Properties

Label 1007.2.a
Level 1007
Weight 2
Character orbit a
Rep. character \(\chi_{1007}(1,\cdot)\)
Character field \(\Q\)
Dimension 79
Newform subspaces 5
Sturm bound 180
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 1007 = 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1007.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(180\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1007))\).

Total New Old
Modular forms 92 79 13
Cusp forms 89 79 10
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(19\)\(53\)FrickeDim.
\(+\)\(+\)\(+\)\(13\)
\(+\)\(-\)\(-\)\(28\)
\(-\)\(+\)\(-\)\(26\)
\(-\)\(-\)\(+\)\(12\)
Plus space\(+\)\(25\)
Minus space\(-\)\(54\)

Trace form

\( 79q + 3q^{2} + 4q^{3} + 85q^{4} - 8q^{6} + 6q^{7} + 3q^{8} + 75q^{9} + O(q^{10}) \) \( 79q + 3q^{2} + 4q^{3} + 85q^{4} - 8q^{6} + 6q^{7} + 3q^{8} + 75q^{9} + 2q^{10} + 2q^{11} + 14q^{13} + 16q^{15} + 77q^{16} + 16q^{17} + 27q^{18} - 3q^{19} + 6q^{20} - 20q^{21} - 4q^{22} - 8q^{23} - 12q^{24} + 107q^{25} + 2q^{26} - 20q^{27} - 8q^{28} - 2q^{29} - 16q^{30} + 16q^{31} + 39q^{32} + 36q^{33} - 34q^{34} - 14q^{35} + 101q^{36} + 26q^{37} + 3q^{38} - 40q^{39} - 10q^{40} + 10q^{41} + 28q^{42} + 2q^{43} + 28q^{44} + 32q^{45} + 12q^{46} + 34q^{47} - 28q^{48} + 101q^{49} + 41q^{50} - 4q^{51} + 2q^{52} + q^{53} - 56q^{54} - 22q^{55} - 8q^{56} + 4q^{57} + 18q^{58} - 12q^{59} - 52q^{60} + 16q^{61} - 44q^{62} + 54q^{63} + 69q^{64} + 4q^{65} - 56q^{66} + 52q^{67} + 2q^{68} + 28q^{69} + 28q^{70} - 12q^{71} + 23q^{72} + 20q^{73} + 2q^{74} - 16q^{75} - 3q^{76} - 2q^{77} - 12q^{78} + 8q^{79} + 46q^{80} + 79q^{81} + 26q^{82} - 44q^{83} - 24q^{84} + 6q^{85} + 16q^{86} - 16q^{87} + 36q^{88} - 6q^{89} - 82q^{90} - 40q^{91} - 12q^{92} + 16q^{93} - 44q^{94} + 4q^{95} - 116q^{96} + 42q^{97} + 99q^{98} - 6q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1007))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 19 53
1007.2.a.a \(1\) \(8.041\) \(\Q\) None \(2\) \(0\) \(-3\) \(-1\) \(+\) \(+\) \(q+2q^{2}+2q^{4}-3q^{5}-q^{7}-3q^{9}+\cdots\)
1007.2.a.b \(12\) \(8.041\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-5\) \(1\) \(-4\) \(-8\) \(+\) \(+\) \(q-\beta _{1}q^{2}-\beta _{10}q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots\)
1007.2.a.c \(12\) \(8.041\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-3\) \(-5\) \(-2\) \(-8\) \(-\) \(-\) \(q-\beta _{1}q^{2}-\beta _{9}q^{3}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots\)
1007.2.a.d \(26\) \(8.041\) None \(6\) \(9\) \(4\) \(8\) \(-\) \(+\)
1007.2.a.e \(28\) \(8.041\) None \(3\) \(-1\) \(5\) \(15\) \(+\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1007))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1007)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(53))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 2 T^{2} \))(\( 1 + 5 T + 22 T^{2} + 68 T^{3} + 193 T^{4} + 460 T^{5} + 1028 T^{6} + 2046 T^{7} + 3865 T^{8} + 6674 T^{9} + 11023 T^{10} + 16849 T^{11} + 24723 T^{12} + 33698 T^{13} + 44092 T^{14} + 53392 T^{15} + 61840 T^{16} + 65472 T^{17} + 65792 T^{18} + 58880 T^{19} + 49408 T^{20} + 34816 T^{21} + 22528 T^{22} + 10240 T^{23} + 4096 T^{24} \))(\( 1 + 3 T + 12 T^{2} + 28 T^{3} + 71 T^{4} + 140 T^{5} + 284 T^{6} + 498 T^{7} + 885 T^{8} + 1416 T^{9} + 2287 T^{10} + 3363 T^{11} + 4987 T^{12} + 6726 T^{13} + 9148 T^{14} + 11328 T^{15} + 14160 T^{16} + 15936 T^{17} + 18176 T^{18} + 17920 T^{19} + 18176 T^{20} + 14336 T^{21} + 12288 T^{22} + 6144 T^{23} + 4096 T^{24} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 - T + 17 T^{2} - 15 T^{3} + 155 T^{4} - 117 T^{5} + 981 T^{6} - 638 T^{7} + 4793 T^{8} - 2745 T^{9} + 19011 T^{10} - 9790 T^{11} + 62483 T^{12} - 29370 T^{13} + 171099 T^{14} - 74115 T^{15} + 388233 T^{16} - 155034 T^{17} + 715149 T^{18} - 255879 T^{19} + 1016955 T^{20} - 295245 T^{21} + 1003833 T^{22} - 177147 T^{23} + 531441 T^{24} \))(\( 1 + 5 T + 31 T^{2} + 115 T^{3} + 437 T^{4} + 1301 T^{5} + 3799 T^{6} + 9498 T^{7} + 23027 T^{8} + 49591 T^{9} + 103341 T^{10} + 194314 T^{11} + 353519 T^{12} + 582942 T^{13} + 930069 T^{14} + 1338957 T^{15} + 1865187 T^{16} + 2308014 T^{17} + 2769471 T^{18} + 2845287 T^{19} + 2867157 T^{20} + 2263545 T^{21} + 1830519 T^{22} + 885735 T^{23} + 531441 T^{24} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 4 T + 48 T^{2} + 156 T^{3} + 1072 T^{4} + 2967 T^{5} + 15069 T^{6} + 36367 T^{7} + 149756 T^{8} + 318430 T^{9} + 1111456 T^{10} + 2086788 T^{11} + 6322765 T^{12} + 10433940 T^{13} + 27786400 T^{14} + 39803750 T^{15} + 93597500 T^{16} + 113646875 T^{17} + 235453125 T^{18} + 231796875 T^{19} + 418750000 T^{20} + 304687500 T^{21} + 468750000 T^{22} + 195312500 T^{23} + 244140625 T^{24} \))(\( 1 + 2 T + 42 T^{2} + 88 T^{3} + 876 T^{4} + 1821 T^{5} + 11907 T^{6} + 23585 T^{7} + 116548 T^{8} + 213398 T^{9} + 860010 T^{10} + 1419994 T^{11} + 4885699 T^{12} + 7099970 T^{13} + 21500250 T^{14} + 26674750 T^{15} + 72842500 T^{16} + 73703125 T^{17} + 186046875 T^{18} + 142265625 T^{19} + 342187500 T^{20} + 171875000 T^{21} + 410156250 T^{22} + 97656250 T^{23} + 244140625 T^{24} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 + 8 T + 83 T^{2} + 485 T^{3} + 3008 T^{4} + 14019 T^{5} + 65548 T^{6} + 254844 T^{7} + 974379 T^{8} + 3238813 T^{9} + 10508086 T^{10} + 30235415 T^{11} + 84700363 T^{12} + 211647905 T^{13} + 514896214 T^{14} + 1110912859 T^{15} + 2339483979 T^{16} + 4283163108 T^{17} + 7711656652 T^{18} + 11545249317 T^{19} + 17340521408 T^{20} + 19571499395 T^{21} + 23445445667 T^{22} + 15818613944 T^{23} + 13841287201 T^{24} \))(\( 1 + 8 T + 75 T^{2} + 417 T^{3} + 2280 T^{4} + 9663 T^{5} + 38676 T^{6} + 131900 T^{7} + 422819 T^{8} + 1223333 T^{9} + 3398966 T^{10} + 9043087 T^{11} + 24018207 T^{12} + 63301609 T^{13} + 166549334 T^{14} + 419603219 T^{15} + 1015188419 T^{16} + 2216843300 T^{17} + 4550192724 T^{18} + 7957896009 T^{19} + 13143746280 T^{20} + 16827454119 T^{21} + 21185643675 T^{22} + 15818613944 T^{23} + 13841287201 T^{24} \))
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 + 4 T + 99 T^{2} + 343 T^{3} + 4676 T^{4} + 14256 T^{5} + 140571 T^{6} + 380801 T^{7} + 3012289 T^{8} + 7278074 T^{9} + 48602536 T^{10} + 104501915 T^{11} + 605903385 T^{12} + 1149521065 T^{13} + 5880906856 T^{14} + 9687116494 T^{15} + 44102923249 T^{16} + 61328381851 T^{17} + 249030101331 T^{18} + 277809109776 T^{19} + 1002342127556 T^{20} + 808776058013 T^{21} + 2567805035499 T^{22} + 1141246682444 T^{23} + 3138428376721 T^{24} \))(\( 1 + 4 T + 71 T^{2} + 299 T^{3} + 2704 T^{4} + 11012 T^{5} + 69919 T^{6} + 269601 T^{7} + 1351553 T^{8} + 4858646 T^{9} + 20532272 T^{10} + 67745431 T^{11} + 250826497 T^{12} + 745199741 T^{13} + 2484404912 T^{14} + 6466857826 T^{15} + 19788087473 T^{16} + 43419510651 T^{17} + 123865773559 T^{18} + 214592727052 T^{19} + 579626414224 T^{20} + 705026359609 T^{21} + 1841557146671 T^{22} + 1141246682444 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 + 13 T^{2} \))(\( 1 + 5 T + 82 T^{2} + 329 T^{3} + 3251 T^{4} + 11047 T^{5} + 84955 T^{6} + 249377 T^{7} + 1656422 T^{8} + 4271104 T^{9} + 26257971 T^{10} + 61572663 T^{11} + 361649455 T^{12} + 800444619 T^{13} + 4437597099 T^{14} + 9383615488 T^{15} + 47309068742 T^{16} + 92591934461 T^{17} + 410061558595 T^{18} + 693182867299 T^{19} + 2651940573971 T^{20} + 3488880293717 T^{21} + 11304396331618 T^{22} + 8960801970185 T^{23} + 23298085122481 T^{24} \))(\( 1 + 17 T + 252 T^{2} + 2519 T^{3} + 22495 T^{4} + 164251 T^{5} + 1094437 T^{6} + 6349449 T^{7} + 34051970 T^{8} + 163411270 T^{9} + 730161455 T^{10} + 2955615789 T^{11} + 11176079817 T^{12} + 38423005257 T^{13} + 123397285895 T^{14} + 359014560190 T^{15} + 972558315170 T^{16} + 2357505967557 T^{17} + 5282638361533 T^{18} + 10306506665767 T^{19} + 18349862568895 T^{20} + 26712733920587 T^{21} + 34740339945948 T^{22} + 30466726698629 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 + 4 T + 113 T^{2} + 449 T^{3} + 6563 T^{4} + 25592 T^{5} + 257505 T^{6} + 971187 T^{7} + 7592909 T^{8} + 27202863 T^{9} + 176880459 T^{10} + 588427067 T^{11} + 3332431899 T^{12} + 10003260139 T^{13} + 51118452651 T^{14} + 133647665919 T^{15} + 634167352589 T^{16} + 1378946660259 T^{17} + 6215544705345 T^{18} + 10501387319416 T^{19} + 45781896085283 T^{20} + 53245956547153 T^{21} + 227807310750737 T^{22} + 137087585230532 T^{23} + 582622237229761 T^{24} \))(\( 1 + 8 T + 149 T^{2} + 1001 T^{3} + 10435 T^{4} + 61740 T^{5} + 464561 T^{6} + 2471203 T^{7} + 14810993 T^{8} + 71356871 T^{9} + 359567307 T^{10} + 1563935703 T^{11} + 6860695183 T^{12} + 26586906951 T^{13} + 103914951723 T^{14} + 350576307223 T^{15} + 1237028946353 T^{16} + 3508754877971 T^{17} + 11213373192209 T^{18} + 25334309671020 T^{19} + 72792028896835 T^{20} + 118706464373497 T^{21} + 300383091166901 T^{22} + 274175170461064 T^{23} + 582622237229761 T^{24} \))
$19$ (\( 1 + T \))(\( ( 1 + T )^{12} \))(\( ( 1 - T )^{12} \))
$23$ (\( 1 + 8 T + 23 T^{2} \))(\( 1 + 8 T + 220 T^{2} + 1552 T^{3} + 23187 T^{4} + 144423 T^{5} + 1544172 T^{6} + 8512886 T^{7} + 72293219 T^{8} + 352827516 T^{9} + 2505520490 T^{10} + 10786482149 T^{11} + 65902170667 T^{12} + 248089089427 T^{13} + 1325420339210 T^{14} + 4292852387172 T^{15} + 20230606698179 T^{16} + 54791854215898 T^{17} + 228592874788908 T^{18} + 491735105532081 T^{19} + 1815796815710547 T^{20} + 2795388930590576 T^{21} + 9113832467002780 T^{22} + 7622478063311416 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 2 T + 170 T^{2} + 482 T^{3} + 14083 T^{4} + 50017 T^{5} + 769646 T^{6} + 3074782 T^{7} + 31394909 T^{8} + 128247472 T^{9} + 1007768190 T^{10} + 3903854119 T^{11} + 25880454373 T^{12} + 89788644737 T^{13} + 533109372510 T^{14} + 1560386991824 T^{15} + 8785582729469 T^{16} + 19790351602226 T^{17} + 113935229825294 T^{18} + 170299154382599 T^{19} + 1102853605712323 T^{20} + 868155582825166 T^{21} + 7042506906320330 T^{22} + 1905619515827854 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 238 T^{2} - 559 T^{3} + 27394 T^{4} - 70153 T^{5} + 2039874 T^{6} - 5343814 T^{7} + 110261711 T^{8} - 279910607 T^{9} + 4570388014 T^{10} - 10751219026 T^{11} + 148946410597 T^{12} - 311785351754 T^{13} + 3843696319774 T^{14} - 6826739794123 T^{15} + 77986013217791 T^{16} - 109607765182286 T^{17} + 1213364627101554 T^{18} - 1210130572705277 T^{19} + 13703750236653634 T^{20} - 8109494600510771 T^{21} + 100128321525447838 T^{22} - 24401019531411658 T^{23} + 353814783205469041 T^{24} \))(\( 1 + 10 T + 216 T^{2} + 1943 T^{3} + 22974 T^{4} + 184231 T^{5} + 1611164 T^{6} + 11479470 T^{7} + 83715705 T^{8} + 530178491 T^{9} + 3398999888 T^{10} + 19192747898 T^{11} + 110095337119 T^{12} + 556589689042 T^{13} + 2858558905808 T^{14} + 12930523216999 T^{15} + 59210527548105 T^{16} + 235457119611030 T^{17} + 958357921155644 T^{18} + 3177961962283379 T^{19} + 11492661091366014 T^{20} + 28187384631113467 T^{21} + 90872762392843416 T^{22} + 122005097657058290 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( 1 + 2 T + 31 T^{2} \))(\( 1 + 4 T + 225 T^{2} + 896 T^{3} + 25013 T^{4} + 101592 T^{5} + 1830605 T^{6} + 7572545 T^{7} + 98915308 T^{8} + 407487773 T^{9} + 4190655854 T^{10} + 16505785977 T^{11} + 143685314213 T^{12} + 511679365287 T^{13} + 4027220275694 T^{14} + 12139468245443 T^{15} + 91350364159468 T^{16} + 216795534259295 T^{17} + 1624668675957005 T^{18} + 2795061492764712 T^{19} + 21333363519511733 T^{20} + 23689901455961216 T^{21} + 184416364570680225 T^{22} + 101633907585619324 T^{23} + 787662783788549761 T^{24} \))(\( 1 + 8 T + 243 T^{2} + 1684 T^{3} + 29815 T^{4} + 182852 T^{5} + 2399889 T^{6} + 13158825 T^{7} + 140540492 T^{8} + 690769363 T^{9} + 6282780278 T^{10} + 27616729505 T^{11} + 219555485373 T^{12} + 856118614655 T^{13} + 6037751847158 T^{14} + 20578710093133 T^{15} + 129792095712332 T^{16} + 376725987907575 T^{17} + 2129910321491409 T^{18} + 5030736515424572 T^{19} + 25428946281303415 T^{20} + 44524323718569964 T^{21} + 199169673736334643 T^{22} + 203267815171238648 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( 1 + 4 T + 37 T^{2} \))(\( 1 + 35 T + 842 T^{2} + 14658 T^{3} + 211174 T^{4} + 2567200 T^{5} + 27472000 T^{6} + 261630877 T^{7} + 2261774181 T^{8} + 17868930846 T^{9} + 130414897872 T^{10} + 882243069875 T^{11} + 5561122710247 T^{12} + 32642993585375 T^{13} + 178537995186768 T^{14} + 905114954142438 T^{15} + 4238928960837141 T^{16} + 18142520284560289 T^{17} + 70485635908048000 T^{18} + 243709114975837600 T^{19} + 741744336202313254 T^{20} + 1904979181916238666 T^{21} + 4048828041575828858 T^{22} + 6227116762281114455 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 37 T + 932 T^{2} + 16976 T^{3} + 255648 T^{4} + 3241650 T^{5} + 36137390 T^{6} + 357097655 T^{7} + 3192728635 T^{8} + 25919901340 T^{9} + 193270036528 T^{10} + 1324035682345 T^{11} + 8387135562673 T^{12} + 48989320246765 T^{13} + 264586680006832 T^{14} + 1312920762575020 T^{15} + 5983687491300235 T^{16} + 24762564433120835 T^{17} + 92718655875332510 T^{18} + 307735919508189450 T^{19} + 897958347435995808 T^{20} + 2206230494761227152 T^{21} + 4481600635093435268 T^{22} + 6582952005840035281 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 24 T + 523 T^{2} + 7585 T^{3} + 100729 T^{4} + 1087127 T^{5} + 10963667 T^{6} + 96351274 T^{7} + 806762051 T^{8} + 6103998258 T^{9} + 44776746857 T^{10} + 303172223531 T^{11} + 2012971915559 T^{12} + 12430061164771 T^{13} + 75269711466617 T^{14} + 420693663939618 T^{15} + 2279716739995811 T^{16} + 11162892567150074 T^{17} + 52078561113611747 T^{18} + 211722629501429887 T^{19} + 804313533404129209 T^{20} + 2483191972378194185 T^{21} + 7020050819209705723 T^{22} + 13207896761189962584 T^{23} + 22563490300366186081 T^{24} \))(\( 1 - 14 T + 409 T^{2} - 4355 T^{3} + 71279 T^{4} - 608547 T^{5} + 7278771 T^{6} - 51675612 T^{7} + 506559517 T^{8} - 3099129674 T^{9} + 26833457879 T^{10} - 147537459463 T^{11} + 1183046456607 T^{12} - 6049035837983 T^{13} + 45107042694599 T^{14} - 213595116261754 T^{15} + 1431416127317437 T^{16} - 5986940090670012 T^{17} + 34574920996367811 T^{18} - 118517129107460907 T^{19} + 569157485406515759 T^{20} - 1425748324285700155 T^{21} + 5489867657852332009 T^{22} - 7704606444027478174 T^{23} + 22563490300366186081 T^{24} \))
$43$ (\( 1 + 5 T + 43 T^{2} \))(\( 1 + 22 T + 493 T^{2} + 7656 T^{3} + 108910 T^{4} + 1312729 T^{5} + 14555039 T^{6} + 144252409 T^{7} + 1329691717 T^{8} + 11192213951 T^{9} + 88162074982 T^{10} + 640663618470 T^{11} + 4366997579725 T^{12} + 27548535594210 T^{13} + 163011676641718 T^{14} + 889859354602157 T^{15} + 4545951371771317 T^{16} + 21206322046089187 T^{17} + 92007685711353911 T^{18} + 356824173539881003 T^{19} + 1272961892233524910 T^{20} + 3847849036988470008 T^{21} + 10654460780449134757 T^{22} + 20444462268366899554 T^{23} + 39959630797262576401 T^{24} \))(\( 1 + 26 T + 633 T^{2} + 9788 T^{3} + 143358 T^{4} + 1653749 T^{5} + 18413435 T^{6} + 174013773 T^{7} + 1604396697 T^{8} + 12970426859 T^{9} + 102825291150 T^{10} + 725406698238 T^{11} + 5031085898373 T^{12} + 31192488024234 T^{13} + 190123963336350 T^{14} + 1031239728278513 T^{15} + 5485113032100297 T^{16} + 25581493829285439 T^{17} + 116398007614163315 T^{18} + 449519756299590143 T^{19} + 1675597015396324158 T^{20} + 4919376485637819284 T^{21} + 13680068304308929617 T^{22} + 24161637226251790382 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 - 13 T + 47 T^{2} \))(\( 1 + 4 T + 258 T^{2} + 1371 T^{3} + 35117 T^{4} + 212065 T^{5} + 3467251 T^{6} + 20767278 T^{7} + 269081619 T^{8} + 1520631566 T^{9} + 16756569064 T^{10} + 89028945061 T^{11} + 860339432901 T^{12} + 4184360417867 T^{13} + 37015261062376 T^{14} + 157876531076818 T^{15} + 1313032463683539 T^{16} + 4762871518280946 T^{17} + 37374245128690579 T^{18} + 107437032040986095 T^{19} + 836180953701061037 T^{20} + 1534327878623893557 T^{21} + 13570576116844152642 T^{22} + 9888636860336049212 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 4 T + 466 T^{2} + 1731 T^{3} + 103201 T^{4} + 353397 T^{5} + 14350807 T^{6} + 44881758 T^{7} + 1394508299 T^{8} + 3936592494 T^{9} + 99701140008 T^{10} + 250245919041 T^{11} + 5375872573717 T^{12} + 11761558194927 T^{13} + 220239818277672 T^{14} + 408708842504562 T^{15} + 6804755650972619 T^{16} + 10293407102682306 T^{17} + 154690438797920503 T^{18} + 179039090902262811 T^{19} + 2457348594780396961 T^{20} + 1937214848940889677 T^{21} + 24511195621896802834 T^{22} + 9888636860336049212 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + T \))(\( ( 1 + T )^{12} \))(\( ( 1 - T )^{12} \))
$59$ (\( 1 + 14 T + 59 T^{2} \))(\( 1 - 24 T + 584 T^{2} - 8176 T^{3} + 114782 T^{4} - 1126917 T^{5} + 11810279 T^{6} - 92943347 T^{7} + 892896330 T^{8} - 6775674482 T^{9} + 66794902828 T^{10} - 498910243670 T^{11} + 4485542519955 T^{12} - 29435704376530 T^{13} + 232513056744268 T^{14} - 1391581249438678 T^{15} + 10819547166185130 T^{16} - 66447457200688753 T^{17} + 498163870669095839 T^{18} - 2804503665317773023 T^{19} + 16853491289099173022 T^{20} - 70828653813322781264 T^{21} + \)\(29\!\cdots\!84\)\( T^{22} - \)\(72\!\cdots\!16\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 + 16 T + 620 T^{2} + 8496 T^{3} + 181268 T^{4} + 2139587 T^{5} + 32929843 T^{6} + 336845025 T^{7} + 4133106730 T^{8} + 36790143556 T^{9} + 377321898888 T^{10} + 2924327119978 T^{11} + 25684663160505 T^{12} + 172535300078702 T^{13} + 1313457530029128 T^{14} + 7555922893387724 T^{15} + 50082346298939530 T^{16} + 240818693369762475 T^{17} + 1388998350454348363 T^{18} + 5324686364449429753 T^{19} + 26615659763660059028 T^{20} + 73600812475292361744 T^{21} + \)\(31\!\cdots\!20\)\( T^{22} + \)\(48\!\cdots\!44\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 - T + 61 T^{2} \))(\( 1 + 9 T + 250 T^{2} + 1900 T^{3} + 33440 T^{4} + 209041 T^{5} + 3019228 T^{6} + 17152940 T^{7} + 232859124 T^{8} + 1307486833 T^{9} + 16709105087 T^{10} + 94028669419 T^{11} + 1093064383311 T^{12} + 5735748834559 T^{13} + 62174580028727 T^{14} + 296774668841173 T^{15} + 3224130406303284 T^{16} + 14487309675274940 T^{17} + 155551756841213308 T^{18} + 656962105184665861 T^{19} + 6410692546629076640 T^{20} + 22218877576384867900 T^{21} + \)\(17\!\cdots\!50\)\( T^{22} + \)\(39\!\cdots\!49\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 5 T + 448 T^{2} + 2466 T^{3} + 100064 T^{4} + 592719 T^{5} + 14829586 T^{6} + 90727278 T^{7} + 1626212100 T^{8} + 9809837573 T^{9} + 138890470519 T^{10} + 788106680783 T^{11} + 9457221830557 T^{12} + 48074507527763 T^{13} + 516811440801199 T^{14} + 2226646742157113 T^{15} + 22516274168876100 T^{16} + 76627923398598678 T^{17} + 764025822338644546 T^{18} + 1862763391023531099 T^{19} + 19183000567759925984 T^{20} + 28837764264928991706 T^{21} + \)\(31\!\cdots\!48\)\( T^{22} + \)\(21\!\cdots\!05\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 + 28 T + 925 T^{2} + 18656 T^{3} + 366850 T^{4} + 5768611 T^{5} + 85084091 T^{6} + 1092232863 T^{7} + 13062854495 T^{8} + 140662455671 T^{9} + 1411330699614 T^{10} + 12935264930790 T^{11} + 110552236253451 T^{12} + 866662750362930 T^{13} + 6335463510567246 T^{14} + 42306062154976973 T^{15} + 263231161534138895 T^{16} + 1474651011026791341 T^{17} + 7696569220179973379 T^{18} + 34961887634293916353 T^{19} + \)\(14\!\cdots\!50\)\( T^{20} + \)\(50\!\cdots\!32\)\( T^{21} + \)\(16\!\cdots\!25\)\( T^{22} + \)\(34\!\cdots\!24\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 40 T + 1219 T^{2} + 26154 T^{3} + 479112 T^{4} + 7319663 T^{5} + 100250239 T^{6} + 1216108609 T^{7} + 13611327457 T^{8} + 139205973061 T^{9} + 1337762403020 T^{10} + 11945956120284 T^{11} + 101162177718663 T^{12} + 800379060059028 T^{13} + 6005215427156780 T^{14} + 41868006075745543 T^{15} + 274283506556629297 T^{16} + 1641898765849746163 T^{17} + 9068474431995588391 T^{18} + 44362366491153366149 T^{19} + \)\(19\!\cdots\!92\)\( T^{20} + \)\(71\!\cdots\!38\)\( T^{21} + \)\(22\!\cdots\!31\)\( T^{22} + \)\(48\!\cdots\!20\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 - 10 T + 71 T^{2} \))(\( 1 - 9 T + 556 T^{2} - 4742 T^{3} + 150220 T^{4} - 1209564 T^{5} + 26400366 T^{6} - 199980193 T^{7} + 3395034911 T^{8} - 23990778680 T^{9} + 338356985760 T^{10} - 2194011074505 T^{11} + 26831377904865 T^{12} - 155774786289855 T^{13} + 1705657565216160 T^{14} - 8586563588137480 T^{15} + 86273544142195391 T^{16} - 360810133829244743 T^{17} + 3381894380218315086 T^{18} - 11001129919264051524 T^{19} + 97005095463738217420 T^{20} - \)\(21\!\cdots\!02\)\( T^{21} + \)\(18\!\cdots\!56\)\( T^{22} - \)\(20\!\cdots\!39\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 + 19 T + 556 T^{2} + 7926 T^{3} + 144544 T^{4} + 1719454 T^{5} + 24526838 T^{6} + 254553539 T^{7} + 3077727445 T^{8} + 28502814780 T^{9} + 302726688286 T^{10} + 2521582539277 T^{11} + 23906540054861 T^{12} + 179032360288667 T^{13} + 1526045235649726 T^{14} + 10201470940724580 T^{15} + 78210228037285045 T^{16} + 459272966464723189 T^{17} + 3141894911484371798 T^{18} + 15638640736826038514 T^{19} + 93339798420387277984 T^{20} + \)\(36\!\cdots\!06\)\( T^{21} + \)\(18\!\cdots\!56\)\( T^{22} + \)\(43\!\cdots\!49\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 - 11 T + 73 T^{2} \))(\( 1 + 68 T + 2740 T^{2} + 79526 T^{3} + 1841163 T^{4} + 35603165 T^{5} + 594003520 T^{6} + 8716898326 T^{7} + 114190707239 T^{8} + 1348031729014 T^{9} + 14445984116657 T^{10} + 141133749361084 T^{11} + 1260705380238639 T^{12} + 10302763703359132 T^{13} + 76982649357665153 T^{14} + 524407259125839238 T^{15} + 3242815224133566599 T^{16} + 18070754298699853318 T^{17} + 89893063112142537280 T^{18} + \)\(39\!\cdots\!05\)\( T^{19} + \)\(14\!\cdots\!03\)\( T^{20} + \)\(46\!\cdots\!38\)\( T^{21} + \)\(11\!\cdots\!60\)\( T^{22} + \)\(21\!\cdots\!36\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 26 T + 916 T^{2} + 17408 T^{3} + 362549 T^{4} + 5493235 T^{5} + 85464292 T^{6} + 1078179718 T^{7} + 13607651009 T^{8} + 146288278184 T^{9} + 1555538285065 T^{10} + 14411290849072 T^{11} + 131476024413129 T^{12} + 1052024231982256 T^{13} + 8289463521111385 T^{14} + 56908627114305128 T^{15} + 386433352797475169 T^{16} + 2235143745534550774 T^{17} + 12933672505157172388 T^{18} + 60685956204051808795 T^{19} + \)\(29\!\cdots\!69\)\( T^{20} + \)\(10\!\cdots\!04\)\( T^{21} + \)\(39\!\cdots\!84\)\( T^{22} + \)\(81\!\cdots\!02\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 2 T + 79 T^{2} \))(\( 1 + 9 T + 445 T^{2} + 2972 T^{3} + 93209 T^{4} + 487713 T^{5} + 13045956 T^{6} + 57819757 T^{7} + 1427298239 T^{8} + 5621110389 T^{9} + 131141631025 T^{10} + 472980175707 T^{11} + 10760736520889 T^{12} + 37365433880853 T^{13} + 818454919227025 T^{14} + 2771426645082171 T^{15} + 55593382020207359 T^{16} + 177914653265475043 T^{17} + 3171308248878923076 T^{18} + 9365996063366564367 T^{19} + \)\(14\!\cdots\!49\)\( T^{20} + \)\(35\!\cdots\!68\)\( T^{21} + \)\(42\!\cdots\!45\)\( T^{22} + \)\(67\!\cdots\!11\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 3 T + 437 T^{2} + 766 T^{3} + 105467 T^{4} + 85919 T^{5} + 17860688 T^{6} + 3392481 T^{7} + 2334878345 T^{8} - 448570881 T^{9} + 246215524883 T^{10} - 86272016067 T^{11} + 21356858306493 T^{12} - 6815489269293 T^{13} + 1536631090794803 T^{14} - 221162938597359 T^{15} + 90943700662895945 T^{16} + 10438855369535919 T^{17} + 4341709199774458448 T^{18} + 1649980656181795121 T^{19} + \)\(16\!\cdots\!87\)\( T^{20} + 91806322522685632354 T^{21} + \)\(41\!\cdots\!37\)\( T^{22} + \)\(22\!\cdots\!37\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 + 10 T + 561 T^{2} + 4284 T^{3} + 139612 T^{4} + 820033 T^{5} + 21534801 T^{6} + 103460393 T^{7} + 2526414560 T^{8} + 11363483203 T^{9} + 257235262247 T^{10} + 1151185952412 T^{11} + 23032480765585 T^{12} + 95548434050196 T^{13} + 1772093721619583 T^{14} + 6497491970193761 T^{15} + 119899393167553760 T^{16} + 407534692967752699 T^{17} + 7040595879367114569 T^{18} + 22252457301176797691 T^{19} + \)\(31\!\cdots\!92\)\( T^{20} + \)\(80\!\cdots\!52\)\( T^{21} + \)\(87\!\cdots\!89\)\( T^{22} + \)\(12\!\cdots\!70\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 10 T + 533 T^{2} + 4030 T^{3} + 127462 T^{4} + 636505 T^{5} + 17587669 T^{6} + 28228105 T^{7} + 1472384246 T^{8} - 6051824595 T^{9} + 74694243687 T^{10} - 1221291941290 T^{11} + 3669601849071 T^{12} - 101367231127070 T^{13} + 514568644759743 T^{14} - 3460354629701265 T^{15} + 69876884182010966 T^{16} + 111191652869871515 T^{17} + 5750119069550386861 T^{18} + 17272232135152533635 T^{19} + \)\(28\!\cdots\!42\)\( T^{20} + \)\(75\!\cdots\!90\)\( T^{21} + \)\(82\!\cdots\!17\)\( T^{22} + \)\(12\!\cdots\!70\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 - 16 T + 89 T^{2} \))(\( 1 + 17 T + 571 T^{2} + 7323 T^{3} + 156672 T^{4} + 1716031 T^{5} + 29222913 T^{6} + 284966947 T^{7} + 4177272124 T^{8} + 36946000313 T^{9} + 484171847345 T^{10} + 3926231105907 T^{11} + 46934120440183 T^{12} + 349434568425723 T^{13} + 3835125202819745 T^{14} + 26045784894655297 T^{15} + 262091414326589884 T^{16} + 1591272373048032203 T^{17} + 14523241028380989393 T^{18} + 75902341852109525399 T^{19} + \)\(61\!\cdots\!32\)\( T^{20} + \)\(25\!\cdots\!07\)\( T^{21} + \)\(17\!\cdots\!71\)\( T^{22} + \)\(47\!\cdots\!13\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 + 31 T + 1213 T^{2} + 26455 T^{3} + 613850 T^{4} + 10471963 T^{5} + 181508461 T^{6} + 2550779273 T^{7} + 35823712446 T^{8} + 426286002355 T^{9} + 5036027217747 T^{10} + 51437382423205 T^{11} + 520333716095317 T^{12} + 4577927035665245 T^{13} + 39890371591773987 T^{14} + 300518416794201995 T^{15} + 2247659999801631486 T^{16} + 14243703101709000577 T^{17} + 90206309268124321021 T^{18} + \)\(46\!\cdots\!27\)\( T^{19} + \)\(24\!\cdots\!50\)\( T^{20} + \)\(92\!\cdots\!95\)\( T^{21} + \)\(37\!\cdots\!13\)\( T^{22} + \)\(86\!\cdots\!59\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 + 53 T + 1965 T^{2} + 50659 T^{3} + 1070582 T^{4} + 18291142 T^{5} + 268477735 T^{6} + 3318765135 T^{7} + 35890994106 T^{8} + 331588096432 T^{9} + 2785787952402 T^{10} + 21855007586449 T^{11} + 197629991382345 T^{12} + 2119935735885553 T^{13} + 26211478844150418 T^{14} + 302631502734882736 T^{15} + 3177403902579417786 T^{16} + 28499365447313539695 T^{17} + \)\(22\!\cdots\!15\)\( T^{18} + \)\(14\!\cdots\!46\)\( T^{19} + \)\(83\!\cdots\!02\)\( T^{20} + \)\(38\!\cdots\!03\)\( T^{21} + \)\(14\!\cdots\!85\)\( T^{22} + \)\(37\!\cdots\!09\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 69 T + 2721 T^{2} + 77149 T^{3} + 1745876 T^{4} + 33295750 T^{5} + 554981367 T^{6} + 8286583629 T^{7} + 112813976168 T^{8} + 1416878290644 T^{9} + 16539262243536 T^{10} + 180152730223765 T^{11} + 1834595057152359 T^{12} + 17474814831705205 T^{13} + 155617918449430224 T^{14} + 1293146560156931412 T^{15} + 9987340196904175208 T^{16} + 71159713190308852653 T^{17} + \)\(46\!\cdots\!43\)\( T^{18} + \)\(26\!\cdots\!50\)\( T^{19} + \)\(13\!\cdots\!36\)\( T^{20} + \)\(58\!\cdots\!33\)\( T^{21} + \)\(20\!\cdots\!29\)\( T^{22} + \)\(49\!\cdots\!57\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
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