Properties

Label 450.2.h
Level 450
Weight 2
Character orbit h
Rep. character \(\chi_{450}(91,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 52
Newform subspaces 7
Sturm bound 180
Trace bound 7

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Defining parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.h (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 7 \)
Sturm bound: \(180\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(450, [\chi])\).

Total New Old
Modular forms 392 52 340
Cusp forms 328 52 276
Eisenstein series 64 0 64

Trace form

\( 52q - q^{2} - 13q^{4} - 11q^{5} - q^{8} + O(q^{10}) \) \( 52q - q^{2} - 13q^{4} - 11q^{5} - q^{8} + q^{10} + 8q^{11} + 6q^{13} + 4q^{14} - 13q^{16} + 16q^{17} - 14q^{19} + 4q^{20} - 8q^{22} + 30q^{23} + 7q^{25} - 30q^{26} - 10q^{28} + 12q^{29} - 18q^{31} + 4q^{32} + 5q^{34} + 14q^{35} - 13q^{37} + 4q^{38} + q^{40} + 20q^{41} + 28q^{43} - 2q^{44} - 16q^{46} - 30q^{47} + 68q^{49} - q^{50} + 6q^{52} + 39q^{53} + 38q^{55} + 4q^{56} - 18q^{58} - 40q^{61} - 8q^{62} - 13q^{64} - 47q^{65} + 2q^{67} - 54q^{68} - 12q^{70} - 2q^{71} - 62q^{73} - 54q^{74} + 16q^{76} - 40q^{77} - 40q^{79} - q^{80} - 86q^{82} - 18q^{83} + 13q^{85} - 2q^{86} + 2q^{88} - q^{89} - 36q^{91} - 30q^{92} + 8q^{94} + 44q^{95} + 14q^{97} + 31q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
450.2.h.a \(4\) \(3.593\) \(\Q(\zeta_{10})\) None \(-1\) \(0\) \(-5\) \(-12\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
450.2.h.b \(4\) \(3.593\) \(\Q(\zeta_{10})\) None \(-1\) \(0\) \(-5\) \(8\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
450.2.h.c \(4\) \(3.593\) \(\Q(\zeta_{10})\) None \(1\) \(0\) \(-5\) \(6\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{3}q^{4}+\cdots\)
450.2.h.d \(8\) \(3.593\) 8.0.1064390625.3 None \(-2\) \(0\) \(4\) \(-2\) \(q-\beta _{3}q^{2}+(-1-\beta _{2}+\beta _{3}-\beta _{5})q^{4}+\cdots\)
450.2.h.e \(8\) \(3.593\) 8.0.58140625.2 None \(2\) \(0\) \(0\) \(4\) \(q+\beta _{6}q^{2}-\beta _{3}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
450.2.h.f \(12\) \(3.593\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-3\) \(0\) \(1\) \(-2\) \(q-\beta _{7}q^{2}+(-1+\beta _{4}-\beta _{6}+\beta _{7})q^{4}+\cdots\)
450.2.h.g \(12\) \(3.593\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(3\) \(0\) \(-1\) \(-2\) \(q+\beta _{7}q^{2}+(-1+\beta _{4}-\beta _{6}+\beta _{7})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \))
$3$ 1
$5$ (\( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} \))(\( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} \))(\( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} \))(\( 1 - 4 T + T^{2} + 16 T^{3} - 39 T^{4} + 80 T^{5} + 25 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 - 15 T^{2} + 105 T^{4} - 375 T^{6} + 625 T^{8} \))(\( 1 - T + 6 T^{2} - 26 T^{3} + 61 T^{4} - 120 T^{5} + 465 T^{6} - 600 T^{7} + 1525 T^{8} - 3250 T^{9} + 3750 T^{10} - 3125 T^{11} + 15625 T^{12} \))(\( 1 + T + 6 T^{2} + 26 T^{3} + 61 T^{4} + 120 T^{5} + 465 T^{6} + 600 T^{7} + 1525 T^{8} + 3250 T^{9} + 3750 T^{10} + 3125 T^{11} + 15625 T^{12} \))
$7$ (\( ( 1 + 3 T + 7 T^{2} )^{4} \))(\( ( 1 - 2 T + 7 T^{2} )^{4} \))(\( ( 1 - 3 T + 15 T^{2} - 21 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 + T + 9 T^{2} + 17 T^{3} + 104 T^{4} + 119 T^{5} + 441 T^{6} + 343 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} - 30 T^{3} + 156 T^{4} - 210 T^{5} + 833 T^{6} - 686 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 + T + 13 T^{2} + 17 T^{3} + 47 T^{4} + 88 T^{5} + 34 T^{6} + 616 T^{7} + 2303 T^{8} + 5831 T^{9} + 31213 T^{10} + 16807 T^{11} + 117649 T^{12} )^{2} \))(\( ( 1 + T + 13 T^{2} + 17 T^{3} + 47 T^{4} + 88 T^{5} + 34 T^{6} + 616 T^{7} + 2303 T^{8} + 5831 T^{9} + 31213 T^{10} + 16807 T^{11} + 117649 T^{12} )^{2} \))
$11$ (\( 1 + 3 T + 8 T^{2} + 51 T^{3} + 265 T^{4} + 561 T^{5} + 968 T^{6} + 3993 T^{7} + 14641 T^{8} \))(\( 1 - 2 T + 13 T^{2} - 34 T^{3} + 225 T^{4} - 374 T^{5} + 1573 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 - 5 T - T^{2} + 55 T^{3} - 184 T^{4} + 605 T^{5} - 121 T^{6} - 6655 T^{7} + 14641 T^{8} \))(\( 1 - 5 T + 38 T^{2} - 210 T^{3} + 1048 T^{4} - 4455 T^{5} + 19301 T^{6} - 69050 T^{7} + 243280 T^{8} - 759550 T^{9} + 2335421 T^{10} - 5929605 T^{11} + 15343768 T^{12} - 33820710 T^{13} + 67319318 T^{14} - 97435855 T^{15} + 214358881 T^{16} \))(\( 1 + T - 25 T^{2} - 40 T^{3} + 160 T^{4} + 653 T^{5} + 2838 T^{6} - 3940 T^{7} - 61905 T^{8} - 43340 T^{9} + 343398 T^{10} + 869143 T^{11} + 2342560 T^{12} - 6442040 T^{13} - 44289025 T^{14} + 19487171 T^{15} + 214358881 T^{16} \))(\( 1 - T - 41 T^{2} + 17 T^{3} + 759 T^{4} + 230 T^{5} - 9528 T^{6} - 5068 T^{7} + 111330 T^{8} - 16692 T^{9} - 1322666 T^{10} + 402304 T^{11} + 14962106 T^{12} + 4425344 T^{13} - 160042586 T^{14} - 22217052 T^{15} + 1629982530 T^{16} - 816206468 T^{17} - 16879433208 T^{18} + 4482049330 T^{19} + 162698390679 T^{20} + 40085110747 T^{21} - 1063434408641 T^{22} - 285311670611 T^{23} + 3138428376721 T^{24} \))(\( 1 + T - 41 T^{2} - 17 T^{3} + 759 T^{4} - 230 T^{5} - 9528 T^{6} + 5068 T^{7} + 111330 T^{8} + 16692 T^{9} - 1322666 T^{10} - 402304 T^{11} + 14962106 T^{12} - 4425344 T^{13} - 160042586 T^{14} + 22217052 T^{15} + 1629982530 T^{16} + 816206468 T^{17} - 16879433208 T^{18} - 4482049330 T^{19} + 162698390679 T^{20} - 40085110747 T^{21} - 1063434408641 T^{22} + 285311670611 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 + T - 12 T^{2} - 25 T^{3} + 131 T^{4} - 325 T^{5} - 2028 T^{6} + 2197 T^{7} + 28561 T^{8} \))(\( 1 + 6 T + 23 T^{2} + 120 T^{3} + 601 T^{4} + 1560 T^{5} + 3887 T^{6} + 13182 T^{7} + 28561 T^{8} \))(\( 1 - 12 T + 51 T^{2} - 76 T^{3} + 9 T^{4} - 988 T^{5} + 8619 T^{6} - 26364 T^{7} + 28561 T^{8} \))(\( 1 - 6 T + 6 T^{2} + 26 T^{3} + 3 T^{4} - 1366 T^{5} + 5284 T^{6} - 2112 T^{7} - 15227 T^{8} - 27456 T^{9} + 892996 T^{10} - 3001102 T^{11} + 85683 T^{12} + 9653618 T^{13} + 28960854 T^{14} - 376491102 T^{15} + 815730721 T^{16} \))(\( 1 + 13 T + 67 T^{2} + 164 T^{3} + 76 T^{4} - 427 T^{5} + 3190 T^{6} + 37102 T^{7} + 173189 T^{8} + 482326 T^{9} + 539110 T^{10} - 938119 T^{11} + 2170636 T^{12} + 60892052 T^{13} + 323396203 T^{14} + 815730721 T^{15} + 815730721 T^{16} \))(\( 1 - 4 T + 23 T^{2} - 40 T^{3} + 442 T^{4} - 684 T^{5} + 4610 T^{6} - 1048 T^{7} + 98883 T^{8} - 113084 T^{9} + 1314010 T^{10} - 1658428 T^{11} + 23586263 T^{12} - 21559564 T^{13} + 222067690 T^{14} - 248445548 T^{15} + 2824197363 T^{16} - 389115064 T^{17} + 22251589490 T^{18} - 42919985628 T^{19} + 360552978682 T^{20} - 424179974920 T^{21} + 3170745312527 T^{22} - 7168641576148 T^{23} + 23298085122481 T^{24} \))(\( 1 - 4 T + 23 T^{2} - 40 T^{3} + 442 T^{4} - 684 T^{5} + 4610 T^{6} - 1048 T^{7} + 98883 T^{8} - 113084 T^{9} + 1314010 T^{10} - 1658428 T^{11} + 23586263 T^{12} - 21559564 T^{13} + 222067690 T^{14} - 248445548 T^{15} + 2824197363 T^{16} - 389115064 T^{17} + 22251589490 T^{18} - 42919985628 T^{19} + 360552978682 T^{20} - 424179974920 T^{21} + 3170745312527 T^{22} - 7168641576148 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 3 T + 37 T^{2} + 45 T^{3} + 676 T^{4} + 765 T^{5} + 10693 T^{6} + 14739 T^{7} + 83521 T^{8} \))(\( 1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 3060 T^{5} + 10693 T^{6} - 58956 T^{7} + 83521 T^{8} \))(\( 1 + 6 T - T^{2} - 18 T^{3} + 169 T^{4} - 306 T^{5} - 289 T^{6} + 29478 T^{7} + 83521 T^{8} \))(\( 1 - 2 T + 44 T^{2} - 12 T^{3} + 1273 T^{4} + 258 T^{5} + 28916 T^{6} + 13984 T^{7} + 544693 T^{8} + 237728 T^{9} + 8356724 T^{10} + 1267554 T^{11} + 106322233 T^{12} - 17038284 T^{13} + 1062053036 T^{14} - 820677346 T^{15} + 6975757441 T^{16} \))(\( 1 - 11 T + 38 T^{2} + 8 T^{3} - 474 T^{4} + 1639 T^{5} + 2235 T^{6} - 60196 T^{7} + 352544 T^{8} - 1023332 T^{9} + 645915 T^{10} + 8052407 T^{11} - 39588954 T^{12} + 11358856 T^{13} + 917227622 T^{14} - 4513725403 T^{15} + 6975757441 T^{16} \))(\( 1 + 8 T - 15 T^{2} - 294 T^{3} - 306 T^{4} + 5078 T^{5} + 15054 T^{6} - 23070 T^{7} - 83077 T^{8} - 40138 T^{9} - 1667150 T^{10} + 2666170 T^{11} + 61412975 T^{12} + 45324890 T^{13} - 481806350 T^{14} - 197197994 T^{15} - 6938674117 T^{16} - 32756100990 T^{17} + 363366963726 T^{18} + 2083699781494 T^{19} - 2134581776946 T^{20} - 34864835690118 T^{21} - 30239908506735 T^{22} + 274175170461064 T^{23} + 582622237229761 T^{24} \))(\( 1 - 8 T - 15 T^{2} + 294 T^{3} - 306 T^{4} - 5078 T^{5} + 15054 T^{6} + 23070 T^{7} - 83077 T^{8} + 40138 T^{9} - 1667150 T^{10} - 2666170 T^{11} + 61412975 T^{12} - 45324890 T^{13} - 481806350 T^{14} + 197197994 T^{15} - 6938674117 T^{16} + 32756100990 T^{17} + 363366963726 T^{18} - 2083699781494 T^{19} - 2134581776946 T^{20} + 34864835690118 T^{21} - 30239908506735 T^{22} - 274175170461064 T^{23} + 582622237229761 T^{24} \))
$19$ (\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 + 21 T^{2} - 10 T^{3} + 381 T^{4} - 190 T^{5} + 7581 T^{6} + 130321 T^{8} \))(\( 1 + 16 T + 117 T^{2} + 578 T^{3} + 2525 T^{4} + 10982 T^{5} + 42237 T^{6} + 109744 T^{7} + 130321 T^{8} \))(\( ( 1 - 4 T - 3 T^{2} - 62 T^{3} + 605 T^{4} - 1178 T^{5} - 1083 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 15 T + 93 T^{2} - 285 T^{3} + 361 T^{4} )^{2}( 1 + 10 T + 41 T^{2} - 10 T^{3} - 509 T^{4} - 190 T^{5} + 14801 T^{6} + 68590 T^{7} + 130321 T^{8} ) \))(\( 1 + 8 T + 39 T^{2} + 342 T^{3} + 2286 T^{4} + 14478 T^{5} + 82544 T^{6} + 405542 T^{7} + 2240021 T^{8} + 11413658 T^{9} + 55478694 T^{10} + 251121822 T^{11} + 1042225791 T^{12} + 4771314618 T^{13} + 20027808534 T^{14} + 78286280222 T^{15} + 291921776741 T^{16} + 1004162140658 T^{17} + 3883355201264 T^{18} + 12941475037242 T^{19} + 38824425111726 T^{20} + 110359192640418 T^{21} + 239111584054239 T^{22} + 931922071185752 T^{23} + 2213314919066161 T^{24} \))(\( 1 + 8 T + 39 T^{2} + 342 T^{3} + 2286 T^{4} + 14478 T^{5} + 82544 T^{6} + 405542 T^{7} + 2240021 T^{8} + 11413658 T^{9} + 55478694 T^{10} + 251121822 T^{11} + 1042225791 T^{12} + 4771314618 T^{13} + 20027808534 T^{14} + 78286280222 T^{15} + 291921776741 T^{16} + 1004162140658 T^{17} + 3883355201264 T^{18} + 12941475037242 T^{19} + 38824425111726 T^{20} + 110359192640418 T^{21} + 239111584054239 T^{22} + 931922071185752 T^{23} + 2213314919066161 T^{24} \))
$23$ (\( 1 + 9 T + 8 T^{2} - 195 T^{3} - 1259 T^{4} - 4485 T^{5} + 4232 T^{6} + 109503 T^{7} + 279841 T^{8} \))(\( 1 - 6 T + 13 T^{2} + 60 T^{3} - 659 T^{4} + 1380 T^{5} + 6877 T^{6} - 73002 T^{7} + 279841 T^{8} \))(\( 1 - 10 T + 37 T^{2} - 200 T^{3} + 1389 T^{4} - 4600 T^{5} + 19573 T^{6} - 121670 T^{7} + 279841 T^{8} \))(\( ( 1 - 10 T + 37 T^{2} - 200 T^{3} + 1389 T^{4} - 4600 T^{5} + 19573 T^{6} - 121670 T^{7} + 279841 T^{8} )^{2} \))(\( 1 - 3 T + 17 T^{2} + 26 T^{3} + 866 T^{4} + 2257 T^{5} - 7690 T^{6} + 132948 T^{7} + 158519 T^{8} + 3057804 T^{9} - 4068010 T^{10} + 27460919 T^{11} + 242342306 T^{12} + 167344918 T^{13} + 2516610113 T^{14} - 10214476341 T^{15} + 78310985281 T^{16} \))(\( 1 + T^{2} - 50 T^{3} + 604 T^{4} + 4450 T^{5} - 3500 T^{6} - 55450 T^{7} - 131975 T^{8} + 1781050 T^{9} + 11647136 T^{10} - 36925050 T^{11} - 137469589 T^{12} - 849276150 T^{13} + 6161334944 T^{14} + 21670035350 T^{15} - 36932015975 T^{16} - 356895219350 T^{17} - 518125611500 T^{18} + 15151473239150 T^{19} + 47299835109724 T^{20} - 90057633073150 T^{21} + 41426511213649 T^{22} + 21914624432020321 T^{24} \))(\( 1 + T^{2} + 50 T^{3} + 604 T^{4} - 4450 T^{5} - 3500 T^{6} + 55450 T^{7} - 131975 T^{8} - 1781050 T^{9} + 11647136 T^{10} + 36925050 T^{11} - 137469589 T^{12} + 849276150 T^{13} + 6161334944 T^{14} - 21670035350 T^{15} - 36932015975 T^{16} + 356895219350 T^{17} - 518125611500 T^{18} - 15151473239150 T^{19} + 47299835109724 T^{20} + 90057633073150 T^{21} + 41426511213649 T^{22} + 21914624432020321 T^{24} \))
$29$ (\( 1 + 15 T + 56 T^{2} - 315 T^{3} - 3629 T^{4} - 9135 T^{5} + 47096 T^{6} + 365835 T^{7} + 707281 T^{8} \))(\( 1 - 19 T^{2} - 120 T^{3} + 721 T^{4} - 3480 T^{5} - 15979 T^{6} + 707281 T^{8} \))(\( 1 + 6 T + 47 T^{2} + 288 T^{3} + 2365 T^{4} + 8352 T^{5} + 39527 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 18 T + 160 T^{2} - 1050 T^{3} + 5925 T^{4} - 26514 T^{5} + 92732 T^{6} - 300180 T^{7} + 1280525 T^{8} - 8705220 T^{9} + 77987612 T^{10} - 646649946 T^{11} + 4190639925 T^{12} - 21536706450 T^{13} + 95171731360 T^{14} - 310497773562 T^{15} + 500246412961 T^{16} \))(\( 1 - 15 T + 47 T^{2} + 480 T^{3} - 4732 T^{4} + 16485 T^{5} + 2494 T^{6} - 526050 T^{7} + 4240405 T^{8} - 15255450 T^{9} + 2097454 T^{10} + 402052665 T^{11} - 3346853692 T^{12} + 9845351520 T^{13} + 27956696087 T^{14} - 258748144635 T^{15} + 500246412961 T^{16} \))(\( 1 + 6 T - 65 T^{2} - 144 T^{3} + 3334 T^{4} + 2454 T^{5} - 34326 T^{6} + 111510 T^{7} - 1136681 T^{8} + 3780366 T^{9} + 101840410 T^{10} + 249990 T^{11} - 2493208365 T^{12} + 7249710 T^{13} + 85647784810 T^{14} + 92199346374 T^{15} - 803952874361 T^{16} + 2287198224990 T^{17} - 20417905316646 T^{18} + 42331196462286 T^{19} + 1667821540811974 T^{20} - 2089029020525136 T^{21} - 27345970164513065 T^{22} + 73203058594234974 T^{23} + 353814783205469041 T^{24} \))(\( 1 - 6 T - 65 T^{2} + 144 T^{3} + 3334 T^{4} - 2454 T^{5} - 34326 T^{6} - 111510 T^{7} - 1136681 T^{8} - 3780366 T^{9} + 101840410 T^{10} - 249990 T^{11} - 2493208365 T^{12} - 7249710 T^{13} + 85647784810 T^{14} - 92199346374 T^{15} - 803952874361 T^{16} - 2287198224990 T^{17} - 20417905316646 T^{18} - 42331196462286 T^{19} + 1667821540811974 T^{20} + 2089029020525136 T^{21} - 27345970164513065 T^{22} - 73203058594234974 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( 1 - 3 T - 12 T^{2} - 131 T^{3} + 1365 T^{4} - 4061 T^{5} - 11532 T^{6} - 89373 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 113 T^{2} + 834 T^{3} + 5605 T^{4} + 25854 T^{5} + 108593 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( ( 1 - T - 39 T^{2} - 31 T^{3} + 961 T^{4} )( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} ) \))(\( 1 - 9 T - 20 T^{2} + 300 T^{3} + 690 T^{4} - 8217 T^{5} - 2797 T^{6} + 139770 T^{7} - 752500 T^{8} + 4332870 T^{9} - 2687917 T^{10} - 244792647 T^{11} + 637229490 T^{12} + 8588745300 T^{13} - 17750073620 T^{14} - 247613526999 T^{15} + 852891037441 T^{16} \))(\( 1 + 9 T + 25 T^{2} + 330 T^{3} + 2510 T^{4} + 7017 T^{5} + 86378 T^{6} + 461340 T^{7} + 675415 T^{8} + 14301540 T^{9} + 83009258 T^{10} + 209043447 T^{11} + 2318037710 T^{12} + 9447619830 T^{13} + 22187592025 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \))(\( 1 + 3 T - 17 T^{2} + 43 T^{3} + 1721 T^{4} + 6968 T^{5} - 13096 T^{6} + 35514 T^{7} + 2478224 T^{8} + 8032138 T^{9} + 1507368 T^{10} + 182144424 T^{11} + 2061324094 T^{12} + 5646477144 T^{13} + 1448580648 T^{14} + 239285423158 T^{15} + 2288691906704 T^{16} + 1016735668614 T^{17} - 11622748206376 T^{18} + 191707895125448 T^{19} + 1467825475435961 T^{20} + 1136903752908853 T^{21} - 13933680878673617 T^{22} + 76225430689214493 T^{23} + 787662783788549761 T^{24} \))(\( 1 + 3 T - 17 T^{2} + 43 T^{3} + 1721 T^{4} + 6968 T^{5} - 13096 T^{6} + 35514 T^{7} + 2478224 T^{8} + 8032138 T^{9} + 1507368 T^{10} + 182144424 T^{11} + 2061324094 T^{12} + 5646477144 T^{13} + 1448580648 T^{14} + 239285423158 T^{15} + 2288691906704 T^{16} + 1016735668614 T^{17} - 11622748206376 T^{18} + 191707895125448 T^{19} + 1467825475435961 T^{20} + 1136903752908853 T^{21} - 13933680878673617 T^{22} + 76225430689214493 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( 1 + 17 T + 147 T^{2} + 1135 T^{3} + 7976 T^{4} + 41995 T^{5} + 201243 T^{6} + 861101 T^{7} + 1874161 T^{8} \))(\( 1 - 13 T + 27 T^{2} + 445 T^{3} - 4264 T^{4} + 16465 T^{5} + 36963 T^{6} - 658489 T^{7} + 1874161 T^{8} \))(\( 1 + 8 T + 27 T^{2} - 80 T^{3} - 1639 T^{4} - 2960 T^{5} + 36963 T^{6} + 405224 T^{7} + 1874161 T^{8} \))(\( 1 - 21 T + 258 T^{2} - 2462 T^{3} + 19746 T^{4} - 143741 T^{5} + 980785 T^{6} - 6308436 T^{7} + 38864404 T^{8} - 233412132 T^{9} + 1342694665 T^{10} - 7280912873 T^{11} + 37007183106 T^{12} - 170724822134 T^{13} + 661957413522 T^{14} - 1993569419793 T^{15} + 3512479453921 T^{16} \))(\( 1 + 6 T - 27 T^{2} - 318 T^{3} - 99 T^{4} + 14106 T^{5} + 58255 T^{6} - 123234 T^{7} - 1122756 T^{8} - 4559658 T^{9} + 79751095 T^{10} + 714511218 T^{11} - 185541939 T^{12} - 22051378326 T^{13} - 69274613043 T^{14} + 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 + 8 T - 48 T^{2} - 440 T^{3} + 1217 T^{4} + 8308 T^{5} - 96890 T^{6} - 403256 T^{7} + 3422938 T^{8} + 15342208 T^{9} - 44309010 T^{10} - 109014876 T^{11} + 1386361728 T^{12} - 4033550412 T^{13} - 60659034690 T^{14} + 777128861824 T^{15} + 6415136905018 T^{16} - 27963366723992 T^{17} - 248593231768010 T^{18} + 788694035220964 T^{19} + 4274687495421857 T^{20} - 57183165509833880 T^{21} - 230812049876056752 T^{22} + 1423340974235683304 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 8 T - 48 T^{2} - 440 T^{3} + 1217 T^{4} + 8308 T^{5} - 96890 T^{6} - 403256 T^{7} + 3422938 T^{8} + 15342208 T^{9} - 44309010 T^{10} - 109014876 T^{11} + 1386361728 T^{12} - 4033550412 T^{13} - 60659034690 T^{14} + 777128861824 T^{15} + 6415136905018 T^{16} - 27963366723992 T^{17} - 248593231768010 T^{18} + 788694035220964 T^{19} + 4274687495421857 T^{20} - 57183165509833880 T^{21} - 230812049876056752 T^{22} + 1423340974235683304 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 + 13 T + 28 T^{2} - 169 T^{3} - 945 T^{4} - 6929 T^{5} + 47068 T^{6} + 895973 T^{7} + 2825761 T^{8} \))(\( 1 - 12 T + 53 T^{2} - 444 T^{3} + 4405 T^{4} - 18204 T^{5} + 89093 T^{6} - 827052 T^{7} + 2825761 T^{8} \))(\( 1 - 14 T + 95 T^{2} - 786 T^{3} + 6569 T^{4} - 32226 T^{5} + 159695 T^{6} - 964894 T^{7} + 2825761 T^{8} \))(\( 1 + 2 T + 36 T^{2} + 168 T^{3} + 2201 T^{4} + 17286 T^{5} + 84044 T^{6} + 959864 T^{7} + 2420157 T^{8} + 39354424 T^{9} + 141277964 T^{10} + 1191368406 T^{11} + 6219499961 T^{12} + 19463841768 T^{13} + 171003752676 T^{14} + 389508547762 T^{15} + 7984925229121 T^{16} \))(\( 1 - 9 T - 55 T^{2} + 1030 T^{3} - 2470 T^{4} - 35497 T^{5} + 265658 T^{6} + 359550 T^{7} - 11328415 T^{8} + 14741550 T^{9} + 446571098 T^{10} - 2446488737 T^{11} - 6979629670 T^{12} + 119331887030 T^{13} - 261255733255 T^{14} - 1752788464929 T^{15} + 7984925229121 T^{16} \))(\( 1 - 20 T + 57 T^{2} + 1770 T^{3} - 17394 T^{4} - 11990 T^{5} + 1073350 T^{6} - 5109010 T^{7} - 22264525 T^{8} + 321080810 T^{9} - 860930798 T^{10} - 6513044050 T^{11} + 74131341039 T^{12} - 267034806050 T^{13} - 1447224671438 T^{14} + 22129210506010 T^{15} - 62914226428525 T^{16} - 591910489471010 T^{17} + 5098524387077350 T^{18} - 2335103743833190 T^{19} - 138889789435330674 T^{20} + 579466023877310970 T^{21} + 765091580678686857 T^{22} - 11006580634324968820 T^{23} + 22563490300366186081 T^{24} \))(\( 1 + 20 T + 57 T^{2} - 1770 T^{3} - 17394 T^{4} + 11990 T^{5} + 1073350 T^{6} + 5109010 T^{7} - 22264525 T^{8} - 321080810 T^{9} - 860930798 T^{10} + 6513044050 T^{11} + 74131341039 T^{12} + 267034806050 T^{13} - 1447224671438 T^{14} - 22129210506010 T^{15} - 62914226428525 T^{16} + 591910489471010 T^{17} + 5098524387077350 T^{18} + 2335103743833190 T^{19} - 138889789435330674 T^{20} - 579466023877310970 T^{21} + 765091580678686857 T^{22} + 11006580634324968820 T^{23} + 22563490300366186081 T^{24} \))
$43$ (\( ( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 2 T + 82 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 2 T + 42 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 16 T + 168 T^{2} + 1280 T^{3} + 9326 T^{4} + 55040 T^{5} + 310632 T^{6} + 1272112 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 6 T + 133 T^{2} - 710 T^{3} + 7916 T^{4} - 30530 T^{5} + 245917 T^{6} - 477042 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 16 T + 224 T^{2} - 2332 T^{3} + 21051 T^{4} - 167884 T^{5} + 1143752 T^{6} - 7219012 T^{7} + 38923299 T^{8} - 185410324 T^{9} + 765811424 T^{10} - 2352135088 T^{11} + 6321363049 T^{12} )^{2} \))(\( ( 1 - 16 T + 224 T^{2} - 2332 T^{3} + 21051 T^{4} - 167884 T^{5} + 1143752 T^{6} - 7219012 T^{7} + 38923299 T^{8} - 185410324 T^{9} + 765811424 T^{10} - 2352135088 T^{11} + 6321363049 T^{12} )^{2} \))
$47$ (\( 1 + 23 T + 202 T^{2} + 925 T^{3} + 4101 T^{4} + 43475 T^{5} + 446218 T^{6} + 2387929 T^{7} + 4879681 T^{8} \))(\( 1 - 2 T + 17 T^{2} + 130 T^{3} + 761 T^{4} + 6110 T^{5} + 37553 T^{6} - 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 20 T + 113 T^{2} - 270 T^{3} - 5851 T^{4} - 12690 T^{5} + 249617 T^{6} + 2076460 T^{7} + 4879681 T^{8} \))(\( 1 - 10 T + 66 T^{2} - 210 T^{3} + 1787 T^{4} + 6510 T^{5} - 94072 T^{6} + 974600 T^{7} - 2764935 T^{8} + 45806200 T^{9} - 207805048 T^{10} + 675887730 T^{11} + 8719989947 T^{12} - 48162451470 T^{13} + 711428211714 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 - T - 77 T^{2} + 108 T^{3} + 5436 T^{4} - 2341 T^{5} - 357450 T^{6} + 29584 T^{7} + 16743019 T^{8} + 1390448 T^{9} - 789607050 T^{10} - 243049643 T^{11} + 26525945916 T^{12} + 24769260756 T^{13} - 829999580333 T^{14} - 506623120463 T^{15} + 23811286661761 T^{16} \))(\( 1 - 11 T^{2} - 140 T^{3} + 404 T^{4} + 37780 T^{5} - 40930 T^{6} + 148540 T^{7} - 4073685 T^{8} + 35390020 T^{9} + 536648324 T^{10} - 1335868840 T^{11} + 9098825071 T^{12} - 62785835480 T^{13} + 1185456147716 T^{14} + 3674298046460 T^{15} - 19878283294485 T^{16} + 34066907339780 T^{17} - 441193283415970 T^{18} + 19140221491092140 T^{19} + 9619759811351444 T^{20} - 156678266234387380 T^{21} - 578590454594130539 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 11 T^{2} + 140 T^{3} + 404 T^{4} - 37780 T^{5} - 40930 T^{6} - 148540 T^{7} - 4073685 T^{8} - 35390020 T^{9} + 536648324 T^{10} + 1335868840 T^{11} + 9098825071 T^{12} + 62785835480 T^{13} + 1185456147716 T^{14} - 3674298046460 T^{15} - 19878283294485 T^{16} - 34066907339780 T^{17} - 441193283415970 T^{18} - 19140221491092140 T^{19} + 9619759811351444 T^{20} + 156678266234387380 T^{21} - 578590454594130539 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 - 16 T + 43 T^{2} + 700 T^{3} - 7959 T^{4} + 37100 T^{5} + 120787 T^{6} - 2382032 T^{7} + 7890481 T^{8} \))(\( 1 - 21 T + 253 T^{2} - 2535 T^{3} + 21196 T^{4} - 134355 T^{5} + 710677 T^{6} - 3126417 T^{7} + 7890481 T^{8} \))(\( 1 - 16 T + 53 T^{2} + 200 T^{3} - 1759 T^{4} + 10600 T^{5} + 148877 T^{6} - 2382032 T^{7} + 7890481 T^{8} \))(\( 1 + 7 T + 62 T^{2} + 816 T^{3} + 3586 T^{4} - 23943 T^{5} - 77365 T^{6} - 2388872 T^{7} - 30975296 T^{8} - 126610216 T^{9} - 217318285 T^{10} - 3564562011 T^{11} + 28295264866 T^{12} + 341247522288 T^{13} + 1374190389998 T^{14} + 8222977978859 T^{15} + 62259690411361 T^{16} \))(\( 1 + 7 T + 2 T^{2} + 6 T^{3} - 794 T^{4} - 27393 T^{5} + 64385 T^{6} + 1339888 T^{7} + 6789964 T^{8} + 71014064 T^{9} + 180857465 T^{10} - 4078187661 T^{11} - 6265041914 T^{12} + 2509172958 T^{13} + 44328722258 T^{14} + 8222977978859 T^{15} + 62259690411361 T^{16} \))(\( 1 - 2 T - 218 T^{2} + 154 T^{3} + 20819 T^{4} - 14082 T^{5} - 945826 T^{6} + 2993876 T^{7} + 12643662 T^{8} - 300705398 T^{9} + 702917212 T^{10} + 8975478116 T^{11} - 50340536356 T^{12} + 475700340148 T^{13} + 1974494448508 T^{14} - 44768117538046 T^{15} + 99764574781422 T^{16} + 1252025449800868 T^{17} - 20963629029197554 T^{18} - 16542282271184634 T^{19} + 1296184494674124659 T^{20} + 508163593137528482 T^{21} - 38125468539681844682 T^{22} - 18538071858744383194 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 2 T - 218 T^{2} - 154 T^{3} + 20819 T^{4} + 14082 T^{5} - 945826 T^{6} - 2993876 T^{7} + 12643662 T^{8} + 300705398 T^{9} + 702917212 T^{10} - 8975478116 T^{11} - 50340536356 T^{12} - 475700340148 T^{13} + 1974494448508 T^{14} + 44768117538046 T^{15} + 99764574781422 T^{16} - 1252025449800868 T^{17} - 20963629029197554 T^{18} + 16542282271184634 T^{19} + 1296184494674124659 T^{20} - 508163593137528482 T^{21} - 38125468539681844682 T^{22} + 18538071858744383194 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 - 10 T + T^{2} - 200 T^{3} + 5061 T^{4} - 11800 T^{5} + 3481 T^{6} - 2053790 T^{7} + 12117361 T^{8} \))(\( 1 + 20 T + 181 T^{2} + 1600 T^{3} + 14601 T^{4} + 94400 T^{5} + 630061 T^{6} + 4107580 T^{7} + 12117361 T^{8} \))(\( 1 + 5 T - 49 T^{2} - 295 T^{3} + 1736 T^{4} - 17405 T^{5} - 170569 T^{6} + 1026895 T^{7} + 12117361 T^{8} \))(\( 1 - 25 T + 212 T^{2} - 150 T^{3} - 12692 T^{4} + 128475 T^{5} - 252121 T^{6} - 7858000 T^{7} + 98067400 T^{8} - 463622000 T^{9} - 877633201 T^{10} + 26386067025 T^{11} - 153793545812 T^{12} - 107238644850 T^{13} + 8942273131892 T^{14} - 62216287120475 T^{15} + 146830437604321 T^{16} \))(\( 1 + 10 T + 142 T^{2} + 730 T^{3} + 11283 T^{4} + 22410 T^{5} + 330384 T^{6} - 1700800 T^{7} + 16244505 T^{8} - 100347200 T^{9} + 1150066704 T^{10} + 4602543390 T^{11} + 136720184163 T^{12} + 521894738270 T^{13} + 5989635777022 T^{14} + 24886514848190 T^{15} + 146830437604321 T^{16} \))(\( 1 + 19 T + 75 T^{2} - 1271 T^{3} - 15671 T^{4} - 73204 T^{5} - 120356 T^{6} + 4773540 T^{7} + 97305244 T^{8} + 638482244 T^{9} - 599172130 T^{10} - 34389514670 T^{11} - 299642137310 T^{12} - 2028981365530 T^{13} - 2085718184530 T^{14} + 131130844790476 T^{15} + 1179082768741084 T^{16} + 3412719738248460 T^{17} - 5076680306896196 T^{18} - 182179243294690076 T^{19} - 2300979787697314391 T^{20} - 11010667685510427469 T^{21} + 38333756497548105075 T^{22} + \)\(57\!\cdots\!21\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 19 T + 75 T^{2} + 1271 T^{3} - 15671 T^{4} + 73204 T^{5} - 120356 T^{6} - 4773540 T^{7} + 97305244 T^{8} - 638482244 T^{9} - 599172130 T^{10} + 34389514670 T^{11} - 299642137310 T^{12} + 2028981365530 T^{13} - 2085718184530 T^{14} - 131130844790476 T^{15} + 1179082768741084 T^{16} - 3412719738248460 T^{17} - 5076680306896196 T^{18} + 182179243294690076 T^{19} - 2300979787697314391 T^{20} + 11010667685510427469 T^{21} + 38333756497548105075 T^{22} - \)\(57\!\cdots\!21\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 25864 T^{5} + 11163 T^{6} + 453962 T^{7} + 13845841 T^{8} \))(\( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 25864 T^{5} + 11163 T^{6} + 453962 T^{7} + 13845841 T^{8} \))(\( 1 - 21 T^{2} - 410 T^{3} + 2901 T^{4} - 25010 T^{5} - 78141 T^{6} + 13845841 T^{8} \))(\( 1 - 10 T - 62 T^{2} + 700 T^{3} + 5543 T^{4} - 26410 T^{5} - 544444 T^{6} + 851800 T^{7} + 32679625 T^{8} + 51959800 T^{9} - 2025876124 T^{10} - 5994568210 T^{11} + 76747496663 T^{12} + 591217410700 T^{13} - 3194263210382 T^{14} - 31427428360210 T^{15} + 191707312997281 T^{16} \))(\( 1 - 6 T - 140 T^{2} + 210 T^{3} + 7045 T^{4} + 27242 T^{5} + 433648 T^{6} - 1657580 T^{7} - 60459295 T^{8} - 101112380 T^{9} + 1613604208 T^{10} + 6183416402 T^{11} + 97543949845 T^{12} + 177365223210 T^{13} - 7212852410540 T^{14} - 18856457016126 T^{15} + 191707312997281 T^{16} \))(\( 1 + 26 T + 331 T^{2} + 2856 T^{3} + 14346 T^{4} + 37146 T^{5} + 629326 T^{6} + 14318326 T^{7} + 195406371 T^{8} + 1764901066 T^{9} + 10336656006 T^{10} + 40426619066 T^{11} + 178740751411 T^{12} + 2466023763026 T^{13} + 38462696998326 T^{14} + 400599008861746 T^{15} + 2705565543253011 T^{16} + 12093205176112126 T^{17} + 32423111115110686 T^{18} + 116740325386836066 T^{19} + 2750233112258993226 T^{20} + 33398481241134306696 T^{21} + \)\(23\!\cdots\!31\)\( T^{22} + \)\(11\!\cdots\!86\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 26 T + 331 T^{2} + 2856 T^{3} + 14346 T^{4} + 37146 T^{5} + 629326 T^{6} + 14318326 T^{7} + 195406371 T^{8} + 1764901066 T^{9} + 10336656006 T^{10} + 40426619066 T^{11} + 178740751411 T^{12} + 2466023763026 T^{13} + 38462696998326 T^{14} + 400599008861746 T^{15} + 2705565543253011 T^{16} + 12093205176112126 T^{17} + 32423111115110686 T^{18} + 116740325386836066 T^{19} + 2750233112258993226 T^{20} + 33398481241134306696 T^{21} + \)\(23\!\cdots\!31\)\( T^{22} + \)\(11\!\cdots\!86\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 - 13 T + 12 T^{2} + 895 T^{3} - 9919 T^{4} + 59965 T^{5} + 53868 T^{6} - 3909919 T^{7} + 20151121 T^{8} \))(\( 1 - 18 T + 77 T^{2} - 30 T^{3} + 961 T^{4} - 2010 T^{5} + 345653 T^{6} - 5413734 T^{7} + 20151121 T^{8} \))(\( 1 - 16 T + 29 T^{2} + 868 T^{3} - 9191 T^{4} + 58156 T^{5} + 130181 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T + 34 T^{2} + 232 T^{3} - 637 T^{4} - 10798 T^{5} + 66956 T^{6} + 2488096 T^{7} - 7257587 T^{8} + 166702432 T^{9} + 300565484 T^{10} - 3247638874 T^{11} - 12836264077 T^{12} + 313229024824 T^{13} + 3075584993746 T^{14} + 12121423210646 T^{15} + 406067677556641 T^{16} \))(\( 1 + 11 T + 103 T^{2} + 1672 T^{3} + 14576 T^{4} + 139271 T^{5} + 1482950 T^{6} + 11696476 T^{7} + 89866399 T^{8} + 783663892 T^{9} + 6656962550 T^{10} + 41887563773 T^{11} + 293722739696 T^{12} + 2257409178904 T^{13} + 9317213363407 T^{14} + 66667827658553 T^{15} + 406067677556641 T^{16} \))(\( 1 + 16 T + 41 T^{2} - 1646 T^{3} - 19888 T^{4} + 9078 T^{5} + 1871366 T^{6} + 13090638 T^{7} - 40906785 T^{8} - 1205696466 T^{9} - 5599022836 T^{10} + 41834098474 T^{11} + 684349156075 T^{12} + 2802884597758 T^{13} - 25134013510804 T^{14} - 362628886203558 T^{15} - 824317574255985 T^{16} + 17673999030448266 T^{17} + 169280740806072854 T^{18} + 55019139953122194 T^{19} - 8075873971246476208 T^{20} - 44781955616301482762 T^{21} + 74736349986622219409 T^{22} + \)\(19\!\cdots\!28\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 16 T + 41 T^{2} - 1646 T^{3} - 19888 T^{4} + 9078 T^{5} + 1871366 T^{6} + 13090638 T^{7} - 40906785 T^{8} - 1205696466 T^{9} - 5599022836 T^{10} + 41834098474 T^{11} + 684349156075 T^{12} + 2802884597758 T^{13} - 25134013510804 T^{14} - 362628886203558 T^{15} - 824317574255985 T^{16} + 17673999030448266 T^{17} + 169280740806072854 T^{18} + 55019139953122194 T^{19} - 8075873971246476208 T^{20} - 44781955616301482762 T^{21} + 74736349986622219409 T^{22} + \)\(19\!\cdots\!28\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 + 3 T - 62 T^{2} - 399 T^{3} + 3205 T^{4} - 28329 T^{5} - 312542 T^{6} + 1073733 T^{7} + 25411681 T^{8} \))(\( 1 + 28 T + 313 T^{2} + 2126 T^{3} + 14805 T^{4} + 150946 T^{5} + 1577833 T^{6} + 10021508 T^{7} + 25411681 T^{8} \))(\( 1 - 20 T + 89 T^{2} + 1420 T^{3} - 22639 T^{4} + 100820 T^{5} + 448649 T^{6} - 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 182 T^{2} + 750 T^{3} + 7983 T^{4} - 134400 T^{5} + 952256 T^{6} + 5830800 T^{7} - 134438695 T^{8} + 413986800 T^{9} + 4800322496 T^{10} - 48103238400 T^{11} + 202861449423 T^{12} + 1353172013250 T^{13} - 23314251673622 T^{14} + 645753531245761 T^{16} \))(\( 1 - 9 T + 75 T^{2} - 1220 T^{3} + 18360 T^{4} - 165937 T^{5} + 1180298 T^{6} - 14013840 T^{7} + 141940835 T^{8} - 994982640 T^{9} + 5949882218 T^{10} - 59390677607 T^{11} + 466558463160 T^{12} - 2201159808220 T^{13} + 9607521294075 T^{14} - 81856081425519 T^{15} + 645753531245761 T^{16} \))(\( 1 - 48 T + 1075 T^{2} - 14632 T^{3} + 121834 T^{4} - 217632 T^{5} - 11827404 T^{6} + 218641680 T^{7} - 2132764051 T^{8} + 10733037072 T^{9} + 34603163190 T^{10} - 1365818396500 T^{11} + 15372122385915 T^{12} - 96973106151500 T^{13} + 174434545640790 T^{14} + 3841472031476592 T^{15} - 54197119712279731 T^{16} + 394479736407949680 T^{17} - 1515093810448371084 T^{18} - 1979389190310950112 T^{19} + 78674735725796045674 T^{20} - \)\(67\!\cdots\!92\)\( T^{21} + \)\(34\!\cdots\!75\)\( T^{22} - \)\(11\!\cdots\!08\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 + 48 T + 1075 T^{2} + 14632 T^{3} + 121834 T^{4} + 217632 T^{5} - 11827404 T^{6} - 218641680 T^{7} - 2132764051 T^{8} - 10733037072 T^{9} + 34603163190 T^{10} + 1365818396500 T^{11} + 15372122385915 T^{12} + 96973106151500 T^{13} + 174434545640790 T^{14} - 3841472031476592 T^{15} - 54197119712279731 T^{16} - 394479736407949680 T^{17} - 1515093810448371084 T^{18} + 1979389190310950112 T^{19} + 78674735725796045674 T^{20} + \)\(67\!\cdots\!92\)\( T^{21} + \)\(34\!\cdots\!75\)\( T^{22} + \)\(11\!\cdots\!08\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 62050 T^{5} + 335727 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( 1 - 14 T + 3 T^{2} + 980 T^{3} - 9259 T^{4} + 71540 T^{5} + 15987 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( 1 - 2 T + 51 T^{2} - 376 T^{3} + 6389 T^{4} - 27448 T^{5} + 271779 T^{6} - 778034 T^{7} + 28398241 T^{8} \))(\( 1 + 24 T + 226 T^{2} + 1116 T^{3} + 5643 T^{4} + 16584 T^{5} - 622036 T^{6} - 12920832 T^{7} - 137738827 T^{8} - 943220736 T^{9} - 3314829844 T^{10} + 6451457928 T^{11} + 160251273963 T^{12} + 2313547897788 T^{13} + 34201535141314 T^{14} + 265137564458328 T^{15} + 806460091894081 T^{16} \))(\( 1 + 8 T + 62 T^{2} + 1114 T^{3} + 11231 T^{4} + 71208 T^{5} + 863380 T^{6} + 9235272 T^{7} + 61248289 T^{8} + 674174856 T^{9} + 4600952020 T^{10} + 27701122536 T^{11} + 318940644671 T^{12} + 2309401754602 T^{13} + 9382722029918 T^{14} + 88379188152776 T^{15} + 806460091894081 T^{16} \))(\( 1 + 30 T + 361 T^{2} + 2900 T^{3} + 28374 T^{4} + 255570 T^{5} + 1562330 T^{6} + 9342990 T^{7} + 52825615 T^{8} + 492207530 T^{9} + 8592376626 T^{10} + 63871721690 T^{11} + 283645412871 T^{12} + 4662635683370 T^{13} + 45788775039954 T^{14} + 191477096698010 T^{15} + 1500154545743215 T^{16} + 19368687162683070 T^{17} + 236434001758093370 T^{18} + 2823383639525620290 T^{19} + 22882498647402654294 T^{20} + \)\(17\!\cdots\!00\)\( T^{21} + \)\(15\!\cdots\!89\)\( T^{22} + \)\(94\!\cdots\!10\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 30 T + 361 T^{2} + 2900 T^{3} + 28374 T^{4} + 255570 T^{5} + 1562330 T^{6} + 9342990 T^{7} + 52825615 T^{8} + 492207530 T^{9} + 8592376626 T^{10} + 63871721690 T^{11} + 283645412871 T^{12} + 4662635683370 T^{13} + 45788775039954 T^{14} + 191477096698010 T^{15} + 1500154545743215 T^{16} + 19368687162683070 T^{17} + 236434001758093370 T^{18} + 2823383639525620290 T^{19} + 22882498647402654294 T^{20} + \)\(17\!\cdots\!00\)\( T^{21} + \)\(15\!\cdots\!89\)\( T^{22} + \)\(94\!\cdots\!10\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 10 T - 39 T^{2} + 790 T^{3} - 2839 T^{4} + 62410 T^{5} - 243399 T^{6} - 4930390 T^{7} + 38950081 T^{8} \))(\( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} \))(\( 1 - 2 T - 55 T^{2} + 578 T^{3} + 3679 T^{4} + 45662 T^{5} - 343255 T^{6} - 986078 T^{7} + 38950081 T^{8} \))(\( 1 + 6 T + 34 T^{2} + 336 T^{3} + 4761 T^{4} - 34122 T^{5} + 60506 T^{6} - 838512 T^{7} + 1917227 T^{8} - 66242448 T^{9} + 377617946 T^{10} - 16823476758 T^{11} + 185441335641 T^{12} + 1033890950064 T^{13} + 8264973487714 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 10 T - 38 T^{2} + 320 T^{3} + 15323 T^{4} + 59210 T^{5} - 186436 T^{6} + 4386400 T^{7} + 89172925 T^{8} + 346525600 T^{9} - 1163547076 T^{10} + 29192839190 T^{11} + 596832091163 T^{12} + 984658047680 T^{13} - 9237323309798 T^{14} + 192039089861590 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 18 T + 39 T^{2} - 778 T^{3} + 7476 T^{4} + 334048 T^{5} + 2263454 T^{6} - 15862408 T^{7} - 150964159 T^{8} + 1962807008 T^{9} + 29425549114 T^{10} - 11043695308 T^{11} - 2130346215359 T^{12} - 872451929332 T^{13} + 183644852020474 T^{14} + 967740404417312 T^{15} - 5880066221146879 T^{16} - 48809524039948792 T^{17} + 550217273548829534 T^{18} + 6415027389008441632 T^{19} + 11341905462861450036 T^{20} - 93244541674477052182 T^{21} + \)\(36\!\cdots\!39\)\( T^{22} + \)\(13\!\cdots\!22\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 18 T + 39 T^{2} - 778 T^{3} + 7476 T^{4} + 334048 T^{5} + 2263454 T^{6} - 15862408 T^{7} - 150964159 T^{8} + 1962807008 T^{9} + 29425549114 T^{10} - 11043695308 T^{11} - 2130346215359 T^{12} - 872451929332 T^{13} + 183644852020474 T^{14} + 967740404417312 T^{15} - 5880066221146879 T^{16} - 48809524039948792 T^{17} + 550217273548829534 T^{18} + 6415027389008441632 T^{19} + 11341905462861450036 T^{20} - 93244541674477052182 T^{21} + \)\(36\!\cdots\!39\)\( T^{22} + \)\(13\!\cdots\!22\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 - T + 58 T^{2} - 335 T^{3} + 7041 T^{4} - 27805 T^{5} + 399562 T^{6} - 571787 T^{7} + 47458321 T^{8} \))(\( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 64740 T^{5} - 323783 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 13 T - 19 T^{2} + 951 T^{3} - 6196 T^{4} + 78933 T^{5} - 130891 T^{6} - 7433231 T^{7} + 47458321 T^{8} \))(\( 1 + 11 T - 4 T^{2} + 264 T^{3} + 5188 T^{4} - 21759 T^{5} + 513329 T^{6} + 7685372 T^{7} + 27478708 T^{8} + 637885876 T^{9} + 3536323481 T^{10} - 12441513333 T^{11} + 246213769348 T^{12} + 1039906729752 T^{13} - 1307761493476 T^{14} + 298496560885897 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 27 T + 237 T^{2} + 126 T^{3} - 12774 T^{4} - 113973 T^{5} - 181210 T^{6} + 11072628 T^{7} + 166729299 T^{8} + 919028124 T^{9} - 1248355690 T^{10} - 65168279751 T^{11} - 606232592454 T^{12} + 496319121018 T^{13} + 77484868488453 T^{14} + 732673376719929 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 29 T + 205 T^{2} - 2483 T^{3} - 45111 T^{4} - 9616 T^{5} + 4478376 T^{6} + 25972790 T^{7} - 144493422 T^{8} - 1200220924 T^{9} + 8072175950 T^{10} + 36109733090 T^{11} - 517891517510 T^{12} + 2997107846470 T^{13} + 55609220119550 T^{14} - 686270721471188 T^{15} - 6857415203664462 T^{16} + 102307875422103970 T^{17} + 1464161921526768744 T^{18} - 260940266316253232 T^{19} - \)\(10\!\cdots\!51\)\( T^{20} - \)\(46\!\cdots\!49\)\( T^{21} + \)\(31\!\cdots\!45\)\( T^{22} + \)\(37\!\cdots\!43\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 - 29 T + 205 T^{2} + 2483 T^{3} - 45111 T^{4} + 9616 T^{5} + 4478376 T^{6} - 25972790 T^{7} - 144493422 T^{8} + 1200220924 T^{9} + 8072175950 T^{10} - 36109733090 T^{11} - 517891517510 T^{12} - 2997107846470 T^{13} + 55609220119550 T^{14} + 686270721471188 T^{15} - 6857415203664462 T^{16} - 102307875422103970 T^{17} + 1464161921526768744 T^{18} + 260940266316253232 T^{19} - \)\(10\!\cdots\!51\)\( T^{20} + \)\(46\!\cdots\!49\)\( T^{21} + \)\(31\!\cdots\!45\)\( T^{22} - \)\(37\!\cdots\!43\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 10 T - 29 T^{2} + 200 T^{3} + 10101 T^{4} + 17800 T^{5} - 229709 T^{6} + 7049690 T^{7} + 62742241 T^{8} \))(\( 1 - 5 T - 79 T^{2} + 445 T^{3} + 5276 T^{4} + 39605 T^{5} - 625759 T^{6} - 3524845 T^{7} + 62742241 T^{8} \))(\( 1 + 2 T + 35 T^{2} + 632 T^{3} + 8789 T^{4} + 56248 T^{5} + 277235 T^{6} + 1409938 T^{7} + 62742241 T^{8} \))(\( 1 + 9 T - 166 T^{2} - 1356 T^{3} + 12786 T^{4} + 119727 T^{5} + 426031 T^{6} - 2824668 T^{7} - 98110168 T^{8} - 251395452 T^{9} + 3374591551 T^{10} + 84403823463 T^{11} + 802222293426 T^{12} - 7571984612844 T^{13} - 82498894299526 T^{14} + 398082014059761 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 15 T - 108 T^{2} + 1640 T^{3} + 7368 T^{4} - 158465 T^{5} + 2297399 T^{6} - 297300 T^{7} - 272740720 T^{8} - 26459700 T^{9} + 18197697479 T^{10} - 111712912585 T^{11} + 462284831688 T^{12} + 9157857496360 T^{13} - 53673979423788 T^{14} - 663470023432935 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 62 T + 1666 T^{2} - 25854 T^{3} + 274609 T^{4} - 2573150 T^{5} + 27410848 T^{6} - 292619604 T^{7} + 2355071920 T^{8} - 12032738534 T^{9} + 54718570046 T^{10} - 770989409628 T^{11} + 10095165651756 T^{12} - 68618057456892 T^{13} + 433425793334366 T^{14} - 8482707651575446 T^{15} + 147762489976972720 T^{16} - 1634005264678838196 T^{17} + 13622678625375744928 T^{18} - \)\(11\!\cdots\!50\)\( T^{19} + \)\(10\!\cdots\!29\)\( T^{20} - \)\(90\!\cdots\!86\)\( T^{21} + \)\(51\!\cdots\!66\)\( T^{22} - \)\(17\!\cdots\!18\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 + 62 T + 1666 T^{2} + 25854 T^{3} + 274609 T^{4} + 2573150 T^{5} + 27410848 T^{6} + 292619604 T^{7} + 2355071920 T^{8} + 12032738534 T^{9} + 54718570046 T^{10} + 770989409628 T^{11} + 10095165651756 T^{12} + 68618057456892 T^{13} + 433425793334366 T^{14} + 8482707651575446 T^{15} + 147762489976972720 T^{16} + 1634005264678838196 T^{17} + 13622678625375744928 T^{18} + \)\(11\!\cdots\!50\)\( T^{19} + \)\(10\!\cdots\!29\)\( T^{20} + \)\(90\!\cdots\!86\)\( T^{21} + \)\(51\!\cdots\!66\)\( T^{22} + \)\(17\!\cdots\!18\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 + 22 T + 207 T^{2} + 2420 T^{3} + 31541 T^{4} + 234740 T^{5} + 1947663 T^{6} + 20078806 T^{7} + 88529281 T^{8} \))(\( 1 - 18 T + 47 T^{2} + 1740 T^{3} - 27059 T^{4} + 168780 T^{5} + 442223 T^{6} - 16428114 T^{7} + 88529281 T^{8} \))(\( 1 - 7 T - 73 T^{2} + 835 T^{3} + 1816 T^{4} + 80995 T^{5} - 686857 T^{6} - 6388711 T^{7} + 88529281 T^{8} \))(\( 1 - T + 28 T^{2} - 632 T^{3} + 5996 T^{4} - 78691 T^{5} + 707075 T^{6} - 3586376 T^{7} + 72919024 T^{8} - 347878472 T^{9} + 6652868675 T^{10} - 71819151043 T^{11} + 530821568876 T^{12} - 5427199042424 T^{13} + 23323216138012 T^{14} - 80798284478113 T^{15} + 7837433594376961 T^{16} \))(\( ( 1 + 18 T + 27 T^{2} - 1060 T^{3} - 9699 T^{4} - 102820 T^{5} + 254043 T^{6} + 16428114 T^{7} + 88529281 T^{8} )^{2} \))(\( 1 - 23 T - 3 T^{2} + 2745 T^{3} - 783 T^{4} - 170628 T^{5} - 91460 T^{6} - 6181364 T^{7} + 190100268 T^{8} - 1739567428 T^{9} + 18395502870 T^{10} + 51462799126 T^{11} - 2827976867282 T^{12} + 4991891515222 T^{13} + 173083286503830 T^{14} - 1587656223215044 T^{15} + 16829440043947308 T^{16} - 53081475920370548 T^{17} - 76183619570806340 T^{18} - 13786449683931464964 T^{19} - 6136710504397160463 T^{20} + \)\(20\!\cdots\!65\)\( T^{21} - \)\(22\!\cdots\!47\)\( T^{22} - \)\(16\!\cdots\!19\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 23 T - 3 T^{2} + 2745 T^{3} - 783 T^{4} - 170628 T^{5} - 91460 T^{6} - 6181364 T^{7} + 190100268 T^{8} - 1739567428 T^{9} + 18395502870 T^{10} + 51462799126 T^{11} - 2827976867282 T^{12} + 4991891515222 T^{13} + 173083286503830 T^{14} - 1587656223215044 T^{15} + 16829440043947308 T^{16} - 53081475920370548 T^{17} - 76183619570806340 T^{18} - 13786449683931464964 T^{19} - 6136710504397160463 T^{20} + \)\(20\!\cdots\!65\)\( T^{21} - \)\(22\!\cdots\!47\)\( T^{22} - \)\(16\!\cdots\!19\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
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