\( T^{30} - \)\(58\!\cdots\!56\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 - 8 T - 78 T^{2} + 1184 T^{3} - 2973 T^{4} - 20320 T^{5} + 566974 T^{6} - 5965256 T^{7} + 2148105 T^{8} + 493398784 T^{9} - 3412297068 T^{10} - 591604736 T^{11} + 160029842478 T^{12} - 1529452153232 T^{13} + 7166583780200 T^{14} + 53834495957056 T^{15} - 875021360553214 T^{16} + 2853228285723968 T^{17} + 20130933838581800 T^{18} - 227700248216720464 T^{19} + 1262712431505651918 T^{20} - 247406434232654848 T^{21} - 75631384494579869772 T^{22} + \)\(57\!\cdots\!08\)\( T^{23} + \)\(13\!\cdots\!05\)\( T^{24} - \)\(19\!\cdots\!48\)\( T^{25} + \)\(99\!\cdots\!26\)\( T^{26} - \)\(18\!\cdots\!40\)\( T^{27} - \)\(14\!\cdots\!93\)\( T^{28} + \)\(30\!\cdots\!32\)\( T^{29} - \)\(10\!\cdots\!82\)\( T^{30} - \)\(58\!\cdots\!56\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 4 T - 278 T^{2} + 744 T^{3} + 39283 T^{4} - 56888 T^{5} - 3899258 T^{6} + 2731764 T^{7} + 316041561 T^{8} - 129333776 T^{9} - 23263019324 T^{10} + 6499521008 T^{11} + 1633005733406 T^{12} - 280959606200 T^{13} - 107397941612568 T^{14} + 6656498449920 T^{15} + 6551823717325442 T^{16} + 392733408545280 T^{17} - 373852234753349208 T^{18} - 57703202961749800 T^{19} + 19787719986750261566 T^{20} + 4646665500480173392 T^{21} - \)\(98\!\cdots\!84\)\( T^{22} - \)\(32\!\cdots\!44\)\( T^{23} + \)\(46\!\cdots\!81\)\( T^{24} + \)\(23\!\cdots\!96\)\( T^{25} - \)\(19\!\cdots\!58\)\( T^{26} - \)\(17\!\cdots\!92\)\( T^{27} + \)\(69\!\cdots\!23\)\( T^{28} + \)\(78\!\cdots\!76\)\( T^{29} - \)\(17\!\cdots\!58\)\( T^{30} - \)\(14\!\cdots\!96\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 4 T - 278 T^{2} - 744 T^{3} + 39283 T^{4} + 56888 T^{5} - 3899258 T^{6} - 2731764 T^{7} + 316041561 T^{8} + 129333776 T^{9} - 23263019324 T^{10} - 6499521008 T^{11} + 1633005733406 T^{12} + 280959606200 T^{13} - 107397941612568 T^{14} - 6656498449920 T^{15} + 6551823717325442 T^{16} - 392733408545280 T^{17} - 373852234753349208 T^{18} + 57703202961749800 T^{19} + 19787719986750261566 T^{20} - 4646665500480173392 T^{21} - \)\(98\!\cdots\!84\)\( T^{22} + \)\(32\!\cdots\!44\)\( T^{23} + \)\(46\!\cdots\!81\)\( T^{24} - \)\(23\!\cdots\!96\)\( T^{25} - \)\(19\!\cdots\!58\)\( T^{26} + \)\(17\!\cdots\!92\)\( T^{27} + \)\(69\!\cdots\!23\)\( T^{28} - \)\(78\!\cdots\!76\)\( T^{29} - \)\(17\!\cdots\!58\)\( T^{30} + \)\(14\!\cdots\!96\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - 24 T + 520 T^{2} - 7872 T^{3} + 105220 T^{4} - 1233096 T^{5} + 13416560 T^{6} - 139381944 T^{7} + 1398589418 T^{8} - 13649424720 T^{9} + 129262229752 T^{10} - 1182300753576 T^{11} + 10462356189200 T^{12} - 89714645688552 T^{13} + 745406748402104 T^{14} - 6054389325199248 T^{15} + 47735892633152755 T^{16} - 369317748837154128 T^{17} + 2773658510804228984 T^{18} - 20363519993033221512 T^{19} + \)\(14\!\cdots\!00\)\( T^{20} - \)\(99\!\cdots\!76\)\( T^{21} + \)\(66\!\cdots\!72\)\( T^{22} - \)\(42\!\cdots\!20\)\( T^{23} + \)\(26\!\cdots\!58\)\( T^{24} - \)\(16\!\cdots\!04\)\( T^{25} + \)\(95\!\cdots\!60\)\( T^{26} - \)\(53\!\cdots\!56\)\( T^{27} + \)\(27\!\cdots\!20\)\( T^{28} - \)\(12\!\cdots\!32\)\( T^{29} + \)\(51\!\cdots\!20\)\( T^{30} - \)\(14\!\cdots\!24\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 - 24 T + 520 T^{2} - 7872 T^{3} + 105220 T^{4} - 1233096 T^{5} + 13416560 T^{6} - 139381944 T^{7} + 1398589418 T^{8} - 13649424720 T^{9} + 129262229752 T^{10} - 1182300753576 T^{11} + 10462356189200 T^{12} - 89714645688552 T^{13} + 745406748402104 T^{14} - 6054389325199248 T^{15} + 47735892633152755 T^{16} - 369317748837154128 T^{17} + 2773658510804228984 T^{18} - 20363519993033221512 T^{19} + \)\(14\!\cdots\!00\)\( T^{20} - \)\(99\!\cdots\!76\)\( T^{21} + \)\(66\!\cdots\!72\)\( T^{22} - \)\(42\!\cdots\!20\)\( T^{23} + \)\(26\!\cdots\!58\)\( T^{24} - \)\(16\!\cdots\!04\)\( T^{25} + \)\(95\!\cdots\!60\)\( T^{26} - \)\(53\!\cdots\!56\)\( T^{27} + \)\(27\!\cdots\!20\)\( T^{28} - \)\(12\!\cdots\!32\)\( T^{29} + \)\(51\!\cdots\!20\)\( T^{30} - \)\(14\!\cdots\!24\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 + 12 T + 354 T^{2} + 3672 T^{3} + 64867 T^{4} + 585624 T^{5} + 7670686 T^{6} + 62761668 T^{7} + 670299289 T^{8} + 5202844464 T^{9} + 49053747588 T^{10} + 389712261936 T^{11} + 3478800861854 T^{12} + 29196505227048 T^{13} + 254841620895224 T^{14} + 2150356193732448 T^{15} + 17864389643451906 T^{16} + 144073864980074016 T^{17} + 1143984036198660536 T^{18} + 8781228501602637624 T^{19} + 70101737102124238334 T^{20} + \)\(52\!\cdots\!52\)\( T^{21} + \)\(44\!\cdots\!72\)\( T^{22} + \)\(31\!\cdots\!72\)\( T^{23} + \)\(27\!\cdots\!49\)\( T^{24} + \)\(17\!\cdots\!96\)\( T^{25} + \)\(13\!\cdots\!14\)\( T^{26} + \)\(71\!\cdots\!92\)\( T^{27} + \)\(53\!\cdots\!87\)\( T^{28} + \)\(20\!\cdots\!64\)\( T^{29} + \)\(13\!\cdots\!66\)\( T^{30} + \)\(29\!\cdots\!16\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 - 12 T + 354 T^{2} - 3672 T^{3} + 64867 T^{4} - 585624 T^{5} + 7670686 T^{6} - 62761668 T^{7} + 670299289 T^{8} - 5202844464 T^{9} + 49053747588 T^{10} - 389712261936 T^{11} + 3478800861854 T^{12} - 29196505227048 T^{13} + 254841620895224 T^{14} - 2150356193732448 T^{15} + 17864389643451906 T^{16} - 144073864980074016 T^{17} + 1143984036198660536 T^{18} - 8781228501602637624 T^{19} + 70101737102124238334 T^{20} - \)\(52\!\cdots\!52\)\( T^{21} + \)\(44\!\cdots\!72\)\( T^{22} - \)\(31\!\cdots\!72\)\( T^{23} + \)\(27\!\cdots\!49\)\( T^{24} - \)\(17\!\cdots\!96\)\( T^{25} + \)\(13\!\cdots\!14\)\( T^{26} - \)\(71\!\cdots\!92\)\( T^{27} + \)\(53\!\cdots\!87\)\( T^{28} - \)\(20\!\cdots\!64\)\( T^{29} + \)\(13\!\cdots\!66\)\( T^{30} - \)\(29\!\cdots\!16\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 - 704 T^{2} + 240152 T^{4} - 53098944 T^{6} + 8598903516 T^{8} - 1093483638336 T^{10} + 114258843245224 T^{12} - 10109733274809664 T^{14} + 770931618303563846 T^{16} - 50963165438315516224 T^{18} + \)\(29\!\cdots\!44\)\( T^{20} - \)\(14\!\cdots\!56\)\( T^{22} + \)\(55\!\cdots\!76\)\( T^{24} - \)\(17\!\cdots\!44\)\( T^{26} + \)\(39\!\cdots\!32\)\( T^{28} - \)\(58\!\cdots\!24\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 - 704 T^{2} + 240152 T^{4} - 53098944 T^{6} + 8598903516 T^{8} - 1093483638336 T^{10} + 114258843245224 T^{12} - 10109733274809664 T^{14} + 770931618303563846 T^{16} - 50963165438315516224 T^{18} + \)\(29\!\cdots\!44\)\( T^{20} - \)\(14\!\cdots\!56\)\( T^{22} + \)\(55\!\cdots\!76\)\( T^{24} - \)\(17\!\cdots\!44\)\( T^{26} + \)\(39\!\cdots\!32\)\( T^{28} - \)\(58\!\cdots\!24\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 12 T + 234 T^{2} - 2232 T^{3} + 22355 T^{4} - 173856 T^{5} + 989254 T^{6} - 11225076 T^{7} + 53667833 T^{8} - 1011835944 T^{9} + 5788877684 T^{10} - 73907745240 T^{11} + 590847685822 T^{12} - 5461132976544 T^{13} + 74257181172216 T^{14} - 566395732865256 T^{15} + 7250352758101906 T^{16} - 41346888499163688 T^{17} + 395716518466739064 T^{18} - 2124473567136217248 T^{19} + 16779034976265439102 T^{20} - \)\(15\!\cdots\!20\)\( T^{21} + \)\(87\!\cdots\!76\)\( T^{22} - \)\(11\!\cdots\!68\)\( T^{23} + \)\(43\!\cdots\!73\)\( T^{24} - \)\(66\!\cdots\!88\)\( T^{25} + \)\(42\!\cdots\!46\)\( T^{26} - \)\(54\!\cdots\!12\)\( T^{27} + \)\(51\!\cdots\!55\)\( T^{28} - \)\(37\!\cdots\!56\)\( T^{29} + \)\(28\!\cdots\!06\)\( T^{30} - \)\(10\!\cdots\!84\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 12 T + 234 T^{2} - 2232 T^{3} + 22355 T^{4} - 173856 T^{5} + 989254 T^{6} - 11225076 T^{7} + 53667833 T^{8} - 1011835944 T^{9} + 5788877684 T^{10} - 73907745240 T^{11} + 590847685822 T^{12} - 5461132976544 T^{13} + 74257181172216 T^{14} - 566395732865256 T^{15} + 7250352758101906 T^{16} - 41346888499163688 T^{17} + 395716518466739064 T^{18} - 2124473567136217248 T^{19} + 16779034976265439102 T^{20} - \)\(15\!\cdots\!20\)\( T^{21} + \)\(87\!\cdots\!76\)\( T^{22} - \)\(11\!\cdots\!68\)\( T^{23} + \)\(43\!\cdots\!73\)\( T^{24} - \)\(66\!\cdots\!88\)\( T^{25} + \)\(42\!\cdots\!46\)\( T^{26} - \)\(54\!\cdots\!12\)\( T^{27} + \)\(51\!\cdots\!55\)\( T^{28} - \)\(37\!\cdots\!56\)\( T^{29} + \)\(28\!\cdots\!06\)\( T^{30} - \)\(10\!\cdots\!84\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 12 T + 452 T^{2} - 4848 T^{3} + 99326 T^{4} - 752460 T^{5} + 12206616 T^{6} - 45128196 T^{7} + 802260297 T^{8} + 2552889420 T^{9} + 31280077608 T^{10} + 585579231756 T^{11} + 3876798962782 T^{12} + 22903847721096 T^{13} + 800190643393852 T^{14} - 1777305262761948 T^{15} + 87562114349854388 T^{16} - 140407115758193892 T^{17} + 4993989805421030332 T^{18} + 11292490176561450744 T^{19} + \)\(15\!\cdots\!42\)\( T^{20} + \)\(18\!\cdots\!44\)\( T^{21} + \)\(76\!\cdots\!68\)\( T^{22} + \)\(49\!\cdots\!80\)\( T^{23} + \)\(12\!\cdots\!17\)\( T^{24} - \)\(54\!\cdots\!24\)\( T^{25} + \)\(11\!\cdots\!16\)\( T^{26} - \)\(56\!\cdots\!40\)\( T^{27} + \)\(58\!\cdots\!66\)\( T^{28} - \)\(22\!\cdots\!72\)\( T^{29} + \)\(16\!\cdots\!12\)\( T^{30} - \)\(34\!\cdots\!88\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 + 12 T + 452 T^{2} + 4848 T^{3} + 99326 T^{4} + 752460 T^{5} + 12206616 T^{6} + 45128196 T^{7} + 802260297 T^{8} - 2552889420 T^{9} + 31280077608 T^{10} - 585579231756 T^{11} + 3876798962782 T^{12} - 22903847721096 T^{13} + 800190643393852 T^{14} + 1777305262761948 T^{15} + 87562114349854388 T^{16} + 140407115758193892 T^{17} + 4993989805421030332 T^{18} - 11292490176561450744 T^{19} + \)\(15\!\cdots\!42\)\( T^{20} - \)\(18\!\cdots\!44\)\( T^{21} + \)\(76\!\cdots\!68\)\( T^{22} - \)\(49\!\cdots\!80\)\( T^{23} + \)\(12\!\cdots\!17\)\( T^{24} + \)\(54\!\cdots\!24\)\( T^{25} + \)\(11\!\cdots\!16\)\( T^{26} + \)\(56\!\cdots\!40\)\( T^{27} + \)\(58\!\cdots\!66\)\( T^{28} + \)\(22\!\cdots\!72\)\( T^{29} + \)\(16\!\cdots\!12\)\( T^{30} + \)\(34\!\cdots\!88\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( ( 1 - 4 T + 510 T^{2} - 1368 T^{3} + 119761 T^{4} - 213952 T^{5} + 17308534 T^{6} - 21965340 T^{7} + 1711141540 T^{8} - 1823123220 T^{9} + 119238490726 T^{10} - 122334972224 T^{11} + 5683655981281 T^{12} - 5388607599624 T^{13} + 166739590418190 T^{14} - 108544203958508 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 4 T + 510 T^{2} + 1368 T^{3} + 119761 T^{4} + 213952 T^{5} + 17308534 T^{6} + 21965340 T^{7} + 1711141540 T^{8} + 1823123220 T^{9} + 119238490726 T^{10} + 122334972224 T^{11} + 5683655981281 T^{12} + 5388607599624 T^{13} + 166739590418190 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 + 272 T^{2} + 26468 T^{4} + 139584 T^{5} + 983008 T^{6} + 40361088 T^{7} + 36104074 T^{8} + 4345742592 T^{9} + 15915680112 T^{10} + 231982822464 T^{11} + 3174421757584 T^{12} + 14601730770624 T^{13} + 335852371900656 T^{14} + 1870227047990784 T^{15} + 28375693200769107 T^{16} + 166450207271179776 T^{17} + 2660286637825096176 T^{18} + 10293767539636030656 T^{19} + \)\(19\!\cdots\!44\)\( T^{20} + \)\(12\!\cdots\!36\)\( T^{21} + \)\(79\!\cdots\!32\)\( T^{22} + \)\(19\!\cdots\!68\)\( T^{23} + \)\(14\!\cdots\!94\)\( T^{24} + \)\(14\!\cdots\!92\)\( T^{25} + \)\(30\!\cdots\!08\)\( T^{26} + \)\(38\!\cdots\!76\)\( T^{27} + \)\(65\!\cdots\!28\)\( T^{28} + \)\(53\!\cdots\!52\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 + 272 T^{2} + 26468 T^{4} + 139584 T^{5} + 983008 T^{6} + 40361088 T^{7} + 36104074 T^{8} + 4345742592 T^{9} + 15915680112 T^{10} + 231982822464 T^{11} + 3174421757584 T^{12} + 14601730770624 T^{13} + 335852371900656 T^{14} + 1870227047990784 T^{15} + 28375693200769107 T^{16} + 166450207271179776 T^{17} + 2660286637825096176 T^{18} + 10293767539636030656 T^{19} + \)\(19\!\cdots\!44\)\( T^{20} + \)\(12\!\cdots\!36\)\( T^{21} + \)\(79\!\cdots\!32\)\( T^{22} + \)\(19\!\cdots\!68\)\( T^{23} + \)\(14\!\cdots\!94\)\( T^{24} + \)\(14\!\cdots\!92\)\( T^{25} + \)\(30\!\cdots\!08\)\( T^{26} + \)\(38\!\cdots\!76\)\( T^{27} + \)\(65\!\cdots\!28\)\( T^{28} + \)\(53\!\cdots\!52\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 - 1060 T^{2} + 554918 T^{4} - 190432200 T^{6} + 47922422193 T^{8} - 9371042858136 T^{10} + 1472049420153334 T^{12} - 189391291052283068 T^{14} + 20162972419702510820 T^{16} - \)\(17\!\cdots\!12\)\( T^{18} + \)\(13\!\cdots\!54\)\( T^{20} - \)\(78\!\cdots\!44\)\( T^{22} + \)\(37\!\cdots\!73\)\( T^{24} - \)\(14\!\cdots\!00\)\( T^{26} + \)\(38\!\cdots\!38\)\( T^{28} - \)\(69\!\cdots\!40\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 - 1060 T^{2} + 554918 T^{4} - 190432200 T^{6} + 47922422193 T^{8} - 9371042858136 T^{10} + 1472049420153334 T^{12} - 189391291052283068 T^{14} + 20162972419702510820 T^{16} - \)\(17\!\cdots\!12\)\( T^{18} + \)\(13\!\cdots\!54\)\( T^{20} - \)\(78\!\cdots\!44\)\( T^{22} + \)\(37\!\cdots\!73\)\( T^{24} - \)\(14\!\cdots\!00\)\( T^{26} + \)\(38\!\cdots\!38\)\( T^{28} - \)\(69\!\cdots\!40\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))
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