Properties

Label 57.2
Level 57
Weight 2
Dimension 71
Nonzero newspaces 6
Newform subspaces 11
Sturm bound 480
Trace bound 3

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 57 = 3 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 11 \)
Sturm bound: \(480\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(57))\).

Total New Old
Modular forms 156 107 49
Cusp forms 85 71 14
Eisenstein series 71 36 35

Trace form

\( 71q - 3q^{2} - 10q^{3} - 25q^{4} - 6q^{5} - 12q^{6} - 26q^{7} - 15q^{8} - 10q^{9} + O(q^{10}) \) \( 71q - 3q^{2} - 10q^{3} - 25q^{4} - 6q^{5} - 12q^{6} - 26q^{7} - 15q^{8} - 10q^{9} - 36q^{10} - 12q^{11} - 4q^{12} - 8q^{13} + 12q^{14} + 3q^{15} + 23q^{16} - 3q^{18} + 5q^{19} + 30q^{20} + 4q^{21} - 6q^{23} + 12q^{24} - 13q^{25} - 6q^{26} + 2q^{27} + 4q^{28} + 6q^{29} + 36q^{30} - 14q^{31} + 27q^{32} + 33q^{33} + 18q^{34} + 24q^{35} + 56q^{36} - 2q^{37} + 69q^{38} + 22q^{39} + 54q^{40} - 6q^{41} + 57q^{42} + 10q^{43} + 6q^{44} + 12q^{45} - 12q^{47} - 40q^{48} - 39q^{49} - 39q^{50} - 63q^{51} - 116q^{52} - 54q^{53} - 66q^{54} - 90q^{55} - 120q^{56} - 73q^{57} - 126q^{58} - 60q^{59} - 114q^{60} - 20q^{61} - 6q^{62} - 41q^{63} - q^{64} + 24q^{65} - 45q^{66} + 58q^{67} + 12q^{69} + 54q^{70} + 39q^{72} + 70q^{73} + 30q^{74} + 53q^{75} + 29q^{76} + 102q^{77} + 93q^{78} + 106q^{79} + 66q^{80} + 26q^{81} + 198q^{82} + 42q^{83} + 151q^{84} + 90q^{85} + 66q^{86} + 105q^{87} + 126q^{88} + 72q^{89} + 36q^{90} + 38q^{91} + 66q^{92} + 61q^{93} + 36q^{94} - 60q^{95} - 9q^{96} - 62q^{97} - 27q^{98} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(57))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
57.2.a \(\chi_{57}(1, \cdot)\) 57.2.a.a 1 1
57.2.a.b 1
57.2.a.c 1
57.2.d \(\chi_{57}(56, \cdot)\) 57.2.d.a 4 1
57.2.e \(\chi_{57}(7, \cdot)\) 57.2.e.a 2 2
57.2.e.b 6
57.2.f \(\chi_{57}(8, \cdot)\) 57.2.f.a 8 2
57.2.i \(\chi_{57}(4, \cdot)\) 57.2.i.a 6 6
57.2.i.b 12
57.2.j \(\chi_{57}(2, \cdot)\) 57.2.j.a 6 6
57.2.j.b 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(57))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(57)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T + T^{3} - 2 T^{4} - T^{5} + 11 T^{6} - 2 T^{7} - 8 T^{8} + 8 T^{9} - 32 T^{11} + 64 T^{12} \))(\( 1 - 4 T^{2} + 7 T^{4} - 4 T^{6} + T^{8} - 16 T^{10} + 112 T^{12} - 256 T^{14} + 256 T^{16} \))(\( 1 - 3 T + 6 T^{2} - 8 T^{3} + 15 T^{4} - 27 T^{5} + 47 T^{6} - 54 T^{7} + 60 T^{8} - 64 T^{9} + 96 T^{10} - 96 T^{11} + 64 T^{12} \))(\( 1 + 3 T + 6 T^{2} + 8 T^{3} + 3 T^{4} - 15 T^{5} - 43 T^{6} - 72 T^{7} - 72 T^{8} - 28 T^{9} + 96 T^{10} + 264 T^{11} + 424 T^{12} + 528 T^{13} + 384 T^{14} - 224 T^{15} - 1152 T^{16} - 2304 T^{17} - 2752 T^{18} - 1920 T^{19} + 768 T^{20} + 4096 T^{21} + 6144 T^{22} + 6144 T^{23} + 4096 T^{24} \))(\( 1 - 8 T^{6} + 64 T^{12} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \))(\( 1 - T^{3} + T^{6} \))(\( ( 1 + T^{3} + T^{6} )^{2} \))(\( 1 - 9 T^{3} + 27 T^{6} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 2 T - 3 T^{2} - 2 T^{3} - 2 T^{4} - 34 T^{5} - 31 T^{6} - 170 T^{7} - 50 T^{8} - 250 T^{9} - 1875 T^{10} + 6250 T^{11} + 15625 T^{12} \))(\( ( 1 + 8 T^{2} + 39 T^{4} + 200 T^{6} + 625 T^{8} )^{2} \))(\( 1 + 6 T + 18 T^{2} + 45 T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 435 T^{7} + 2025 T^{8} + 5625 T^{9} + 11250 T^{10} + 18750 T^{11} + 15625 T^{12} \))(\( 1 - 6 T + 24 T^{2} - 73 T^{3} + 147 T^{4} - 123 T^{5} - 358 T^{6} + 2205 T^{7} - 6057 T^{8} + 9746 T^{9} - 4305 T^{10} - 29499 T^{11} + 105481 T^{12} - 147495 T^{13} - 107625 T^{14} + 1218250 T^{15} - 3785625 T^{16} + 6890625 T^{17} - 5593750 T^{18} - 9609375 T^{19} + 57421875 T^{20} - 142578125 T^{21} + 234375000 T^{22} - 292968750 T^{23} + 244140625 T^{24} \))(\( 1 + 125 T^{6} + 15625 T^{12} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( ( 1 - T + 7 T^{2} )^{4} \))(\( ( 1 - T + 7 T^{2} )^{2} \))(\( ( 1 + T + 12 T^{2} + 17 T^{3} + 84 T^{4} + 49 T^{5} + 343 T^{6} )^{2} \))(\( ( 1 + 4 T + 15 T^{2} + 28 T^{3} + 49 T^{4} )^{4} \))(\( 1 - 3 T - 12 T^{2} + 19 T^{3} + 171 T^{4} - 126 T^{5} - 1161 T^{6} - 882 T^{7} + 8379 T^{8} + 6517 T^{9} - 28812 T^{10} - 50421 T^{11} + 117649 T^{12} \))(\( 1 + 9 T + 24 T^{2} + 9 T^{3} - 3 T^{4} - 162 T^{5} - 2744 T^{6} - 9621 T^{7} - 9045 T^{8} + 1782 T^{9} + 3447 T^{10} + 367587 T^{11} + 1830159 T^{12} + 2573109 T^{13} + 168903 T^{14} + 611226 T^{15} - 21717045 T^{16} - 161700147 T^{17} - 322828856 T^{18} - 133413966 T^{19} - 17294403 T^{20} + 363182463 T^{21} + 6779405976 T^{22} + 17795940687 T^{23} + 13841287201 T^{24} \))(\( ( 1 + 17 T^{3} + 343 T^{6} )^{2} \))
$11$ (\( 1 - T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( ( 1 - 17 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} - 36 T^{3} + 99 T^{4} + 1331 T^{6} )^{2} \))(\( ( 1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2} \))(\( 1 - 3 T - 6 T^{2} + 13 T^{3} - 27 T^{4} + 192 T^{5} - 61 T^{6} + 2112 T^{7} - 3267 T^{8} + 17303 T^{9} - 87846 T^{10} - 483153 T^{11} + 1771561 T^{12} \))(\( 1 + 9 T + 9 T^{2} + 855 T^{4} + 2205 T^{5} - 3823 T^{6} + 27261 T^{7} + 139914 T^{8} - 100440 T^{9} + 483597 T^{10} + 3314745 T^{11} - 178197 T^{12} + 36462195 T^{13} + 58515237 T^{14} - 133685640 T^{15} + 2048480874 T^{16} + 4390411311 T^{17} - 6772677703 T^{18} + 42969212055 T^{19} + 183276843255 T^{20} + 233436821409 T^{22} + 2567805035499 T^{23} + 3138428376721 T^{24} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{3} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( ( 1 - 16 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - T - 17 T^{2} + 40 T^{3} + 61 T^{4} - 223 T^{5} + 854 T^{6} - 2899 T^{7} + 10309 T^{8} + 87880 T^{9} - 485537 T^{10} - 371293 T^{11} + 4826809 T^{12} \))(\( ( 1 - 6 T + 37 T^{2} - 150 T^{3} + 612 T^{4} - 1950 T^{5} + 6253 T^{6} - 13182 T^{7} + 28561 T^{8} )^{2} \))(\( 1 + 6 T + 30 T^{2} + 86 T^{3} + 288 T^{4} + 180 T^{5} + 231 T^{6} + 2340 T^{7} + 48672 T^{8} + 188942 T^{9} + 856830 T^{10} + 2227758 T^{11} + 4826809 T^{12} \))(\( 1 + 3 T - 6 T^{2} - 60 T^{3} - 249 T^{4} - 591 T^{5} + 2128 T^{6} + 5796 T^{7} + 8037 T^{8} + 23904 T^{9} + 27657 T^{10} - 209088 T^{11} + 2319225 T^{12} - 2718144 T^{13} + 4674033 T^{14} + 52517088 T^{15} + 229544757 T^{16} + 2152014228 T^{17} + 10271449552 T^{18} - 37084373547 T^{19} - 203116949529 T^{20} - 636269962380 T^{21} - 827150951094 T^{22} + 5376481182111 T^{23} + 23298085122481 T^{24} \))(\( ( 1 + 5 T + 13 T^{2} )^{3}( 1 - 89 T^{3} + 2197 T^{6} ) \))
$17$ (\( 1 + T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( ( 1 + 11 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{3} \))(\( ( 1 + 10 T^{2} - 189 T^{4} + 2890 T^{6} + 83521 T^{8} )^{2} \))(\( 1 + 12 T + 114 T^{2} + 809 T^{3} + 4965 T^{4} + 25191 T^{5} + 111881 T^{6} + 428247 T^{7} + 1434885 T^{8} + 3974617 T^{9} + 9521394 T^{10} + 17038284 T^{11} + 24137569 T^{12} \))(\( 1 - 12 T + 18 T^{2} + 315 T^{3} - 1485 T^{4} + 555 T^{5} + 16634 T^{6} - 87723 T^{7} - 100071 T^{8} + 2623806 T^{9} - 4546053 T^{10} - 31268241 T^{11} + 197865993 T^{12} - 531560097 T^{13} - 1313809317 T^{14} + 12890758878 T^{15} - 8358029991 T^{16} - 124554115611 T^{17} + 401504322746 T^{18} + 227737963515 T^{19} - 10358999799885 T^{20} + 37355181096555 T^{21} + 36287890208082 T^{22} - 411262755691596 T^{23} + 582622237229761 T^{24} \))(\( 1 + 4913 T^{6} + 24137569 T^{12} \))
$19$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - 6 T + 19 T^{2} )^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 17 T^{2} - 136 T^{3} + 323 T^{4} - 1444 T^{5} + 6859 T^{6} \))(\( ( 1 + 26 T^{2} + 361 T^{4} )^{2} \))(\( 1 - 18 T + 162 T^{2} - 883 T^{3} + 3078 T^{4} - 6498 T^{5} + 6859 T^{6} \))(\( 1 + 9 T + 72 T^{2} + 538 T^{3} + 3195 T^{4} + 16245 T^{5} + 78033 T^{6} + 308655 T^{7} + 1153395 T^{8} + 3690142 T^{9} + 9383112 T^{10} + 22284891 T^{11} + 47045881 T^{12} \))(\( ( 1 - T + 19 T^{2} )^{3} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( ( 1 - 26 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 14 T + 99 T^{2} + 382 T^{3} + 346 T^{4} - 7726 T^{5} - 57613 T^{6} - 177698 T^{7} + 183034 T^{8} + 4647794 T^{9} + 27704259 T^{10} + 90108802 T^{11} + 148035889 T^{12} \))(\( 1 + 64 T^{2} + 2206 T^{4} + 53248 T^{6} + 1125475 T^{8} + 28168192 T^{10} + 617329246 T^{12} + 9474296896 T^{14} + 78310985281 T^{16} \))(\( 1 + 9 T + 18 T^{2} - 130 T^{3} - 603 T^{4} + 4725 T^{5} + 49121 T^{6} + 108675 T^{7} - 318987 T^{8} - 1581710 T^{9} + 5037138 T^{10} + 57927087 T^{11} + 148035889 T^{12} \))(\( 1 - 9 T - 18 T^{2} + 278 T^{3} + 891 T^{4} - 4491 T^{5} - 52096 T^{6} + 68292 T^{7} + 1806471 T^{8} - 2074390 T^{9} - 38985543 T^{10} + 45515898 T^{11} + 603696715 T^{12} + 1046865654 T^{13} - 20623352247 T^{14} - 25239103130 T^{15} + 505524651111 T^{16} + 439550736156 T^{17} - 7712077673344 T^{18} - 15291071082477 T^{19} + 69775087885371 T^{20} + 500720439886714 T^{21} - 745677201845682 T^{22} - 8575287821225343 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 12167 T^{6} + 148035889 T^{12} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( ( 1 + 26 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 8 T + 35 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 4 T - 39 T^{2} - 52 T^{3} + 886 T^{4} - 1916 T^{5} - 33907 T^{6} - 55564 T^{7} + 745126 T^{8} - 1268228 T^{9} - 27583959 T^{10} + 82044596 T^{11} + 594823321 T^{12} \))(\( 1 - 52 T^{2} + 1114 T^{4} + 4784 T^{6} - 453245 T^{8} + 4023344 T^{10} + 787911034 T^{12} - 30930812692 T^{14} + 500246412961 T^{16} \))(\( 1 - 12 T + 60 T^{2} - 26 T^{3} - 1200 T^{4} + 9792 T^{5} - 51073 T^{6} + 283968 T^{7} - 1009200 T^{8} - 634114 T^{9} + 42436860 T^{10} - 246133788 T^{11} + 594823321 T^{12} \))(\( 1 - 6 T^{2} + 310 T^{3} - 894 T^{4} + 6660 T^{5} + 49610 T^{6} - 275760 T^{7} + 2417580 T^{8} - 852920 T^{9} + 7918416 T^{10} + 366039840 T^{11} - 1334165873 T^{12} + 10615155360 T^{13} + 6659387856 T^{14} - 20801865880 T^{15} + 1709908399980 T^{16} - 5656154448240 T^{17} + 29509184954810 T^{18} + 114884176217940 T^{19} - 447220293187134 T^{20} + 4497215252519390 T^{21} - 2524243399801206 T^{22} + 353814783205469041 T^{24} \))(\( 1 - 24389 T^{6} + 594823321 T^{12} \))
$31$ (\( 1 + 6 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( ( 1 - 52 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( ( 1 - 15 T + 144 T^{2} - 899 T^{3} + 4464 T^{4} - 14415 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 98 T^{2} + 4275 T^{4} - 94178 T^{6} + 923521 T^{8} )^{2} \))(\( 1 + 24 T + 294 T^{2} + 2814 T^{3} + 23334 T^{4} + 160962 T^{5} + 949331 T^{6} + 4989822 T^{7} + 22423974 T^{8} + 83831874 T^{9} + 271515174 T^{10} + 687099624 T^{11} + 887503681 T^{12} \))(\( 1 - 150 T^{2} - 20 T^{3} + 12234 T^{4} + 2250 T^{5} - 725114 T^{6} - 99090 T^{7} + 34416180 T^{8} + 2444320 T^{9} - 1360293588 T^{10} - 28518990 T^{11} + 45632108695 T^{12} - 884088690 T^{13} - 1307242138068 T^{14} + 72818737120 T^{15} + 31784064969780 T^{16} - 2836862572590 T^{17} - 643541344144634 T^{18} + 61903381749750 T^{19} + 10434268952053194 T^{20} - 528792443213420 T^{21} - 122944243047120150 T^{22} + 787662783788549761 T^{24} \))(\( ( 1 - 19 T^{3} + 29791 T^{6} )( 1 + 19 T^{3} + 29791 T^{6} ) \))
$37$ (\( 1 + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( ( 1 + 16 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 3 T + 37 T^{2} )^{2} \))(\( ( 1 + T + 37 T^{2} )^{6} \))(\( ( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )^{2}( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 6 T + 102 T^{2} - 393 T^{3} + 3774 T^{4} - 8214 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 + 6 T + 138 T^{2} + 248 T^{3} + 5688 T^{4} - 14688 T^{5} + 146841 T^{6} - 543456 T^{7} + 7786872 T^{8} + 12561944 T^{9} + 258634218 T^{10} + 416063742 T^{11} + 2565726409 T^{12} )^{2} \))(\( ( 1 - 323 T^{3} + 50653 T^{6} )( 1 + 323 T^{3} + 50653 T^{6} ) \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( ( 1 - 16 T^{2} + 1681 T^{4} )^{2} \))(\( 1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 - 4 T - 75 T^{2} + 100 T^{3} + 3622 T^{4} + 2012 T^{5} - 174751 T^{6} + 82492 T^{7} + 6088582 T^{8} + 6892100 T^{9} - 211932075 T^{10} - 463424804 T^{11} + 4750104241 T^{12} \))(\( ( 1 - 50 T^{2} + 819 T^{4} - 84050 T^{6} + 2825761 T^{8} )^{2} \))(\( 1 - 6 T + 81 T^{2} - 351 T^{3} + 4257 T^{4} - 12003 T^{5} + 138106 T^{6} - 492123 T^{7} + 7156017 T^{8} - 24191271 T^{9} + 228886641 T^{10} - 695137206 T^{11} + 4750104241 T^{12} \))(\( 1 + 18 T + 183 T^{2} + 1103 T^{3} + 5469 T^{4} + 34479 T^{5} + 368867 T^{6} + 3114657 T^{7} + 17237808 T^{8} + 76410530 T^{9} + 598462632 T^{10} + 6589044933 T^{11} + 51330742816 T^{12} + 270150842253 T^{13} + 1006015684392 T^{14} + 5266290138130 T^{15} + 48709925571888 T^{16} + 360852327438057 T^{17} + 1752156701064947 T^{18} + 6714932609142999 T^{19} + 43669556078062749 T^{20} + 361102273636538983 T^{21} + 2456346653757889383 T^{22} + 9905922570892471938 T^{23} + 22563490300366186081 T^{24} \))(\( 1 - 68921 T^{6} + 4750104241 T^{12} \))
$43$ (\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{4} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T - 99 T^{2} + 218 T^{3} + 6207 T^{4} - 6951 T^{5} - 285738 T^{6} - 298893 T^{7} + 11476743 T^{8} + 17332526 T^{9} - 338461299 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \))(\( ( 1 + 8 T - 35 T^{2} + 104 T^{3} + 5200 T^{4} + 4472 T^{5} - 64715 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( 1 - 18 T + 144 T^{2} - 520 T^{3} + 2610 T^{4} - 49554 T^{5} + 474393 T^{6} - 2130822 T^{7} + 4825890 T^{8} - 41343640 T^{9} + 492307344 T^{10} - 2646151974 T^{11} + 6321363049 T^{12} \))(\( 1 - 15 T + 90 T^{2} - 302 T^{3} + 63 T^{4} - 4335 T^{5} + 132460 T^{6} - 1016496 T^{7} + 6203205 T^{8} - 45866600 T^{9} + 203493513 T^{10} - 202774140 T^{11} - 881690723 T^{12} - 8719288020 T^{13} + 376259505537 T^{14} - 3646715766200 T^{15} + 21207523457205 T^{16} - 149433494275728 T^{17} + 837327749470540 T^{18} - 1178333679148845 T^{19} + 736356617488863 T^{20} - 151782968804926586 T^{21} + 1945033408195582410 T^{22} - 13939406092068340605 T^{23} + 39959630797262576401 T^{24} \))(\( ( 1 + 13 T + 43 T^{2} )^{3}( 1 - 449 T^{3} + 79507 T^{6} ) \))
$47$ (\( 1 + 9 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( ( 1 - 89 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( ( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} )^{3} \))(\( 1 + 76 T^{2} + 2986 T^{4} - 123728 T^{6} - 9854765 T^{8} - 273315152 T^{10} + 14570727466 T^{12} + 819220365004 T^{14} + 23811286661761 T^{16} \))(\( 1 + 33 T + 477 T^{2} + 3357 T^{3} + 648 T^{4} - 230538 T^{5} - 2370995 T^{6} - 10835286 T^{7} + 1431432 T^{8} + 348533811 T^{9} + 2327607837 T^{10} + 7568385231 T^{11} + 10779215329 T^{12} \))(\( 1 + 9 T + 63 T^{2} - 45 T^{3} - 1710 T^{4} - 32958 T^{5} - 200842 T^{6} - 987525 T^{7} + 5445819 T^{8} + 35605818 T^{9} + 236914821 T^{10} - 330786045 T^{11} + 2833823343 T^{12} - 15546944115 T^{13} + 523344839589 T^{14} + 3696702842214 T^{15} + 26573859503739 T^{16} - 226483928037675 T^{17} - 2164919165107018 T^{18} - 16697284804219554 T^{19} - 40717300191611310 T^{20} - 50360871289624515 T^{21} + 3313745330857293087 T^{22} + 22249432935756110727 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 103823 T^{6} + 10779215329 T^{12} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( ( 1 + 88 T^{2} + 2809 T^{4} )^{2} \))(\( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 51 T^{2} + 102 T^{3} + 1086 T^{4} + 11334 T^{5} - 107183 T^{6} + 600702 T^{7} + 3050574 T^{8} + 15185454 T^{9} - 402414531 T^{10} - 2509172958 T^{11} + 22164361129 T^{12} \))(\( 1 - 56 T^{2} - 1538 T^{4} + 52864 T^{6} + 5146771 T^{8} + 148494976 T^{10} - 12135559778 T^{12} - 1241204223224 T^{14} + 62259690411361 T^{16} \))(\( 1 + 24 T + 375 T^{2} + 4609 T^{3} + 47013 T^{4} + 413955 T^{5} + 3190646 T^{6} + 21939615 T^{7} + 132059517 T^{8} + 686174093 T^{9} + 2958930375 T^{10} + 10036691832 T^{11} + 22164361129 T^{12} \))(\( 1 + 30 T + 531 T^{2} + 6849 T^{3} + 74493 T^{4} + 681915 T^{5} + 5137319 T^{6} + 28302939 T^{7} + 69363450 T^{8} - 837630774 T^{9} - 16059389652 T^{10} - 173893537179 T^{11} - 1420878272748 T^{12} - 9216357470487 T^{13} - 45110825532468 T^{14} - 124703956740798 T^{15} + 547310984319450 T^{16} + 11836161528453927 T^{17} + 113865393550873151 T^{18} + 801053146921947855 T^{19} + 4637911117813514973 T^{20} + 22600080840252808917 T^{21} + 92865246764087429019 T^{22} + \)\(27\!\cdots\!10\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 148877 T^{6} + 22164361129 T^{12} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( ( 1 + 116 T^{2} + 3481 T^{4} )^{2} \))(\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 153 T^{2} + 72 T^{3} + 14382 T^{4} - 5508 T^{5} - 969077 T^{6} - 324972 T^{7} + 50063742 T^{8} + 14787288 T^{9} - 1853956233 T^{10} + 42180533641 T^{12} \))(\( 1 - 40 T^{2} - 4994 T^{4} + 14720 T^{6} + 25793875 T^{8} + 51240320 T^{10} - 60514100834 T^{12} - 1687221345640 T^{14} + 146830437604321 T^{16} \))(\( 1 + 54 T^{2} + 661 T^{3} + 5391 T^{4} + 27351 T^{5} + 485585 T^{6} + 1613709 T^{7} + 18766071 T^{8} + 135755519 T^{9} + 654337494 T^{10} + 42180533641 T^{12} \))(\( 1 + 30 T + 498 T^{2} + 6615 T^{3} + 80079 T^{4} + 880563 T^{5} + 8928170 T^{6} + 86071779 T^{7} + 797956155 T^{8} + 7088465358 T^{9} + 60465430845 T^{10} + 493733665323 T^{11} + 3867087651291 T^{12} + 29130286254057 T^{13} + 210480164771445 T^{14} + 1455821926760682 T^{15} + 9669122792306955 T^{16} + 61534806265257921 T^{17} + 376594975037566970 T^{18} + 2191414417426673097 T^{19} + 11758034612916421359 T^{20} + 57305717340402421485 T^{21} + \)\(25\!\cdots\!98\)\( T^{22} + \)\(90\!\cdots\!70\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 205379 T^{6} + 42180533641 T^{12} \))
$61$ (\( 1 + T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( ( 1 + T + 61 T^{2} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 + 13 T - 25 T^{2} - 504 T^{3} + 6133 T^{4} + 34211 T^{5} - 181514 T^{6} + 2086871 T^{7} + 22820893 T^{8} - 114398424 T^{9} - 346146025 T^{10} + 10979751913 T^{11} + 51520374361 T^{12} \))(\( ( 1 - 14 T + 37 T^{2} - 518 T^{3} + 9604 T^{4} - 31598 T^{5} + 137677 T^{6} - 3177734 T^{7} + 13845841 T^{8} )^{2} \))(\( 1 + 21 T + 204 T^{2} + 872 T^{3} - 5166 T^{4} - 119673 T^{5} - 1122129 T^{6} - 7300053 T^{7} - 19222686 T^{8} + 197927432 T^{9} + 2824551564 T^{10} + 17736522321 T^{11} + 51520374361 T^{12} \))(\( 1 - 21 T + 285 T^{2} - 2944 T^{3} + 33252 T^{4} - 379767 T^{5} + 3866410 T^{6} - 34399233 T^{7} + 307989252 T^{8} - 2862264445 T^{9} + 25217243031 T^{10} - 198720049578 T^{11} + 1528474818418 T^{12} - 12121923024258 T^{13} + 93833361318351 T^{14} - 649679645990545 T^{15} + 4264370212900932 T^{16} - 29053464949037133 T^{17} + 199198890633114010 T^{18} - 1193510018607187107 T^{19} + 6374651571785587812 T^{20} - 34427566097303711104 T^{21} + \)\(20\!\cdots\!85\)\( T^{22} - \)\(91\!\cdots\!81\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( ( 1 + 14 T + 61 T^{2} )^{3}( 1 - 901 T^{3} + 226981 T^{6} ) \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( ( 1 - 94 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 + 9 T - 39 T^{2} - 250 T^{3} + 375 T^{4} - 24519 T^{5} - 284658 T^{6} - 1642773 T^{7} + 1683375 T^{8} - 75190750 T^{9} - 785893719 T^{10} + 12151125963 T^{11} + 90458382169 T^{12} \))(\( ( 1 + 133 T^{2} + 13200 T^{4} + 597037 T^{6} + 20151121 T^{8} )^{2} \))(\( 1 - 30 T + 372 T^{2} - 1966 T^{3} - 7344 T^{4} + 261360 T^{5} - 2850267 T^{6} + 17511120 T^{7} - 32967216 T^{8} - 591300058 T^{9} + 7496217012 T^{10} - 40503753210 T^{11} + 90458382169 T^{12} \))(\( 1 - 3 T - 66 T^{2} + 316 T^{3} - 993 T^{4} - 53211 T^{5} + 342682 T^{6} + 1493820 T^{7} + 3611241 T^{8} + 350243794 T^{9} - 1960151007 T^{10} - 15132364536 T^{11} + 100187638153 T^{12} - 1013868423912 T^{13} - 8799117870423 T^{14} + 105340374214822 T^{15} + 72770554351161 T^{16} + 2016843887338740 T^{17} + 30998459318437258 T^{18} - 322496525230842153 T^{19} - 403225203813744513 T^{20} + 8597264869229203252 T^{21} - \)\(12\!\cdots\!34\)\( T^{22} - \)\(36\!\cdots\!49\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( ( 1 - 11 T + 67 T^{2} )^{3}( 1 - 127 T^{3} + 300763 T^{6} ) \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( ( 1 + 124 T^{2} + 5041 T^{4} )^{2} \))(\( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 - 18 T + 99 T^{2} - 234 T^{3} + 306 T^{4} + 55062 T^{5} - 871373 T^{6} + 3909402 T^{7} + 1542546 T^{8} - 83751174 T^{9} + 2515756419 T^{10} - 32476128318 T^{11} + 128100283921 T^{12} \))(\( 1 - 92 T^{2} + 3178 T^{4} + 441232 T^{6} - 40136909 T^{8} + 2224250512 T^{10} + 80758322218 T^{12} - 11785226120732 T^{14} + 645753531245761 T^{16} \))(\( 1 - 18 T + 144 T^{2} - 180 T^{3} - 4734 T^{4} + 88470 T^{5} - 842507 T^{6} + 6281370 T^{7} - 23864094 T^{8} - 64423980 T^{9} + 3659282064 T^{10} - 32476128318 T^{11} + 128100283921 T^{12} \))(\( 1 + 30 T + 390 T^{2} + 2934 T^{3} + 6846 T^{4} - 198942 T^{5} - 3451366 T^{6} - 31177800 T^{7} - 191632284 T^{8} - 435806892 T^{9} + 9103241592 T^{10} + 158120182512 T^{11} + 1561936451607 T^{12} + 11226532958352 T^{13} + 45889440865272 T^{14} - 155980080522612 T^{15} - 4869698470309404 T^{16} - 56251901859607800 T^{17} - 442120964515286086 T^{18} - 1809401394550622322 T^{19} + 4420828674908479806 T^{20} + \)\(13\!\cdots\!54\)\( T^{21} + \)\(12\!\cdots\!90\)\( T^{22} + \)\(69\!\cdots\!30\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 357911 T^{6} + 128100283921 T^{12} \))
$73$ (\( 1 + 11 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( ( 1 + 3 T + 73 T^{2} )^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 + 19 T + 59 T^{2} + 252 T^{3} + 17041 T^{4} + 91889 T^{5} - 279578 T^{6} + 6707897 T^{7} + 90811489 T^{8} + 98032284 T^{9} + 1675496219 T^{10} + 39388360267 T^{11} + 151334226289 T^{12} \))(\( ( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} )^{4} \))(\( 1 + 27 T + 324 T^{2} + 2356 T^{3} + 4887 T^{4} - 141291 T^{5} - 1960287 T^{6} - 10314243 T^{7} + 26042823 T^{8} + 916524052 T^{9} + 9201030084 T^{10} + 55972933011 T^{11} + 151334226289 T^{12} \))(\( 1 + 24 T + 261 T^{2} + 1346 T^{3} - 4860 T^{4} - 206634 T^{5} - 2593622 T^{6} - 21717126 T^{7} - 147632589 T^{8} - 514741726 T^{9} + 6667065831 T^{10} + 149373266544 T^{11} + 1579140222877 T^{12} + 10904248457712 T^{13} + 35528793813399 T^{14} - 200243282023342 T^{15} - 4192505841875949 T^{16} - 45021156992201718 T^{17} - 392503778656128758 T^{18} - 2282768145595089498 T^{19} - 3919396046605233660 T^{20} + 79241155709328610898 T^{21} + \)\(11\!\cdots\!89\)\( T^{22} + \)\(75\!\cdots\!48\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 17 T + 73 T^{2} )^{3}( 1 + 919 T^{3} + 389017 T^{6} ) \))
$79$ (\( 1 - 16 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( ( 1 + 2 T^{2} + 6241 T^{4} )^{2} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + 11 T - 119 T^{2} - 494 T^{3} + 21403 T^{4} + 27611 T^{5} - 1863994 T^{6} + 2181269 T^{7} + 133576123 T^{8} - 243561266 T^{9} - 4635059639 T^{10} + 33847620389 T^{11} + 243087455521 T^{12} \))(\( ( 1 - 12 T + 169 T^{2} - 1452 T^{3} + 12192 T^{4} - 114708 T^{5} + 1054729 T^{6} - 5916468 T^{7} + 38950081 T^{8} )^{2} \))(\( 1 - 12 T + 168 T^{2} - 1772 T^{3} + 24948 T^{4} - 218700 T^{5} + 2149713 T^{6} - 17277300 T^{7} + 155700468 T^{8} - 873665108 T^{9} + 6543613608 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} \))(\( 1 - 24 T + 513 T^{2} - 7035 T^{3} + 82791 T^{4} - 758175 T^{5} + 5630515 T^{6} - 31509729 T^{7} + 38825604 T^{8} + 1732476276 T^{9} - 35038336140 T^{10} + 425272082301 T^{11} - 4249241399400 T^{12} + 33596494501779 T^{13} - 218674255849740 T^{14} + 854178370642764 T^{15} + 1512260420673924 T^{16} - 96957213250205871 T^{17} + 1368707564622823315 T^{18} - 14559923695581099825 T^{19} + \)\(12\!\cdots\!51\)\( T^{20} - \)\(84\!\cdots\!65\)\( T^{21} + \)\(48\!\cdots\!13\)\( T^{22} - \)\(17\!\cdots\!96\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 4 T + 79 T^{2} )^{3}( 1 - 503 T^{3} + 493039 T^{6} ) \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( ( 1 - 86 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 4 T + 169 T^{2} + 472 T^{3} + 14027 T^{4} + 27556 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 - 268 T^{2} + 30966 T^{4} - 1846252 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 15 T - 60 T^{2} + 603 T^{3} + 20985 T^{4} - 88800 T^{5} - 1199405 T^{6} - 7370400 T^{7} + 144565665 T^{8} + 344787561 T^{9} - 2847499260 T^{10} - 59085609645 T^{11} + 326940373369 T^{12} \))(\( 1 - 3 T - 213 T^{2} + 1826 T^{3} + 11337 T^{4} - 228765 T^{5} + 462329 T^{6} + 3734865 T^{7} + 42465492 T^{8} + 1125412628 T^{9} - 18867606045 T^{10} - 55608801501 T^{11} + 2104181914351 T^{12} - 4615530524583 T^{13} - 129978938044005 T^{14} + 643496310326236 T^{15} + 2015340950758932 T^{16} + 14711785031118195 T^{17} + 151154015879316401 T^{18} - 6207778704642020655 T^{19} + 25534237035760307817 T^{20} + \)\(34\!\cdots\!78\)\( T^{21} - \)\(33\!\cdots\!37\)\( T^{22} - \)\(38\!\cdots\!01\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( ( 1 + 83 T^{2} + 6889 T^{4} )^{3} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( ( 1 + 16 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 16 T - 87 T^{2} + 424 T^{3} + 39826 T^{4} - 163780 T^{5} - 2350663 T^{6} - 14576420 T^{7} + 315461746 T^{8} + 298906856 T^{9} - 5458574967 T^{10} - 89344951184 T^{11} + 496981290961 T^{12} \))(\( ( 1 - 124 T^{2} + 7455 T^{4} - 982204 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 - 45 T + 900 T^{2} - 9000 T^{3} + 10575 T^{4} + 1004445 T^{5} - 14638319 T^{6} + 89395605 T^{7} + 83764575 T^{8} - 6344721000 T^{9} + 56468016900 T^{10} - 251282675205 T^{11} + 496981290961 T^{12} \))(\( 1 - 3 T - 18 T^{2} + 1100 T^{3} - 8883 T^{4} + 3561 T^{5} + 781382 T^{6} - 7400448 T^{7} + 28085625 T^{8} + 353976110 T^{9} - 6587754081 T^{10} - 19263332574 T^{11} + 108084451561 T^{12} - 1714436599086 T^{13} - 52181600075601 T^{14} + 249542184290590 T^{15} + 1762155052385625 T^{16} - 41324541581233152 T^{17} + 388332235093688102 T^{18} + 157507783562978769 T^{19} - 34968718361051585523 T^{20} + \)\(38\!\cdots\!00\)\( T^{21} - \)\(56\!\cdots\!18\)\( T^{22} - \)\(83\!\cdots\!67\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 - 704969 T^{6} + 496981290961 T^{12} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( ( 1 - 184 T^{2} + 9409 T^{4} )^{2} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 2166 T^{3} + 418 T^{4} + 14486 T^{5} + 2859997 T^{6} + 1405142 T^{7} + 3932962 T^{8} - 1976849718 T^{9} - 1682056339 T^{10} - 17174680514 T^{11} + 832972004929 T^{12} \))(\( ( 1 - 12 T + 238 T^{2} - 2280 T^{3} + 31347 T^{4} - 221160 T^{5} + 2239342 T^{6} - 10952076 T^{7} + 88529281 T^{8} )^{2} \))(\( 1 - 21 T + 210 T^{2} - 814 T^{3} - 6003 T^{4} + 250281 T^{5} - 3257067 T^{6} + 24277257 T^{7} - 56482227 T^{8} - 742915822 T^{9} + 18591149010 T^{10} - 180334145397 T^{11} + 832972004929 T^{12} \))(\( 1 + 27 T + 606 T^{2} + 10820 T^{3} + 147963 T^{4} + 1809261 T^{5} + 17276956 T^{6} + 116607672 T^{7} + 327799089 T^{8} - 9137722564 T^{9} - 199457082207 T^{10} - 2736178123296 T^{11} - 30811907053751 T^{12} - 265409277959712 T^{13} - 1876691686485663 T^{14} - 8339752665653572 T^{15} + 29019817661625009 T^{16} + 1001349756040651704 T^{17} + 14391220678390116124 T^{18} + \)\(14\!\cdots\!93\)\( T^{19} + \)\(11\!\cdots\!43\)\( T^{20} + \)\(82\!\cdots\!40\)\( T^{21} + \)\(44\!\cdots\!94\)\( T^{22} + \)\(19\!\cdots\!31\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( ( 1 + 523 T^{3} + 912673 T^{6} )( 1 + 1853 T^{3} + 912673 T^{6} ) \))
show more
show less