Properties

Label 847.2.n
Level 847
Weight 2
Character orbit n
Rep. character \(\chi_{847}(9,\cdot)\)
Character field \(\Q(\zeta_{15})\)
Dimension 512
Newform subspaces 14
Sturm bound 176
Trace bound 6

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 847.n (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 77 \)
Character field: \(\Q(\zeta_{15})\)
Newform subspaces: \( 14 \)
Sturm bound: \(176\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(847, [\chi])\).

Total New Old
Modular forms 800 640 160
Cusp forms 608 512 96
Eisenstein series 192 128 64

Trace form

\( 512q + 5q^{2} + 3q^{3} + 61q^{4} + q^{5} + 12q^{6} + 7q^{7} + 28q^{8} + 45q^{9} + O(q^{10}) \) \( 512q + 5q^{2} + 3q^{3} + 61q^{4} + q^{5} + 12q^{6} + 7q^{7} + 28q^{8} + 45q^{9} - 4q^{10} - 24q^{12} + 4q^{13} - 8q^{14} + 24q^{15} + 35q^{16} + 11q^{17} - 18q^{18} + 7q^{19} + 48q^{20} + 18q^{21} - 42q^{23} + 7q^{24} + 9q^{25} + 43q^{26} + 6q^{27} - 4q^{28} - 24q^{29} - 21q^{30} + 9q^{31} + 12q^{32} - 296q^{34} + 19q^{35} - 42q^{36} - 15q^{37} - 13q^{38} - 33q^{39} - 5q^{40} - 62q^{41} + 32q^{42} + 44q^{43} - 24q^{45} - 16q^{46} - 21q^{47} - 226q^{48} + 15q^{49} - 6q^{50} + q^{51} + 3q^{52} - 23q^{53} - 44q^{54} - 144q^{56} - 20q^{57} + 46q^{58} - 7q^{59} + 95q^{60} - 18q^{61} + 80q^{62} - 22q^{63} - 108q^{64} + 10q^{65} + 76q^{67} + 21q^{68} + 134q^{69} - 37q^{70} + 12q^{71} + 48q^{72} - 26q^{73} + 55q^{74} - 57q^{75} + 144q^{76} + 60q^{78} - 30q^{79} - 105q^{80} + 55q^{81} - 13q^{82} + 28q^{83} + 27q^{84} + 98q^{85} + 4q^{86} - 66q^{87} + 2q^{89} + 18q^{90} + 52q^{91} + 74q^{92} + 24q^{93} - 45q^{94} - 22q^{95} + 55q^{96} + 4q^{97} - 120q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(847, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
847.2.n.a \(8\) \(6.763\) \(\Q(\zeta_{15})\) None \(-3\) \(1\) \(5\) \(5\) \(q+(-1-\zeta_{15}^{5}+\zeta_{15}^{7})q^{2}+(-1+\zeta_{15}+\cdots)q^{3}+\cdots\)
847.2.n.b \(8\) \(6.763\) \(\Q(\zeta_{15})\) None \(2\) \(1\) \(-5\) \(5\) \(q+(1-\zeta_{15}+\zeta_{15}^{5}-\zeta_{15}^{7})q^{2}+(-1+\cdots)q^{3}+\cdots\)
847.2.n.c \(8\) \(6.763\) \(\Q(\zeta_{15})\) None \(3\) \(1\) \(5\) \(-5\) \(q+(1+\zeta_{15}^{5}-\zeta_{15}^{7})q^{2}+(-1+\zeta_{15}+\cdots)q^{3}+\cdots\)
847.2.n.d \(24\) \(6.763\) None \(0\) \(-1\) \(-2\) \(2\)
847.2.n.e \(24\) \(6.763\) None \(0\) \(-1\) \(-2\) \(-2\)
847.2.n.f \(24\) \(6.763\) None \(0\) \(3\) \(6\) \(0\)
847.2.n.g \(24\) \(6.763\) None \(0\) \(3\) \(6\) \(0\)
847.2.n.h \(40\) \(6.763\) None \(-2\) \(1\) \(-6\) \(-13\)
847.2.n.i \(40\) \(6.763\) None \(2\) \(1\) \(-6\) \(13\)
847.2.n.j \(40\) \(6.763\) None \(3\) \(-4\) \(4\) \(2\)
847.2.n.k \(48\) \(6.763\) None \(0\) \(-2\) \(-4\) \(0\)
847.2.n.l \(56\) \(6.763\) None \(0\) \(3\) \(4\) \(2\)
847.2.n.m \(56\) \(6.763\) None \(0\) \(3\) \(4\) \(-2\)
847.2.n.n \(112\) \(6.763\) None \(0\) \(-6\) \(-8\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(847, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(847, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 7 T^{2} + 6 T^{3} + 3 T^{4} - 6 T^{5} + 14 T^{6} + 48 T^{7} + 113 T^{8} + 96 T^{9} + 56 T^{10} - 48 T^{11} + 48 T^{12} + 192 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16} \))(\( 1 - 2 T + 2 T^{2} + 6 T^{3} - 17 T^{4} + 24 T^{5} - T^{6} - 47 T^{7} + 103 T^{8} - 94 T^{9} - 4 T^{10} + 192 T^{11} - 272 T^{12} + 192 T^{13} + 128 T^{14} - 256 T^{15} + 256 T^{16} \))(\( 1 - 3 T + 7 T^{2} - 6 T^{3} + 3 T^{4} + 6 T^{5} + 14 T^{6} - 48 T^{7} + 113 T^{8} - 96 T^{9} + 56 T^{10} + 48 T^{11} + 48 T^{12} - 192 T^{13} + 448 T^{14} - 384 T^{15} + 256 T^{16} \))
$3$ (\( 1 - T + 3 T^{2} - 8 T^{3} + 8 T^{4} + 7 T^{5} + 6 T^{6} + 56 T^{7} - 137 T^{8} + 168 T^{9} + 54 T^{10} + 189 T^{11} + 648 T^{12} - 1944 T^{13} + 2187 T^{14} - 2187 T^{15} + 6561 T^{16} \))(\( 1 - T + 3 T^{2} - 8 T^{3} + 8 T^{4} + 7 T^{5} + 6 T^{6} + 56 T^{7} - 137 T^{8} + 168 T^{9} + 54 T^{10} + 189 T^{11} + 648 T^{12} - 1944 T^{13} + 2187 T^{14} - 2187 T^{15} + 6561 T^{16} \))(\( 1 - T + 3 T^{2} - 8 T^{3} + 8 T^{4} + 7 T^{5} + 6 T^{6} + 56 T^{7} - 137 T^{8} + 168 T^{9} + 54 T^{10} + 189 T^{11} + 648 T^{12} - 1944 T^{13} + 2187 T^{14} - 2187 T^{15} + 6561 T^{16} \))
$5$ (\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 10 T^{2} + 25 T^{3} + 75 T^{4} + 125 T^{5} + 250 T^{6} + 625 T^{7} + 625 T^{8} ) \))(\( ( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2}( 1 - 5 T + 10 T^{2} - 25 T^{3} + 75 T^{4} - 125 T^{5} + 250 T^{6} - 625 T^{7} + 625 T^{8} ) \))(\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 10 T^{2} + 25 T^{3} + 75 T^{4} + 125 T^{5} + 250 T^{6} + 625 T^{7} + 625 T^{8} ) \))
$7$ (\( 1 - 5 T + 18 T^{2} - 55 T^{3} + 149 T^{4} - 385 T^{5} + 882 T^{6} - 1715 T^{7} + 2401 T^{8} \))(\( 1 - 5 T + 18 T^{2} - 55 T^{3} + 149 T^{4} - 385 T^{5} + 882 T^{6} - 1715 T^{7} + 2401 T^{8} \))(\( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 385 T^{5} + 882 T^{6} + 1715 T^{7} + 2401 T^{8} \))
$11$ (\( \))(\( \))(\( \))
$13$ (\( ( 1 + 10 T + 27 T^{2} - 40 T^{3} - 391 T^{4} - 520 T^{5} + 4563 T^{6} + 21970 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 - 5 T + 27 T^{2} - 115 T^{3} + 584 T^{4} - 1495 T^{5} + 4563 T^{6} - 10985 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 - 10 T + 27 T^{2} + 40 T^{3} - 391 T^{4} + 520 T^{5} + 4563 T^{6} - 21970 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 6 T + 37 T^{2} + 30 T^{3} - 60 T^{4} - 2316 T^{5} + 1751 T^{6} + 24090 T^{7} + 292979 T^{8} + 409530 T^{9} + 506039 T^{10} - 11378508 T^{11} - 5011260 T^{12} + 42595710 T^{13} + 893090053 T^{14} + 2462032038 T^{15} + 6975757441 T^{16} \))(\( 1 - 4 T + 17 T^{2} + 120 T^{3} - 740 T^{4} + 3144 T^{5} - 649 T^{6} - 49990 T^{7} + 275299 T^{8} - 849830 T^{9} - 187561 T^{10} + 15446472 T^{11} - 61805540 T^{12} + 170382840 T^{13} + 410338673 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 - 6 T + 37 T^{2} - 30 T^{3} - 60 T^{4} + 2316 T^{5} + 1751 T^{6} - 24090 T^{7} + 292979 T^{8} - 409530 T^{9} + 506039 T^{10} + 11378508 T^{11} - 5011260 T^{12} - 42595710 T^{13} + 893090053 T^{14} - 2462032038 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 12 T + 109 T^{2} + 816 T^{3} + 5382 T^{4} + 32052 T^{5} + 171953 T^{6} + 849906 T^{7} + 3829931 T^{8} + 16148214 T^{9} + 62075033 T^{10} + 219844668 T^{11} + 701387622 T^{12} + 2020496784 T^{13} + 5128001029 T^{14} + 10726460868 T^{15} + 16983563041 T^{16} \))(\( 1 - 3 T - 26 T^{2} + 21 T^{3} + 252 T^{4} + 1302 T^{5} - 1792 T^{6} - 24084 T^{7} + 90161 T^{8} - 457596 T^{9} - 646912 T^{10} + 8930418 T^{11} + 32840892 T^{12} + 51998079 T^{13} - 1223192906 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} \))(\( 1 - 12 T + 109 T^{2} - 816 T^{3} + 5382 T^{4} - 32052 T^{5} + 171953 T^{6} - 849906 T^{7} + 3829931 T^{8} - 16148214 T^{9} + 62075033 T^{10} - 219844668 T^{11} + 701387622 T^{12} - 2020496784 T^{13} + 5128001029 T^{14} - 10726460868 T^{15} + 16983563041 T^{16} \))
$23$ (\( ( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 2024 T^{5} + 3703 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 2024 T^{5} + 3703 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 2024 T^{5} + 3703 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 3 T + 25 T^{2} - 153 T^{3} + 1444 T^{4} - 4437 T^{5} + 21025 T^{6} - 73167 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 + 12 T + 25 T^{2} - 288 T^{3} - 2471 T^{4} - 8352 T^{5} + 21025 T^{6} + 292668 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 + 3 T + 25 T^{2} + 153 T^{3} + 1444 T^{4} + 4437 T^{5} + 21025 T^{6} + 73167 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 4 T - 30 T^{2} + 64 T^{3} + 659 T^{4} + 1984 T^{5} - 28830 T^{6} - 119164 T^{7} + 923521 T^{8} )( 1 + T + 40 T^{2} - 101 T^{3} + 529 T^{4} - 3131 T^{5} + 38440 T^{6} + 29791 T^{7} + 923521 T^{8} ) \))(\( ( 1 - 19 T + 210 T^{2} - 1691 T^{3} + 10649 T^{4} - 52421 T^{5} + 201810 T^{6} - 566029 T^{7} + 923521 T^{8} )( 1 + 11 T + 60 T^{2} - 11 T^{3} - 991 T^{4} - 341 T^{5} + 57660 T^{6} + 327701 T^{7} + 923521 T^{8} ) \))(\( ( 1 - 4 T - 30 T^{2} + 64 T^{3} + 659 T^{4} + 1984 T^{5} - 28830 T^{6} - 119164 T^{7} + 923521 T^{8} )( 1 + T + 40 T^{2} - 101 T^{3} + 529 T^{4} - 3131 T^{5} + 38440 T^{6} + 29791 T^{7} + 923521 T^{8} ) \))
$37$ (\( 1 - 12 T + 87 T^{2} + 196 T^{3} - 5952 T^{4} + 52744 T^{5} - 69191 T^{6} - 1413552 T^{7} + 16477063 T^{8} - 52301424 T^{9} - 94722479 T^{10} + 2671641832 T^{11} - 11155006272 T^{12} + 13591415572 T^{13} + 223218197583 T^{14} - 1139182525596 T^{15} + 3512479453921 T^{16} \))(\( 1 + 13 T + 112 T^{2} + 71 T^{3} - 4102 T^{4} - 46406 T^{5} - 63616 T^{6} + 1263548 T^{7} + 15622363 T^{8} + 46751276 T^{9} - 87090304 T^{10} - 2350603118 T^{11} - 7687808422 T^{12} + 4923420947 T^{13} + 287361357808 T^{14} + 1234114402729 T^{15} + 3512479453921 T^{16} \))(\( 1 - 12 T + 87 T^{2} + 196 T^{3} - 5952 T^{4} + 52744 T^{5} - 69191 T^{6} - 1413552 T^{7} + 16477063 T^{8} - 52301424 T^{9} - 94722479 T^{10} + 2671641832 T^{11} - 11155006272 T^{12} + 13591415572 T^{13} + 223218197583 T^{14} - 1139182525596 T^{15} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 6 T - 25 T^{2} - 276 T^{3} - 371 T^{4} - 11316 T^{5} - 42025 T^{6} + 413526 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + T - 25 T^{2} - 221 T^{3} + 1064 T^{4} - 9061 T^{5} - 42025 T^{6} + 68921 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 6 T - 25 T^{2} + 276 T^{3} - 371 T^{4} + 11316 T^{5} - 42025 T^{6} - 413526 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 + 7 T + 87 T^{2} + 301 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 + 7 T + 87 T^{2} + 301 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 - 7 T + 87 T^{2} - 301 T^{3} + 1849 T^{4} )^{4} \))
$47$ (\( 1 + 14 T + 167 T^{2} + 1110 T^{3} + 8620 T^{4} + 53076 T^{5} + 566321 T^{6} + 4313480 T^{7} + 36265879 T^{8} + 202733560 T^{9} + 1251003089 T^{10} + 5510509548 T^{11} + 42062850220 T^{12} + 254572957770 T^{13} + 1800128959943 T^{14} + 7092723686482 T^{15} + 23811286661761 T^{16} \))(\( 1 - 6 T + 7 T^{2} + 690 T^{3} - 6420 T^{4} + 23196 T^{5} + 57521 T^{6} - 1846680 T^{7} + 13766039 T^{8} - 86793960 T^{9} + 127063889 T^{10} + 2408278308 T^{11} - 31327552020 T^{12} + 158248054830 T^{13} + 75454507303 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16} \))(\( 1 + 14 T + 167 T^{2} + 1110 T^{3} + 8620 T^{4} + 53076 T^{5} + 566321 T^{6} + 4313480 T^{7} + 36265879 T^{8} + 202733560 T^{9} + 1251003089 T^{10} + 5510509548 T^{11} + 42062850220 T^{12} + 254572957770 T^{13} + 1800128959943 T^{14} + 7092723686482 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 13 T + 128 T^{2} + 105 T^{3} - 7190 T^{4} + 92838 T^{5} - 81376 T^{6} - 3553180 T^{7} + 54630859 T^{8} - 188318540 T^{9} - 228585184 T^{10} + 13821442926 T^{11} - 56732558390 T^{12} + 43910526765 T^{13} + 2837038224512 T^{14} - 15271244817881 T^{15} + 62259690411361 T^{16} \))(\( 1 + 12 T + 103 T^{2} - 420 T^{3} - 9840 T^{4} - 102312 T^{5} - 99751 T^{6} + 3945120 T^{7} + 55833959 T^{8} + 209091360 T^{9} - 280200559 T^{10} - 15231903624 T^{11} - 77642333040 T^{12} - 175642107060 T^{13} + 2282929196287 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16} \))(\( 1 - 13 T + 128 T^{2} + 105 T^{3} - 7190 T^{4} + 92838 T^{5} - 81376 T^{6} - 3553180 T^{7} + 54630859 T^{8} - 188318540 T^{9} - 228585184 T^{10} + 13821442926 T^{11} - 56732558390 T^{12} + 43910526765 T^{13} + 2837038224512 T^{14} - 15271244817881 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 12 T + 59 T^{2} + 984 T^{3} - 15948 T^{4} + 123288 T^{5} - 61307 T^{6} - 7136406 T^{7} + 86945531 T^{8} - 421047954 T^{9} - 213409667 T^{10} + 25320766152 T^{11} - 193247673228 T^{12} + 703485510216 T^{13} + 2488651484819 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} \))(\( 1 + 18 T + 239 T^{2} + 1374 T^{3} + 5292 T^{4} - 7812 T^{5} + 470053 T^{6} + 8281674 T^{7} + 95738891 T^{8} + 488618766 T^{9} + 1636254493 T^{10} - 1604420748 T^{11} + 64125074412 T^{12} + 982305986826 T^{13} + 10081147540199 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} \))(\( 1 - 12 T + 59 T^{2} + 984 T^{3} - 15948 T^{4} + 123288 T^{5} - 61307 T^{6} - 7136406 T^{7} + 86945531 T^{8} - 421047954 T^{9} - 213409667 T^{10} + 25320766152 T^{11} - 193247673228 T^{12} + 703485510216 T^{13} + 2488651484819 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 22 T + 301 T^{2} - 1778 T^{3} - 4 T^{4} + 111404 T^{5} - 487141 T^{6} - 4731764 T^{7} + 81018487 T^{8} - 288637604 T^{9} - 1812651661 T^{10} + 25286591324 T^{11} - 55383364 T^{12} - 1501692223178 T^{13} + 15507632682661 T^{14} - 69140342392462 T^{15} + 191707312997281 T^{16} \))(\( 1 + 18 T + 141 T^{2} - 538 T^{3} - 19524 T^{4} - 199396 T^{5} - 460901 T^{6} + 9059916 T^{7} + 127866487 T^{8} + 552654876 T^{9} - 1715012621 T^{10} - 45259103476 T^{11} - 270326199684 T^{12} - 454392809938 T^{13} + 7264372784901 T^{14} + 56569371048378 T^{15} + 191707312997281 T^{16} \))(\( 1 + 22 T + 301 T^{2} + 1778 T^{3} - 4 T^{4} - 111404 T^{5} - 487141 T^{6} + 4731764 T^{7} + 81018487 T^{8} + 288637604 T^{9} - 1812651661 T^{10} - 25286591324 T^{11} - 55383364 T^{12} + 1501692223178 T^{13} + 15507632682661 T^{14} + 69140342392462 T^{15} + 191707312997281 T^{16} \))
$67$ (\( ( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 113297 T^{5} + 619482 T^{6} + 5714497 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 113297 T^{5} + 619482 T^{6} + 5714497 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 113297 T^{5} + 619482 T^{6} + 5714497 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 - 2 T - 47 T^{2} - 394 T^{3} + 5625 T^{4} - 27974 T^{5} - 236927 T^{6} - 715822 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 8 T - 47 T^{2} - 434 T^{3} + 1365 T^{4} - 30814 T^{5} - 236927 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 2 T - 47 T^{2} - 394 T^{3} + 5625 T^{4} - 27974 T^{5} - 236927 T^{6} - 715822 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 17 T^{2} + 840 T^{3} - 3780 T^{4} - 37800 T^{5} + 288413 T^{6} - 666540 T^{7} - 26258761 T^{8} - 48657420 T^{9} + 1536952877 T^{10} - 14704842600 T^{11} - 107345350980 T^{12} + 1741380138120 T^{13} - 2572681846913 T^{14} + 806460091894081 T^{16} \))(\( 1 + 15 T + 208 T^{2} + 2445 T^{3} + 29070 T^{4} + 272130 T^{5} + 2789288 T^{6} + 23441760 T^{7} + 210261239 T^{8} + 1711248480 T^{9} + 14864115752 T^{10} + 105863196210 T^{11} + 825536865870 T^{12} + 5068660044885 T^{13} + 31477519068112 T^{14} + 165710977786455 T^{15} + 806460091894081 T^{16} \))(\( 1 - 17 T^{2} - 840 T^{3} - 3780 T^{4} + 37800 T^{5} + 288413 T^{6} + 666540 T^{7} - 26258761 T^{8} + 48657420 T^{9} + 1536952877 T^{10} + 14704842600 T^{11} - 107345350980 T^{12} - 1741380138120 T^{13} - 2572681846913 T^{14} + 806460091894081 T^{16} \))
$79$ (\( 1 - 21 T + 349 T^{2} - 3666 T^{3} + 39546 T^{4} - 392493 T^{5} + 4522886 T^{6} - 46176768 T^{7} + 449468687 T^{8} - 3647964672 T^{9} + 28227331526 T^{10} - 193514356227 T^{11} + 1540319903226 T^{12} - 11280488758734 T^{13} + 84837521976829 T^{14} - 403282088709339 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 9 T - 11 T^{2} - 1446 T^{3} - 17334 T^{4} - 85743 T^{5} + 200726 T^{6} + 10774872 T^{7} + 128238887 T^{8} + 851214888 T^{9} + 1252730966 T^{10} - 42274642977 T^{11} - 675160704054 T^{12} - 4449423552954 T^{13} - 2673962010731 T^{14} + 172835180875431 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 21 T + 349 T^{2} + 3666 T^{3} + 39546 T^{4} + 392493 T^{5} + 4522886 T^{6} + 46176768 T^{7} + 449468687 T^{8} + 3647964672 T^{9} + 28227331526 T^{10} + 193514356227 T^{11} + 1540319903226 T^{12} + 11280488758734 T^{13} + 84837521976829 T^{14} + 403282088709339 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 + 9 T - 2 T^{2} - 765 T^{3} - 6719 T^{4} - 63495 T^{5} - 13778 T^{6} + 5146083 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 9 T - 2 T^{2} - 765 T^{3} - 6719 T^{4} - 63495 T^{5} - 13778 T^{6} + 5146083 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 9 T - 2 T^{2} + 765 T^{3} - 6719 T^{4} + 63495 T^{5} - 13778 T^{6} - 5146083 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 12 T - 50 T^{2} + 192 T^{3} + 16899 T^{4} + 17088 T^{5} - 396050 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 12 T - 50 T^{2} + 192 T^{3} + 16899 T^{4} + 17088 T^{5} - 396050 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 12 T - 50 T^{2} + 192 T^{3} + 16899 T^{4} + 17088 T^{5} - 396050 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 8 T - 63 T^{2} + 20 T^{3} + 9821 T^{4} + 1940 T^{5} - 592767 T^{6} - 7301384 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 7 T - 63 T^{2} + 185 T^{3} + 11276 T^{4} + 17945 T^{5} - 592767 T^{6} + 6388711 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 8 T - 63 T^{2} + 20 T^{3} + 9821 T^{4} + 1940 T^{5} - 592767 T^{6} - 7301384 T^{7} + 88529281 T^{8} )^{2} \))
show more
show less