Properties

Label 150.6.g.a
Level $150$
Weight $6$
Character orbit 150.g
Analytic conductor $24.058$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0575729719\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 10 x^{19} + 67300 x^{18} - 605415 x^{17} + 1505002471 x^{16} - 12026298320 x^{15} + 15161344295580 x^{14} - 105918997871590 x^{13} + 72521519492275041 x^{12} - 433752716999522710 x^{11} + 150954604904063839080 x^{10} - 750799480917962811625 x^{9} + 100084099910299358028076 x^{8} - 395836369710607771806480 x^{7} + 22144208384329171616972475 x^{6} - 65050346450081546072582845 x^{5} + 865137734840778899106150416 x^{4} - 1622318535480468844324766865 x^{3} + 11299132807307021058314145735 x^{2} - 10498749516125580450644510315 x + 45940690246932155189634577205\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 \beta_{6} q^{2} -9 \beta_{7} q^{3} + 16 \beta_{7} q^{4} + ( -5 + 5 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} - \beta_{13} + \beta_{16} - \beta_{18} ) q^{5} + ( -36 + 36 \beta_{5} - 36 \beta_{6} - 36 \beta_{7} ) q^{6} + ( 20 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 22 \beta_{5} + 22 \beta_{6} + \beta_{9} - \beta_{12} + \beta_{13} - \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{7} + ( 64 - 64 \beta_{5} + 64 \beta_{6} + 64 \beta_{7} ) q^{8} -81 \beta_{5} q^{9} +O(q^{10})\) \( q -4 \beta_{6} q^{2} -9 \beta_{7} q^{3} + 16 \beta_{7} q^{4} + ( -5 + 5 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} - \beta_{13} + \beta_{16} - \beta_{18} ) q^{5} + ( -36 + 36 \beta_{5} - 36 \beta_{6} - 36 \beta_{7} ) q^{6} + ( 20 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 22 \beta_{5} + 22 \beta_{6} + \beta_{9} - \beta_{12} + \beta_{13} - \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{7} + ( 64 - 64 \beta_{5} + 64 \beta_{6} + 64 \beta_{7} ) q^{8} -81 \beta_{5} q^{9} + ( 32 + 4 \beta_{2} + 4 \beta_{3} - 12 \beta_{5} + 32 \beta_{6} + 40 \beta_{7} - 4 \beta_{13} + 4 \beta_{15} + 4 \beta_{16} + 4 \beta_{17} - 4 \beta_{18} ) q^{10} + ( 8 - 2 \beta_{1} - 7 \beta_{2} - 52 \beta_{6} + 8 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} + 7 \beta_{12} + 9 \beta_{13} - \beta_{14} - 9 \beta_{15} + 9 \beta_{18} + 2 \beta_{19} ) q^{11} + 144 \beta_{5} q^{12} + ( -6 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 64 \beta_{5} + 55 \beta_{6} + 55 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} - 4 \beta_{16} + 6 \beta_{17} - 2 \beta_{18} + 6 \beta_{19} ) q^{13} + ( -88 + 4 \beta_{1} - 8 \beta_{3} - 80 \beta_{6} - 88 \beta_{7} - 4 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} - 8 \beta_{17} - 4 \beta_{18} - 4 \beta_{19} ) q^{14} + ( -63 - 9 \beta_{3} + 90 \beta_{5} - 18 \beta_{6} - 18 \beta_{7} - 9 \beta_{13} ) q^{15} -256 \beta_{5} q^{16} + ( -68 - 13 \beta_{1} - 16 \beta_{2} + 14 \beta_{3} + 389 \beta_{5} - 68 \beta_{6} - 389 \beta_{7} + \beta_{8} - 12 \beta_{9} + 12 \beta_{10} + 12 \beta_{11} - \beta_{12} - 5 \beta_{13} + \beta_{15} - 5 \beta_{16} + 4 \beta_{18} ) q^{17} -324 q^{18} + ( 292 - 12 \beta_{1} + 4 \beta_{2} + 26 \beta_{3} + 4 \beta_{4} - 282 \beta_{5} + 292 \beta_{6} + 282 \beta_{7} - 5 \beta_{8} - 5 \beta_{9} + 17 \beta_{10} + 17 \beta_{11} - 14 \beta_{12} - 34 \beta_{13} - 4 \beta_{14} + 14 \beta_{15} + 21 \beta_{16} - 21 \beta_{18} ) q^{19} + ( 112 + 16 \beta_{3} - 160 \beta_{5} + 32 \beta_{6} + 32 \beta_{7} + 16 \beta_{13} ) q^{20} + ( 198 + 9 \beta_{3} - 9 \beta_{4} - 198 \beta_{5} + 18 \beta_{7} - 18 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} - 18 \beta_{12} + 9 \beta_{14} + 9 \beta_{15} - 9 \beta_{16} - 9 \beta_{18} ) q^{21} + ( 32 + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 32 \beta_{5} + 240 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - 8 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} - 4 \beta_{14} + 12 \beta_{17} + 4 \beta_{18} - 12 \beta_{19} ) q^{22} + ( 250 + 7 \beta_{1} + 7 \beta_{2} - 28 \beta_{3} + 598 \beta_{6} + 250 \beta_{7} + 10 \beta_{8} + 10 \beta_{9} + 18 \beta_{10} - 7 \beta_{12} - 13 \beta_{13} - 18 \beta_{14} - 4 \beta_{15} - 28 \beta_{17} - 13 \beta_{18} - 7 \beta_{19} ) q^{23} + 576 q^{24} + ( 340 - 50 \beta_{1} - 20 \beta_{2} + 40 \beta_{3} + 25 \beta_{4} - 270 \beta_{5} + 405 \beta_{6} + 340 \beta_{7} + 25 \beta_{8} + 25 \beta_{10} + 25 \beta_{11} + 25 \beta_{12} - 20 \beta_{13} - 10 \beta_{15} + 20 \beta_{16} + 5 \beta_{17} + 5 \beta_{18} ) q^{25} + ( 476 - 24 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} - 220 \beta_{5} + 220 \beta_{6} + 8 \beta_{11} - 12 \beta_{12} + 12 \beta_{13} - 8 \beta_{14} - 8 \beta_{15} - 12 \beta_{16} - 20 \beta_{17} ) q^{26} -729 \beta_{6} q^{27} + ( -352 - 16 \beta_{3} + 16 \beta_{4} + 352 \beta_{5} - 32 \beta_{7} + 32 \beta_{8} + 16 \beta_{9} - 16 \beta_{10} + 32 \beta_{12} - 16 \beta_{14} - 16 \beta_{15} + 16 \beta_{16} + 16 \beta_{18} ) q^{28} + ( 961 - 31 \beta_{2} - 24 \beta_{3} + 2 \beta_{4} - 961 \beta_{5} + 483 \beta_{7} + 37 \beta_{8} + 24 \beta_{9} - 24 \beta_{10} - 22 \beta_{11} + 12 \beta_{12} + 53 \beta_{13} - 24 \beta_{14} - 55 \beta_{15} - 29 \beta_{16} - 29 \beta_{17} + 10 \beta_{18} - 45 \beta_{19} ) q^{29} + ( 288 + 72 \beta_{5} + 180 \beta_{6} - 36 \beta_{13} + 36 \beta_{15} ) q^{30} + ( 366 - 65 \beta_{1} - 26 \beta_{2} + 61 \beta_{3} - 9 \beta_{4} - 720 \beta_{5} + 366 \beta_{6} + 720 \beta_{7} + 34 \beta_{8} - 19 \beta_{9} + 31 \beta_{10} + 31 \beta_{11} + 4 \beta_{12} - 19 \beta_{13} + 9 \beta_{14} - 4 \beta_{15} + 11 \beta_{16} - 5 \beta_{18} ) q^{31} -1024 q^{32} + ( -468 - 18 \beta_{1} + 27 \beta_{3} - 18 \beta_{4} + 540 \beta_{5} - 468 \beta_{6} - 540 \beta_{7} + 9 \beta_{8} + 18 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} - 9 \beta_{12} + 9 \beta_{13} + 18 \beta_{14} + 9 \beta_{15} + 45 \beta_{16} - 9 \beta_{18} ) q^{33} + ( -52 \beta_{1} - 60 \beta_{2} + 32 \beta_{3} + 1556 \beta_{5} - 1284 \beta_{6} - 1284 \beta_{7} - 52 \beta_{9} + 48 \beta_{11} - 8 \beta_{12} - 44 \beta_{13} - 24 \beta_{15} - 20 \beta_{16} - 24 \beta_{17} - 8 \beta_{18} ) q^{34} + ( -1484 + 30 \beta_{1} + 33 \beta_{2} - 82 \beta_{3} + 20 \beta_{4} - 226 \beta_{5} - 1630 \beta_{6} - 44 \beta_{7} + 35 \beta_{8} - 20 \beta_{9} - 20 \beta_{10} + 15 \beta_{11} + 20 \beta_{12} - 24 \beta_{13} - 50 \beta_{14} - 58 \beta_{15} - 5 \beta_{16} - 77 \beta_{17} - 25 \beta_{18} - 15 \beta_{19} ) q^{35} + 1296 \beta_{6} q^{36} + ( 20 \beta_{1} + 19 \beta_{2} + 108 \beta_{3} - 91 \beta_{4} - 1265 \beta_{5} - 73 \beta_{6} - 73 \beta_{7} - 112 \beta_{8} - 92 \beta_{9} + 91 \beta_{10} + 63 \beta_{11} - 144 \beta_{12} - 187 \beta_{13} + 153 \beta_{15} + 37 \beta_{16} + 41 \beta_{17} - 144 \beta_{18} - 21 \beta_{19} ) q^{37} + ( -40 \beta_{2} + 68 \beta_{3} + 16 \beta_{4} - 1128 \beta_{5} - 40 \beta_{6} - 40 \beta_{7} - 32 \beta_{8} - 32 \beta_{9} - 16 \beta_{10} + 68 \beta_{11} - 52 \beta_{13} - 4 \beta_{15} + 84 \beta_{16} - 36 \beta_{17} - 48 \beta_{19} ) q^{38} + ( 495 - 18 \beta_{2} + 45 \beta_{3} + 1071 \beta_{6} + 495 \beta_{7} + 36 \beta_{8} + 36 \beta_{9} + 18 \beta_{10} + 18 \beta_{12} - 18 \beta_{13} + 9 \beta_{14} + 27 \beta_{15} + 45 \beta_{17} - 18 \beta_{18} ) q^{39} + ( -512 - 128 \beta_{5} - 320 \beta_{6} + 64 \beta_{13} - 64 \beta_{15} ) q^{40} + ( 9 \beta_{1} + 102 \beta_{2} + 157 \beta_{3} - 34 \beta_{4} + 1554 \beta_{5} + 1059 \beta_{6} + 1059 \beta_{7} - 61 \beta_{8} - 52 \beta_{9} + 34 \beta_{10} + 70 \beta_{11} - 131 \beta_{12} - 116 \beta_{13} + 121 \beta_{15} + 132 \beta_{16} + 60 \beta_{17} - 131 \beta_{18} - 27 \beta_{19} ) q^{41} + ( -720 + 36 \beta_{1} - 36 \beta_{3} - 72 \beta_{5} - 720 \beta_{6} + 72 \beta_{7} - 72 \beta_{8} - 36 \beta_{9} + 36 \beta_{10} + 36 \beta_{11} - 72 \beta_{13} - 36 \beta_{18} ) q^{42} + ( 3562 + 23 \beta_{1} + 41 \beta_{2} - 191 \beta_{3} + 36 \beta_{4} - 522 \beta_{5} + 522 \beta_{6} + 82 \beta_{9} + 79 \beta_{11} + 31 \beta_{12} - 31 \beta_{13} - 79 \beta_{14} - 79 \beta_{15} + 36 \beta_{16} - 125 \beta_{17} - 82 \beta_{19} ) q^{43} + ( 832 + 32 \beta_{1} - 48 \beta_{3} + 32 \beta_{4} - 960 \beta_{5} + 832 \beta_{6} + 960 \beta_{7} - 16 \beta_{8} - 32 \beta_{9} - 16 \beta_{10} - 16 \beta_{11} + 16 \beta_{12} - 16 \beta_{13} - 32 \beta_{14} - 16 \beta_{15} - 80 \beta_{16} + 16 \beta_{18} ) q^{44} + ( -162 + 81 \beta_{2} + 648 \beta_{6} + 405 \beta_{7} + 81 \beta_{17} ) q^{45} + ( 1000 + 12 \beta_{2} - 72 \beta_{3} + 144 \beta_{4} - 1000 \beta_{5} - 1392 \beta_{7} + 100 \beta_{8} + 72 \beta_{9} - 72 \beta_{10} + 72 \beta_{11} + 56 \beta_{12} - 84 \beta_{13} - 72 \beta_{14} - 60 \beta_{15} + 156 \beta_{16} - 96 \beta_{17} + 32 \beta_{18} - 112 \beta_{19} ) q^{46} + ( -1618 - 78 \beta_{2} - 81 \beta_{3} + 205 \beta_{4} + 1618 \beta_{5} - 928 \beta_{7} + 132 \beta_{8} + 81 \beta_{9} - 81 \beta_{10} + 124 \beta_{11} + 285 \beta_{12} - 46 \beta_{13} - 81 \beta_{14} - 159 \beta_{15} + 127 \beta_{16} + 37 \beta_{17} + 127 \beta_{18} - 109 \beta_{19} ) q^{47} -2304 \beta_{6} q^{48} + ( -2473 + 11 \beta_{1} - 258 \beta_{2} - 67 \beta_{3} - 2 \beta_{4} - 1232 \beta_{5} + 1232 \beta_{6} + 176 \beta_{9} - 29 \beta_{11} - 34 \beta_{12} + 34 \beta_{13} + 29 \beta_{14} + 29 \beta_{15} - 130 \beta_{16} - 277 \beta_{17} - 176 \beta_{19} ) q^{49} + ( 280 - 100 \beta_{1} + 20 \beta_{2} + 180 \beta_{3} - 1360 \beta_{5} - 260 \beta_{7} - 100 \beta_{8} - 200 \beta_{9} + 100 \beta_{11} - 100 \beta_{12} - 100 \beta_{13} + 100 \beta_{14} + 120 \beta_{15} + 60 \beta_{16} + 20 \beta_{17} - 60 \beta_{18} - 100 \beta_{19} ) q^{50} + ( -612 + 117 \beta_{1} + 81 \beta_{2} - 72 \beta_{3} + 108 \beta_{4} - 2889 \beta_{5} + 2889 \beta_{6} + 117 \beta_{9} - 45 \beta_{12} + 45 \beta_{13} + 99 \beta_{16} - 18 \beta_{17} - 117 \beta_{19} ) q^{51} + ( -880 + 32 \beta_{2} - 80 \beta_{3} - 1904 \beta_{6} - 880 \beta_{7} - 64 \beta_{8} - 64 \beta_{9} - 32 \beta_{10} - 32 \beta_{12} + 32 \beta_{13} - 16 \beta_{14} - 48 \beta_{15} - 80 \beta_{17} + 32 \beta_{18} ) q^{52} + ( 4675 - 7 \beta_{2} - 2 \beta_{3} + 134 \beta_{4} - 4675 \beta_{5} + 1478 \beta_{7} - 125 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 132 \beta_{11} + 131 \beta_{12} - 125 \beta_{13} - 2 \beta_{14} - 9 \beta_{15} + 127 \beta_{16} - 206 \beta_{17} + 18 \beta_{18} - 159 \beta_{19} ) q^{53} + 2916 \beta_{7} q^{54} + ( 1594 - 95 \beta_{1} + 87 \beta_{2} + 180 \beta_{3} - 25 \beta_{4} + 6482 \beta_{5} + 1398 \beta_{6} - 1858 \beta_{7} - 130 \beta_{8} - 135 \beta_{9} + 120 \beta_{10} + 50 \beta_{11} - 125 \beta_{12} - 290 \beta_{13} - 165 \beta_{14} - 39 \beta_{15} + 104 \beta_{16} - 53 \beta_{17} - 289 \beta_{18} + 10 \beta_{19} ) q^{55} + ( 1280 - 64 \beta_{1} + 64 \beta_{3} + 128 \beta_{5} + 1280 \beta_{6} - 128 \beta_{7} + 128 \beta_{8} + 64 \beta_{9} - 64 \beta_{10} - 64 \beta_{11} + 128 \beta_{13} + 64 \beta_{18} ) q^{56} + ( 2628 + 81 \beta_{2} - 189 \beta_{3} + 189 \beta_{4} - 90 \beta_{5} + 90 \beta_{6} + 108 \beta_{9} + 36 \beta_{11} + 63 \beta_{12} - 63 \beta_{13} - 36 \beta_{14} - 36 \beta_{15} + 27 \beta_{16} - 117 \beta_{17} - 108 \beta_{19} ) q^{57} + ( -1912 - 52 \beta_{1} + 56 \beta_{2} - 160 \beta_{3} + 88 \beta_{4} - 1932 \beta_{5} - 1912 \beta_{6} + 1932 \beta_{7} + 148 \beta_{8} - 84 \beta_{9} - 96 \beta_{10} - 96 \beta_{11} + 212 \beta_{12} + 48 \beta_{13} - 88 \beta_{14} - 212 \beta_{15} - 188 \beta_{16} + 40 \beta_{18} ) q^{58} + ( -284 \beta_{1} - 198 \beta_{2} + 33 \beta_{3} + 99 \beta_{4} + 6864 \beta_{5} - 7078 \beta_{6} - 7078 \beta_{7} + 5 \beta_{8} - 279 \beta_{9} - 99 \beta_{10} + 69 \beta_{11} + 16 \beta_{12} - 284 \beta_{13} + 3 \beta_{15} - 152 \beta_{16} + 8 \beta_{17} + 16 \beta_{18} - 94 \beta_{19} ) q^{59} + ( 288 - 144 \beta_{2} - 1152 \beta_{6} - 720 \beta_{7} - 144 \beta_{17} ) q^{60} + ( 159 - 207 \beta_{1} + 290 \beta_{2} + 616 \beta_{3} - 6481 \beta_{6} + 159 \beta_{7} - 278 \beta_{8} - 278 \beta_{9} + 176 \beta_{10} - 290 \beta_{12} - 42 \beta_{13} - 140 \beta_{14} + 282 \beta_{15} + 616 \beta_{17} - 42 \beta_{18} + 207 \beta_{19} ) q^{61} + ( -212 \beta_{1} + 40 \beta_{2} + 292 \beta_{3} - 36 \beta_{4} - 2880 \beta_{5} + 1416 \beta_{6} + 1416 \beta_{7} - 84 \beta_{8} - 296 \beta_{9} + 36 \beta_{10} + 124 \beta_{11} - 128 \beta_{12} - 184 \beta_{13} + 132 \beta_{15} + 44 \beta_{16} + 48 \beta_{17} - 128 \beta_{18} - 48 \beta_{19} ) q^{62} + ( -81 \beta_{1} + 81 \beta_{2} + 81 \beta_{3} + 162 \beta_{5} - 1782 \beta_{6} - 1782 \beta_{7} - 81 \beta_{9} - 81 \beta_{11} - 81 \beta_{12} - 81 \beta_{18} ) q^{63} + 4096 \beta_{6} q^{64} + ( 1969 + 85 \beta_{1} + 145 \beta_{2} - 41 \beta_{3} + 40 \beta_{4} - 16761 \beta_{5} - 458 \beta_{6} + 442 \beta_{7} - 255 \beta_{8} + 135 \beta_{9} - 40 \beta_{10} + 80 \beta_{11} - 210 \beta_{12} - 116 \beta_{13} - 50 \beta_{14} - 63 \beta_{15} + 303 \beta_{16} - 275 \beta_{17} - 138 \beta_{18} - 180 \beta_{19} ) q^{65} + ( 36 \beta_{1} + 288 \beta_{2} + 216 \beta_{3} - 72 \beta_{4} + 2160 \beta_{5} - 288 \beta_{6} - 288 \beta_{7} - 180 \beta_{8} - 144 \beta_{9} + 72 \beta_{10} + 36 \beta_{11} - 324 \beta_{12} - 252 \beta_{13} + 288 \beta_{15} + 180 \beta_{16} + 108 \beta_{17} - 324 \beta_{18} - 108 \beta_{19} ) q^{66} + ( 510 - 226 \beta_{1} - 153 \beta_{2} - 275 \beta_{3} + 245 \beta_{4} - 2712 \beta_{5} + 510 \beta_{6} + 2712 \beta_{7} + 669 \beta_{8} + 714 \beta_{9} - 443 \beta_{10} - 443 \beta_{11} + 501 \beta_{12} + 882 \beta_{13} - 245 \beta_{14} - 501 \beta_{15} + 190 \beta_{16} + 596 \beta_{18} ) q^{67} + ( 1088 - 208 \beta_{1} - 144 \beta_{2} + 128 \beta_{3} - 192 \beta_{4} + 5136 \beta_{5} - 5136 \beta_{6} - 208 \beta_{9} + 80 \beta_{12} - 80 \beta_{13} - 176 \beta_{16} + 32 \beta_{17} + 208 \beta_{19} ) q^{68} + ( 5382 + 63 \beta_{1} - 90 \beta_{2} - 252 \beta_{3} + 162 \beta_{4} - 3132 \beta_{5} + 5382 \beta_{6} + 3132 \beta_{7} - 225 \beta_{8} + 90 \beta_{9} + 162 \beta_{10} + 162 \beta_{11} + 189 \beta_{12} - 126 \beta_{13} - 162 \beta_{14} - 189 \beta_{15} + 9 \beta_{16} - 72 \beta_{18} ) q^{69} + ( -1080 - 220 \beta_{1} - 8 \beta_{2} + 60 \beta_{4} + 176 \beta_{5} + 5760 \beta_{6} + 6344 \beta_{7} + 540 \beta_{8} + 260 \beta_{9} - 200 \beta_{10} - 80 \beta_{11} + 280 \beta_{12} + 364 \beta_{13} - 120 \beta_{14} - 212 \beta_{15} + 88 \beta_{16} + 32 \beta_{17} + 372 \beta_{18} + 160 \beta_{19} ) q^{70} + ( -8998 + 376 \beta_{2} - 60 \beta_{3} + 158 \beta_{4} + 8998 \beta_{5} - 4650 \beta_{7} - 62 \beta_{8} + 60 \beta_{9} - 60 \beta_{10} + 98 \beta_{11} - 704 \beta_{12} - 474 \beta_{13} - 60 \beta_{14} + 316 \beta_{15} + 534 \beta_{16} - 20 \beta_{17} - 147 \beta_{18} - 139 \beta_{19} ) q^{71} -5184 \beta_{7} q^{72} + ( -8801 - 366 \beta_{1} - 621 \beta_{2} - 180 \beta_{3} + 13363 \beta_{6} - 8801 \beta_{7} + 67 \beta_{8} + 67 \beta_{9} + 280 \beta_{10} + 621 \beta_{12} - 267 \beta_{13} + 66 \beta_{14} - 411 \beta_{15} - 180 \beta_{17} - 267 \beta_{18} + 366 \beta_{19} ) q^{73} + ( -5352 + 80 \beta_{1} - 536 \beta_{2} + 68 \beta_{3} + 112 \beta_{4} + 292 \beta_{5} - 292 \beta_{6} - 4 \beta_{9} + 364 \beta_{11} + 248 \beta_{12} - 248 \beta_{13} - 364 \beta_{14} - 364 \beta_{15} + 12 \beta_{16} - 348 \beta_{17} + 4 \beta_{19} ) q^{74} + ( 3645 + 225 \beta_{1} - 90 \beta_{2} - 585 \beta_{3} + 450 \beta_{4} - 585 \beta_{5} + 1215 \beta_{6} + 585 \beta_{7} + 225 \beta_{8} + 675 \beta_{9} - 225 \beta_{10} + 450 \beta_{12} + 135 \beta_{13} - 225 \beta_{14} - 405 \beta_{15} + 360 \beta_{16} - 315 \beta_{17} + 315 \beta_{18} - 450 \beta_{19} ) q^{75} + ( -4672 - 144 \beta_{2} + 336 \beta_{3} - 336 \beta_{4} + 160 \beta_{5} - 160 \beta_{6} - 192 \beta_{9} - 64 \beta_{11} - 112 \beta_{12} + 112 \beta_{13} + 64 \beta_{14} + 64 \beta_{15} - 48 \beta_{16} + 208 \beta_{17} + 192 \beta_{19} ) q^{76} + ( 17420 - 307 \beta_{1} + 214 \beta_{2} + 643 \beta_{3} + 23312 \beta_{6} + 17420 \beta_{7} + 173 \beta_{8} + 173 \beta_{9} + 168 \beta_{10} - 214 \beta_{12} - 206 \beta_{13} + 70 \beta_{14} + 628 \beta_{15} + 643 \beta_{17} - 206 \beta_{18} + 307 \beta_{19} ) q^{77} + ( 1980 - 144 \beta_{2} + 36 \beta_{3} + 36 \beta_{4} - 1980 \beta_{5} - 2304 \beta_{7} - 36 \beta_{8} - 36 \beta_{9} + 36 \beta_{10} + 72 \beta_{11} + 108 \beta_{12} + 72 \beta_{13} + 36 \beta_{14} - 108 \beta_{15} - 108 \beta_{16} - 216 \beta_{17} - 36 \beta_{18} - 216 \beta_{19} ) q^{78} + ( 6542 - 460 \beta_{2} + 77 \beta_{3} - 170 \beta_{4} - 6542 \beta_{5} + 5384 \beta_{7} - 138 \beta_{8} - 77 \beta_{9} + 77 \beta_{10} - 93 \beta_{11} + 568 \beta_{12} + 553 \beta_{13} + 77 \beta_{14} - 383 \beta_{15} - 630 \beta_{16} + 1114 \beta_{17} + 380 \beta_{18} + 883 \beta_{19} ) q^{79} + ( -512 + 256 \beta_{2} + 2048 \beta_{6} + 1280 \beta_{7} + 256 \beta_{17} ) q^{80} + ( -6561 + 6561 \beta_{5} - 6561 \beta_{6} - 6561 \beta_{7} ) q^{81} + ( 10452 + 36 \beta_{1} - 76 \beta_{2} + 492 \beta_{3} - 144 \beta_{4} - 4236 \beta_{5} + 4236 \beta_{6} - 72 \beta_{9} + 136 \beta_{11} - 164 \beta_{12} + 164 \beta_{13} - 136 \beta_{14} - 136 \beta_{15} + 36 \beta_{16} - 28 \beta_{17} + 72 \beta_{19} ) q^{82} + ( -21494 + 336 \beta_{1} - 159 \beta_{2} - 1208 \beta_{3} - 179 \beta_{4} + 3372 \beta_{5} - 21494 \beta_{6} - 3372 \beta_{7} - 360 \beta_{8} - 293 \beta_{9} + 24 \beta_{10} + 24 \beta_{11} + 872 \beta_{12} + 89 \beta_{13} + 179 \beta_{14} - 872 \beta_{15} - 801 \beta_{16} + 135 \beta_{18} ) q^{83} + ( 144 \beta_{1} - 144 \beta_{2} - 144 \beta_{3} - 288 \beta_{5} + 3168 \beta_{6} + 3168 \beta_{7} + 144 \beta_{9} + 144 \beta_{11} + 144 \beta_{12} + 144 \beta_{18} ) q^{84} + ( 14213 + 310 \beta_{1} + 1059 \beta_{2} - 434 \beta_{3} - 420 \beta_{4} - 2584 \beta_{5} - 8126 \beta_{6} - 5353 \beta_{7} - 175 \beta_{8} + 115 \beta_{9} + 280 \beta_{10} - 315 \beta_{11} - 775 \beta_{12} - 107 \beta_{13} + 330 \beta_{14} + 773 \beta_{15} + 360 \beta_{16} - 186 \beta_{17} - 505 \beta_{18} + 395 \beta_{19} ) q^{85} + ( -2088 + 328 \beta_{1} + 376 \beta_{2} - 500 \beta_{3} - 14248 \beta_{6} - 2088 \beta_{7} + 80 \beta_{8} + 80 \beta_{9} - 316 \beta_{10} - 376 \beta_{12} + 104 \beta_{13} - 172 \beta_{14} + 184 \beta_{15} - 500 \beta_{17} + 104 \beta_{18} - 328 \beta_{19} ) q^{86} + ( 522 \beta_{1} + 126 \beta_{2} + 99 \beta_{3} - 198 \beta_{4} + 4347 \beta_{5} - 8649 \beta_{6} - 8649 \beta_{7} - 603 \beta_{8} - 81 \beta_{9} + 198 \beta_{10} + 216 \beta_{11} - 729 \beta_{12} - 747 \beta_{13} + 342 \beta_{15} + 423 \beta_{16} - 261 \beta_{17} - 729 \beta_{18} - 405 \beta_{19} ) q^{87} + ( -64 \beta_{1} - 512 \beta_{2} - 384 \beta_{3} + 128 \beta_{4} - 3840 \beta_{5} + 512 \beta_{6} + 512 \beta_{7} + 320 \beta_{8} + 256 \beta_{9} - 128 \beta_{10} - 64 \beta_{11} + 576 \beta_{12} + 448 \beta_{13} - 512 \beta_{15} - 320 \beta_{16} - 192 \beta_{17} + 576 \beta_{18} + 192 \beta_{19} ) q^{88} + ( 6051 + 650 \beta_{1} + 89 \beta_{2} + 63 \beta_{3} - 4692 \beta_{6} + 6051 \beta_{7} - 371 \beta_{8} - 371 \beta_{9} + 296 \beta_{10} - 89 \beta_{12} - 576 \beta_{13} + 390 \beta_{14} + 762 \beta_{15} + 63 \beta_{17} - 576 \beta_{18} - 650 \beta_{19} ) q^{89} + ( 1620 - 1620 \beta_{5} + 2268 \beta_{6} - 972 \beta_{7} + 324 \beta_{13} - 324 \beta_{16} + 324 \beta_{18} ) q^{90} + ( -403 \beta_{1} - 213 \beta_{2} + 629 \beta_{3} + 398 \beta_{4} + 8944 \beta_{5} + 20052 \beta_{6} + 20052 \beta_{7} + 1235 \beta_{8} + 832 \beta_{9} - 398 \beta_{10} - 256 \beta_{11} + 1051 \beta_{12} + 425 \beta_{13} + 15 \beta_{15} + 624 \beta_{16} + 1250 \beta_{17} + 1051 \beta_{18} + 837 \beta_{19} ) q^{91} + ( -9568 - 112 \beta_{1} + 160 \beta_{2} + 448 \beta_{3} - 288 \beta_{4} + 5568 \beta_{5} - 9568 \beta_{6} - 5568 \beta_{7} + 400 \beta_{8} - 160 \beta_{9} - 288 \beta_{10} - 288 \beta_{11} - 336 \beta_{12} + 224 \beta_{13} + 288 \beta_{14} + 336 \beta_{15} - 16 \beta_{16} + 128 \beta_{18} ) q^{92} + ( 3294 + 477 \beta_{1} + 207 \beta_{2} - 576 \beta_{3} + 198 \beta_{4} + 3186 \beta_{5} - 3186 \beta_{6} + 585 \beta_{9} - 81 \beta_{11} + 81 \beta_{12} - 81 \beta_{13} + 81 \beta_{14} + 81 \beta_{15} + 315 \beta_{16} - 189 \beta_{17} - 585 \beta_{19} ) q^{93} + ( 2760 - 204 \beta_{1} - 184 \beta_{2} + 388 \beta_{3} - 496 \beta_{4} + 3712 \beta_{5} + 2760 \beta_{6} - 3712 \beta_{7} + 528 \beta_{8} - 112 \beta_{9} - 324 \beta_{10} - 324 \beta_{11} - 184 \beta_{12} + 1140 \beta_{13} + 496 \beta_{14} + 184 \beta_{15} - 896 \beta_{16} + 508 \beta_{18} ) q^{94} + ( -35068 + 410 \beta_{1} + 1066 \beta_{2} - 228 \beta_{3} - 525 \beta_{4} + 16040 \beta_{5} - 13782 \beta_{6} - 27070 \beta_{7} - 260 \beta_{8} - 245 \beta_{9} + 390 \beta_{10} - 50 \beta_{11} - 345 \beta_{12} + 245 \beta_{13} + 770 \beta_{14} + 750 \beta_{15} + 397 \beta_{16} - 54 \beta_{17} - 297 \beta_{18} - 380 \beta_{19} ) q^{95} + 9216 \beta_{7} q^{96} + ( -25785 + 155 \beta_{2} + 982 \beta_{3} - 1396 \beta_{4} + 25785 \beta_{5} + 12039 \beta_{7} - 325 \beta_{8} - 982 \beta_{9} + 982 \beta_{10} - 414 \beta_{11} - 2263 \beta_{12} + 259 \beta_{13} + 982 \beta_{14} + 1137 \beta_{15} - 1241 \beta_{16} + 354 \beta_{17} - 357 \beta_{18} + 1449 \beta_{19} ) q^{97} + ( -4928 + 704 \beta_{1} - 44 \beta_{2} - 1108 \beta_{3} + 9892 \beta_{6} - 4928 \beta_{7} - 776 \beta_{8} - 776 \beta_{9} + 116 \beta_{10} + 44 \beta_{12} - 1104 \beta_{13} + 108 \beta_{14} - 64 \beta_{15} - 1108 \beta_{17} - 1104 \beta_{18} - 704 \beta_{19} ) q^{98} + ( -4212 - 81 \beta_{1} - 324 \beta_{3} - 81 \beta_{4} - 648 \beta_{5} + 648 \beta_{6} + 162 \beta_{9} - 162 \beta_{11} + 162 \beta_{14} + 162 \beta_{15} - 162 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 20q^{2} + 45q^{3} - 80q^{4} - 55q^{5} - 180q^{6} + 180q^{7} + 320q^{8} - 405q^{9} + O(q^{10}) \) \( 20q + 20q^{2} + 45q^{3} - 80q^{4} - 55q^{5} - 180q^{6} + 180q^{7} + 320q^{8} - 405q^{9} + 220q^{10} + 380q^{11} + 720q^{12} - 230q^{13} - 920q^{14} - 630q^{15} - 1280q^{16} + 2870q^{17} - 6480q^{18} + 1560q^{19} + 1120q^{20} + 2880q^{21} - 720q^{22} + 760q^{23} + 11520q^{24} + 1725q^{25} + 7320q^{26} + 3645q^{27} - 5120q^{28} + 12000q^{29} + 5220q^{30} - 1710q^{31} - 20480q^{32} - 1620q^{33} + 20620q^{34} - 22440q^{35} - 6480q^{36} - 5595q^{37} - 5240q^{38} + 2070q^{39} - 9280q^{40} - 2820q^{41} - 11520q^{42} + 66020q^{43} + 2880q^{44} - 8505q^{45} + 21960q^{46} - 19630q^{47} + 11520q^{48} - 61780q^{49} + 100q^{50} - 41130q^{51} - 3680q^{52} + 62735q^{53} - 14580q^{54} + 66590q^{55} + 20480q^{56} + 51660q^{57} - 48000q^{58} + 105100q^{59} + 15120q^{60} + 34790q^{61} - 28560q^{62} + 18630q^{63} - 20480q^{64} - 44345q^{65} + 13680q^{66} - 19470q^{67} + 73120q^{68} + 49410q^{69} - 81240q^{70} - 111720q^{71} + 25920q^{72} - 198830q^{73} - 104120q^{74} + 60975q^{75} - 91840q^{76} + 144740q^{77} + 41220q^{78} + 71210q^{79} - 26880q^{80} - 32805q^{81} + 166680q^{82} - 288690q^{83} - 33120q^{84} + 338735q^{85} + 39920q^{86} + 108225q^{87} - 24320q^{88} + 114225q^{89} + 17820q^{90} - 155800q^{91} - 87840q^{92} + 97740q^{93} + 78520q^{94} - 416900q^{95} - 46080q^{96} - 446970q^{97} - 123380q^{98} - 90720q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 10 x^{19} + 67300 x^{18} - 605415 x^{17} + 1505002471 x^{16} - 12026298320 x^{15} + 15161344295580 x^{14} - 105918997871590 x^{13} + 72521519492275041 x^{12} - 433752716999522710 x^{11} + 150954604904063839080 x^{10} - 750799480917962811625 x^{9} + 100084099910299358028076 x^{8} - 395836369710607771806480 x^{7} + 22144208384329171616972475 x^{6} - 65050346450081546072582845 x^{5} + 865137734840778899106150416 x^{4} - 1622318535480468844324766865 x^{3} + 11299132807307021058314145735 x^{2} - 10498749516125580450644510315 x + 45940690246932155189634577205\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(53\!\cdots\!68\)\( \nu^{18} - \)\(48\!\cdots\!12\)\( \nu^{17} + \)\(36\!\cdots\!31\)\( \nu^{16} - \)\(29\!\cdots\!76\)\( \nu^{15} + \)\(81\!\cdots\!18\)\( \nu^{14} - \)\(56\!\cdots\!98\)\( \nu^{13} + \)\(81\!\cdots\!00\)\( \nu^{12} - \)\(48\!\cdots\!98\)\( \nu^{11} + \)\(38\!\cdots\!80\)\( \nu^{10} - \)\(19\!\cdots\!88\)\( \nu^{9} + \)\(80\!\cdots\!67\)\( \nu^{8} - \)\(32\!\cdots\!86\)\( \nu^{7} + \)\(52\!\cdots\!14\)\( \nu^{6} - \)\(15\!\cdots\!42\)\( \nu^{5} + \)\(11\!\cdots\!27\)\( \nu^{4} - \)\(21\!\cdots\!90\)\( \nu^{3} + \)\(29\!\cdots\!35\)\( \nu^{2} - \)\(28\!\cdots\!50\)\( \nu + \)\(20\!\cdots\!50\)\(\)\()/ \)\(39\!\cdots\!85\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(15\!\cdots\!66\)\( \nu^{18} - \)\(14\!\cdots\!94\)\( \nu^{17} + \)\(10\!\cdots\!57\)\( \nu^{16} - \)\(85\!\cdots\!92\)\( \nu^{15} + \)\(23\!\cdots\!41\)\( \nu^{14} - \)\(16\!\cdots\!51\)\( \nu^{13} + \)\(23\!\cdots\!85\)\( \nu^{12} - \)\(14\!\cdots\!51\)\( \nu^{11} + \)\(11\!\cdots\!70\)\( \nu^{10} - \)\(57\!\cdots\!06\)\( \nu^{9} + \)\(23\!\cdots\!14\)\( \nu^{8} - \)\(94\!\cdots\!07\)\( \nu^{7} + \)\(15\!\cdots\!93\)\( \nu^{6} - \)\(46\!\cdots\!99\)\( \nu^{5} + \)\(33\!\cdots\!94\)\( \nu^{4} - \)\(65\!\cdots\!00\)\( \nu^{3} + \)\(99\!\cdots\!35\)\( \nu^{2} - \)\(96\!\cdots\!55\)\( \nu + \)\(71\!\cdots\!00\)\(\)\()/ \)\(39\!\cdots\!85\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(19\!\cdots\!69\)\( \nu^{18} + \)\(17\!\cdots\!21\)\( \nu^{17} - \)\(12\!\cdots\!48\)\( \nu^{16} + \)\(10\!\cdots\!08\)\( \nu^{15} - \)\(29\!\cdots\!04\)\( \nu^{14} + \)\(20\!\cdots\!04\)\( \nu^{13} - \)\(29\!\cdots\!50\)\( \nu^{12} + \)\(17\!\cdots\!74\)\( \nu^{11} - \)\(13\!\cdots\!85\)\( \nu^{10} + \)\(69\!\cdots\!89\)\( \nu^{9} - \)\(28\!\cdots\!51\)\( \nu^{8} + \)\(11\!\cdots\!18\)\( \nu^{7} - \)\(18\!\cdots\!72\)\( \nu^{6} + \)\(56\!\cdots\!86\)\( \nu^{5} - \)\(39\!\cdots\!76\)\( \nu^{4} + \)\(78\!\cdots\!50\)\( \nu^{3} - \)\(10\!\cdots\!65\)\( \nu^{2} + \)\(10\!\cdots\!70\)\( \nu - \)\(67\!\cdots\!70\)\(\)\()/ \)\(39\!\cdots\!85\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(39\!\cdots\!96\)\( \nu^{18} - \)\(35\!\cdots\!64\)\( \nu^{17} + \)\(26\!\cdots\!62\)\( \nu^{16} - \)\(21\!\cdots\!12\)\( \nu^{15} + \)\(59\!\cdots\!21\)\( \nu^{14} - \)\(41\!\cdots\!31\)\( \nu^{13} + \)\(60\!\cdots\!35\)\( \nu^{12} - \)\(36\!\cdots\!11\)\( \nu^{11} + \)\(28\!\cdots\!60\)\( \nu^{10} - \)\(14\!\cdots\!36\)\( \nu^{9} + \)\(59\!\cdots\!24\)\( \nu^{8} - \)\(23\!\cdots\!27\)\( \nu^{7} + \)\(38\!\cdots\!53\)\( \nu^{6} - \)\(11\!\cdots\!69\)\( \nu^{5} + \)\(82\!\cdots\!04\)\( \nu^{4} - \)\(16\!\cdots\!00\)\( \nu^{3} + \)\(22\!\cdots\!75\)\( \nu^{2} - \)\(21\!\cdots\!80\)\( \nu + \)\(14\!\cdots\!30\)\(\)\()/ \)\(39\!\cdots\!85\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(40\!\cdots\!33\)\( \nu^{19} + \)\(36\!\cdots\!21\)\( \nu^{18} - \)\(27\!\cdots\!92\)\( \nu^{17} + \)\(21\!\cdots\!79\)\( \nu^{16} - \)\(60\!\cdots\!26\)\( \nu^{15} + \)\(42\!\cdots\!12\)\( \nu^{14} - \)\(61\!\cdots\!64\)\( \nu^{13} + \)\(36\!\cdots\!98\)\( \nu^{12} - \)\(29\!\cdots\!49\)\( \nu^{11} + \)\(14\!\cdots\!28\)\( \nu^{10} - \)\(60\!\cdots\!96\)\( \nu^{9} + \)\(23\!\cdots\!17\)\( \nu^{8} - \)\(39\!\cdots\!17\)\( \nu^{7} + \)\(11\!\cdots\!99\)\( \nu^{6} - \)\(83\!\cdots\!73\)\( \nu^{5} + \)\(16\!\cdots\!06\)\( \nu^{4} - \)\(23\!\cdots\!65\)\( \nu^{3} + \)\(25\!\cdots\!75\)\( \nu^{2} - \)\(16\!\cdots\!80\)\( \nu + \)\(43\!\cdots\!80\)\(\)\()/ \)\(16\!\cdots\!25\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(40\!\cdots\!33\)\( \nu^{19} + \)\(40\!\cdots\!06\)\( \nu^{18} - \)\(27\!\cdots\!57\)\( \nu^{17} + \)\(24\!\cdots\!49\)\( \nu^{16} - \)\(60\!\cdots\!46\)\( \nu^{15} + \)\(48\!\cdots\!22\)\( \nu^{14} - \)\(61\!\cdots\!74\)\( \nu^{13} + \)\(42\!\cdots\!48\)\( \nu^{12} - \)\(29\!\cdots\!09\)\( \nu^{11} + \)\(17\!\cdots\!53\)\( \nu^{10} - \)\(60\!\cdots\!31\)\( \nu^{9} + \)\(30\!\cdots\!82\)\( \nu^{8} - \)\(39\!\cdots\!37\)\( \nu^{7} + \)\(15\!\cdots\!04\)\( \nu^{6} - \)\(84\!\cdots\!88\)\( \nu^{5} + \)\(25\!\cdots\!21\)\( \nu^{4} - \)\(25\!\cdots\!90\)\( \nu^{3} + \)\(44\!\cdots\!25\)\( \nu^{2} - \)\(18\!\cdots\!05\)\( \nu + \)\(12\!\cdots\!80\)\(\)\()/ \)\(16\!\cdots\!25\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(74\!\cdots\!49\)\( \nu^{19} - \)\(73\!\cdots\!58\)\( \nu^{18} + \)\(50\!\cdots\!81\)\( \nu^{17} - \)\(44\!\cdots\!77\)\( \nu^{16} + \)\(11\!\cdots\!18\)\( \nu^{15} - \)\(87\!\cdots\!56\)\( \nu^{14} + \)\(11\!\cdots\!62\)\( \nu^{13} - \)\(76\!\cdots\!19\)\( \nu^{12} + \)\(53\!\cdots\!67\)\( \nu^{11} - \)\(31\!\cdots\!34\)\( \nu^{10} + \)\(11\!\cdots\!83\)\( \nu^{9} - \)\(53\!\cdots\!31\)\( \nu^{8} + \)\(73\!\cdots\!91\)\( \nu^{7} - \)\(27\!\cdots\!57\)\( \nu^{6} + \)\(15\!\cdots\!24\)\( \nu^{5} - \)\(42\!\cdots\!23\)\( \nu^{4} + \)\(45\!\cdots\!20\)\( \nu^{3} - \)\(72\!\cdots\!00\)\( \nu^{2} + \)\(31\!\cdots\!65\)\( \nu - \)\(19\!\cdots\!15\)\(\)\()/ \)\(16\!\cdots\!25\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(37\!\cdots\!51\)\( \nu^{19} + \)\(15\!\cdots\!27\)\( \nu^{18} - \)\(25\!\cdots\!09\)\( \nu^{17} + \)\(10\!\cdots\!18\)\( \nu^{16} - \)\(56\!\cdots\!02\)\( \nu^{15} + \)\(21\!\cdots\!04\)\( \nu^{14} - \)\(57\!\cdots\!98\)\( \nu^{13} + \)\(21\!\cdots\!81\)\( \nu^{12} - \)\(28\!\cdots\!43\)\( \nu^{11} + \)\(99\!\cdots\!41\)\( \nu^{10} - \)\(59\!\cdots\!27\)\( \nu^{9} + \)\(20\!\cdots\!09\)\( \nu^{8} - \)\(42\!\cdots\!54\)\( \nu^{7} + \)\(12\!\cdots\!73\)\( \nu^{6} - \)\(10\!\cdots\!41\)\( \nu^{5} + \)\(25\!\cdots\!42\)\( \nu^{4} - \)\(55\!\cdots\!05\)\( \nu^{3} + \)\(65\!\cdots\!75\)\( \nu^{2} - \)\(55\!\cdots\!60\)\( \nu + \)\(41\!\cdots\!10\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(85\!\cdots\!28\)\( \nu^{19} + \)\(89\!\cdots\!66\)\( \nu^{18} - \)\(57\!\cdots\!17\)\( \nu^{17} + \)\(54\!\cdots\!49\)\( \nu^{16} - \)\(12\!\cdots\!51\)\( \nu^{15} + \)\(10\!\cdots\!22\)\( \nu^{14} - \)\(12\!\cdots\!04\)\( \nu^{13} + \)\(96\!\cdots\!68\)\( \nu^{12} - \)\(61\!\cdots\!64\)\( \nu^{11} + \)\(39\!\cdots\!48\)\( \nu^{10} - \)\(12\!\cdots\!16\)\( \nu^{9} + \)\(69\!\cdots\!92\)\( \nu^{8} - \)\(83\!\cdots\!07\)\( \nu^{7} + \)\(37\!\cdots\!24\)\( \nu^{6} - \)\(17\!\cdots\!13\)\( \nu^{5} + \)\(62\!\cdots\!66\)\( \nu^{4} - \)\(49\!\cdots\!15\)\( \nu^{3} + \)\(17\!\cdots\!00\)\( \nu^{2} - \)\(43\!\cdots\!05\)\( \nu + \)\(11\!\cdots\!80\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(60\!\cdots\!27\)\( \nu^{19} + \)\(18\!\cdots\!99\)\( \nu^{18} + \)\(40\!\cdots\!96\)\( \nu^{17} + \)\(12\!\cdots\!40\)\( \nu^{16} + \)\(90\!\cdots\!48\)\( \nu^{15} + \)\(29\!\cdots\!20\)\( \nu^{14} + \)\(89\!\cdots\!58\)\( \nu^{13} + \)\(30\!\cdots\!93\)\( \nu^{12} + \)\(41\!\cdots\!73\)\( \nu^{11} + \)\(15\!\cdots\!13\)\( \nu^{10} + \)\(82\!\cdots\!16\)\( \nu^{9} + \)\(32\!\cdots\!39\)\( \nu^{8} + \)\(45\!\cdots\!42\)\( \nu^{7} + \)\(21\!\cdots\!53\)\( \nu^{6} + \)\(58\!\cdots\!85\)\( \nu^{5} + \)\(47\!\cdots\!78\)\( \nu^{4} - \)\(57\!\cdots\!30\)\( \nu^{3} + \)\(13\!\cdots\!40\)\( \nu^{2} - \)\(10\!\cdots\!75\)\( \nu + \)\(94\!\cdots\!40\)\(\)\()/ \)\(24\!\cdots\!95\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(23\!\cdots\!21\)\( \nu^{19} + \)\(12\!\cdots\!87\)\( \nu^{18} - \)\(16\!\cdots\!69\)\( \nu^{17} + \)\(66\!\cdots\!18\)\( \nu^{16} - \)\(35\!\cdots\!82\)\( \nu^{15} + \)\(11\!\cdots\!54\)\( \nu^{14} - \)\(35\!\cdots\!78\)\( \nu^{13} + \)\(77\!\cdots\!26\)\( \nu^{12} - \)\(17\!\cdots\!23\)\( \nu^{11} + \)\(19\!\cdots\!11\)\( \nu^{10} - \)\(35\!\cdots\!87\)\( \nu^{9} + \)\(49\!\cdots\!94\)\( \nu^{8} - \)\(22\!\cdots\!49\)\( \nu^{7} - \)\(20\!\cdots\!32\)\( \nu^{6} - \)\(46\!\cdots\!16\)\( \nu^{5} - \)\(92\!\cdots\!88\)\( \nu^{4} - \)\(96\!\cdots\!80\)\( \nu^{3} - \)\(39\!\cdots\!25\)\( \nu^{2} - \)\(35\!\cdots\!35\)\( \nu - \)\(34\!\cdots\!65\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(30\!\cdots\!78\)\( \nu^{19} + \)\(19\!\cdots\!96\)\( \nu^{18} - \)\(20\!\cdots\!37\)\( \nu^{17} + \)\(11\!\cdots\!09\)\( \nu^{16} - \)\(45\!\cdots\!86\)\( \nu^{15} + \)\(20\!\cdots\!27\)\( \nu^{14} - \)\(45\!\cdots\!84\)\( \nu^{13} + \)\(15\!\cdots\!68\)\( \nu^{12} - \)\(21\!\cdots\!19\)\( \nu^{11} + \)\(52\!\cdots\!73\)\( \nu^{10} - \)\(45\!\cdots\!71\)\( \nu^{9} + \)\(62\!\cdots\!62\)\( \nu^{8} - \)\(29\!\cdots\!67\)\( \nu^{7} + \)\(11\!\cdots\!89\)\( \nu^{6} - \)\(61\!\cdots\!08\)\( \nu^{5} - \)\(36\!\cdots\!89\)\( \nu^{4} - \)\(15\!\cdots\!15\)\( \nu^{3} - \)\(22\!\cdots\!50\)\( \nu^{2} - \)\(98\!\cdots\!80\)\( \nu - \)\(25\!\cdots\!70\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(30\!\cdots\!78\)\( \nu^{19} + \)\(38\!\cdots\!86\)\( \nu^{18} - \)\(20\!\cdots\!47\)\( \nu^{17} + \)\(23\!\cdots\!14\)\( \nu^{16} - \)\(45\!\cdots\!66\)\( \nu^{15} + \)\(48\!\cdots\!67\)\( \nu^{14} - \)\(46\!\cdots\!24\)\( \nu^{13} + \)\(44\!\cdots\!93\)\( \nu^{12} - \)\(22\!\cdots\!09\)\( \nu^{11} + \)\(18\!\cdots\!23\)\( \nu^{10} - \)\(45\!\cdots\!36\)\( \nu^{9} + \)\(34\!\cdots\!72\)\( \nu^{8} - \)\(30\!\cdots\!47\)\( \nu^{7} + \)\(19\!\cdots\!09\)\( \nu^{6} - \)\(66\!\cdots\!93\)\( \nu^{5} + \)\(35\!\cdots\!96\)\( \nu^{4} - \)\(23\!\cdots\!15\)\( \nu^{3} + \)\(78\!\cdots\!00\)\( \nu^{2} - \)\(19\!\cdots\!05\)\( \nu + \)\(39\!\cdots\!30\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(37\!\cdots\!18\)\( \nu^{19} + \)\(63\!\cdots\!71\)\( \nu^{18} - \)\(25\!\cdots\!27\)\( \nu^{17} + \)\(39\!\cdots\!69\)\( \nu^{16} - \)\(57\!\cdots\!31\)\( \nu^{15} + \)\(83\!\cdots\!32\)\( \nu^{14} - \)\(57\!\cdots\!74\)\( \nu^{13} + \)\(78\!\cdots\!58\)\( \nu^{12} - \)\(27\!\cdots\!59\)\( \nu^{11} + \)\(34\!\cdots\!13\)\( \nu^{10} - \)\(57\!\cdots\!71\)\( \nu^{9} + \)\(66\!\cdots\!77\)\( \nu^{8} - \)\(38\!\cdots\!17\)\( \nu^{7} + \)\(39\!\cdots\!19\)\( \nu^{6} - \)\(86\!\cdots\!78\)\( \nu^{5} + \)\(75\!\cdots\!21\)\( \nu^{4} - \)\(32\!\cdots\!15\)\( \nu^{3} + \)\(18\!\cdots\!25\)\( \nu^{2} - \)\(28\!\cdots\!30\)\( \nu + \)\(11\!\cdots\!05\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(41\!\cdots\!39\)\( \nu^{19} + \)\(43\!\cdots\!43\)\( \nu^{18} - \)\(28\!\cdots\!36\)\( \nu^{17} + \)\(26\!\cdots\!07\)\( \nu^{16} - \)\(62\!\cdots\!08\)\( \nu^{15} + \)\(53\!\cdots\!71\)\( \nu^{14} - \)\(63\!\cdots\!37\)\( \nu^{13} + \)\(47\!\cdots\!84\)\( \nu^{12} - \)\(30\!\cdots\!17\)\( \nu^{11} + \)\(19\!\cdots\!24\)\( \nu^{10} - \)\(62\!\cdots\!18\)\( \nu^{9} + \)\(34\!\cdots\!11\)\( \nu^{8} - \)\(41\!\cdots\!86\)\( \nu^{7} + \)\(18\!\cdots\!42\)\( \nu^{6} - \)\(89\!\cdots\!09\)\( \nu^{5} + \)\(29\!\cdots\!73\)\( \nu^{4} - \)\(28\!\cdots\!20\)\( \nu^{3} + \)\(57\!\cdots\!00\)\( \nu^{2} - \)\(20\!\cdots\!65\)\( \nu + \)\(19\!\cdots\!15\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(55\!\cdots\!36\)\( \nu^{19} + \)\(50\!\cdots\!27\)\( \nu^{18} - \)\(37\!\cdots\!94\)\( \nu^{17} + \)\(30\!\cdots\!08\)\( \nu^{16} - \)\(84\!\cdots\!57\)\( \nu^{15} + \)\(59\!\cdots\!24\)\( \nu^{14} - \)\(84\!\cdots\!58\)\( \nu^{13} + \)\(51\!\cdots\!41\)\( \nu^{12} - \)\(40\!\cdots\!78\)\( \nu^{11} + \)\(20\!\cdots\!51\)\( \nu^{10} - \)\(83\!\cdots\!27\)\( \nu^{9} + \)\(33\!\cdots\!44\)\( \nu^{8} - \)\(54\!\cdots\!29\)\( \nu^{7} + \)\(16\!\cdots\!68\)\( \nu^{6} - \)\(11\!\cdots\!96\)\( \nu^{5} + \)\(23\!\cdots\!82\)\( \nu^{4} - \)\(32\!\cdots\!80\)\( \nu^{3} + \)\(30\!\cdots\!00\)\( \nu^{2} - \)\(21\!\cdots\!10\)\( \nu - \)\(12\!\cdots\!90\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(55\!\cdots\!36\)\( \nu^{19} - \)\(55\!\cdots\!57\)\( \nu^{18} + \)\(37\!\cdots\!64\)\( \nu^{17} - \)\(33\!\cdots\!43\)\( \nu^{16} + \)\(84\!\cdots\!17\)\( \nu^{15} - \)\(66\!\cdots\!79\)\( \nu^{14} + \)\(84\!\cdots\!63\)\( \nu^{13} - \)\(58\!\cdots\!16\)\( \nu^{12} + \)\(40\!\cdots\!83\)\( \nu^{11} - \)\(23\!\cdots\!26\)\( \nu^{10} + \)\(83\!\cdots\!82\)\( \nu^{9} - \)\(41\!\cdots\!14\)\( \nu^{8} + \)\(54\!\cdots\!64\)\( \nu^{7} - \)\(21\!\cdots\!58\)\( \nu^{6} + \)\(11\!\cdots\!16\)\( \nu^{5} - \)\(34\!\cdots\!77\)\( \nu^{4} + \)\(34\!\cdots\!80\)\( \nu^{3} - \)\(64\!\cdots\!00\)\( \nu^{2} + \)\(25\!\cdots\!85\)\( \nu - \)\(23\!\cdots\!10\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(62\!\cdots\!14\)\( \nu^{19} - \)\(51\!\cdots\!68\)\( \nu^{18} + \)\(41\!\cdots\!61\)\( \nu^{17} - \)\(30\!\cdots\!07\)\( \nu^{16} + \)\(93\!\cdots\!83\)\( \nu^{15} - \)\(58\!\cdots\!46\)\( \nu^{14} + \)\(94\!\cdots\!12\)\( \nu^{13} - \)\(49\!\cdots\!09\)\( \nu^{12} + \)\(44\!\cdots\!67\)\( \nu^{11} - \)\(19\!\cdots\!24\)\( \nu^{10} + \)\(92\!\cdots\!43\)\( \nu^{9} - \)\(30\!\cdots\!86\)\( \nu^{8} + \)\(60\!\cdots\!86\)\( \nu^{7} - \)\(13\!\cdots\!92\)\( \nu^{6} + \)\(12\!\cdots\!84\)\( \nu^{5} - \)\(16\!\cdots\!23\)\( \nu^{4} + \)\(34\!\cdots\!45\)\( \nu^{3} - \)\(60\!\cdots\!25\)\( \nu^{2} + \)\(22\!\cdots\!15\)\( \nu + \)\(21\!\cdots\!85\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(64\!\cdots\!64\)\( \nu^{19} + \)\(62\!\cdots\!23\)\( \nu^{18} - \)\(43\!\cdots\!81\)\( \nu^{17} + \)\(38\!\cdots\!17\)\( \nu^{16} - \)\(96\!\cdots\!68\)\( \nu^{15} + \)\(75\!\cdots\!26\)\( \nu^{14} - \)\(97\!\cdots\!42\)\( \nu^{13} + \)\(65\!\cdots\!59\)\( \nu^{12} - \)\(46\!\cdots\!22\)\( \nu^{11} + \)\(26\!\cdots\!49\)\( \nu^{10} - \)\(96\!\cdots\!23\)\( \nu^{9} + \)\(45\!\cdots\!81\)\( \nu^{8} - \)\(63\!\cdots\!96\)\( \nu^{7} + \)\(23\!\cdots\!32\)\( \nu^{6} - \)\(13\!\cdots\!29\)\( \nu^{5} + \)\(36\!\cdots\!18\)\( \nu^{4} - \)\(38\!\cdots\!95\)\( \nu^{3} + \)\(60\!\cdots\!25\)\( \nu^{2} - \)\(27\!\cdots\!15\)\( \nu + \)\(14\!\cdots\!15\)\(\)\()/ \)\(37\!\cdots\!25\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{19} - 6 \beta_{18} - 2 \beta_{17} - 2 \beta_{16} + \beta_{15} + \beta_{14} - 5 \beta_{13} - 5 \beta_{12} + \beta_{11} + 4 \beta_{10} - 5 \beta_{9} - 8 \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} + 3\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(479 \beta_{19} - 6 \beta_{18} + 715 \beta_{17} + 164 \beta_{16} + 72 \beta_{15} + 72 \beta_{14} - 95 \beta_{13} + 85 \beta_{12} - 70 \beta_{11} + 4 \beta_{10} - 485 \beta_{9} - 8 \beta_{8} + \beta_{7} - 15121 \beta_{6} + 15124 \beta_{5} - 135 \beta_{4} + 52 \beta_{3} + 512 \beta_{2} + 62 \beta_{1} - 41186\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(73144 \beta_{19} + 127678 \beta_{18} + 94620 \beta_{17} + 30914 \beta_{16} + 5273 \beta_{15} - 2547 \beta_{14} + 101173 \beta_{13} + 101443 \beta_{12} - 2760 \beta_{11} - 46619 \beta_{10} + 118323 \beta_{9} + 159075 \beta_{8} + 2857518 \beta_{7} + 3259355 \beta_{6} + 447206 \beta_{5} + 23110 \beta_{4} + 57564 \beta_{3} - 35853 \beta_{2} - 56138 \beta_{1} + 1366978\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(-12782616 \beta_{19} + 255362 \beta_{18} - 19400826 \beta_{17} - 7643713 \beta_{16} + 1054925 \beta_{15} + 1039285 \beta_{14} + 13577353 \beta_{13} - 13172111 \beta_{12} - 1049901 \beta_{11} - 93242 \beta_{10} + 13165556 \beta_{9} + 318158 \beta_{8} + 5715035 \beta_{7} + 560360360 \beta_{6} - 552947241 \beta_{5} - 1945033 \beta_{4} + 7992869 \beta_{3} - 10660682 \beta_{2} - 5340193 \beta_{1} + 1038724838\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(-2875933971 \beta_{19} - 3770461473 \beta_{18} - 3925309999 \beta_{17} - 637850009 \beta_{16} - 824345837 \beta_{15} - 112290440 \beta_{14} - 2777741908 \beta_{13} - 2844616018 \beta_{12} - 117513050 \beta_{11} + 824354145 \beta_{10} - 3635648391 \beta_{9} - 4558628464 \beta_{8} - 108779611750 \beta_{7} - 134631890345 \beta_{6} - 28621335625 \beta_{5} - 417155875 \beta_{4} - 2318371988 \beta_{3} + 1090556083 \beta_{2} + 1854067672 \beta_{1} - 51800034605\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(407271823029 \beta_{19} - 11312022827 \beta_{18} + 596082402258 \beta_{17} + 266865262882 \beta_{16} - 79293216662 \beta_{15} - 77157011371 \beta_{14} - 564375523258 \beta_{13} + 547507436370 \beta_{12} + 76467627442 \beta_{11} + 2473295542 \beta_{10} - 426807527468 \beta_{9} - 13676680791 \beta_{8} - 326353122837 \beta_{7} - 20023674749860 \beta_{6} + 19533896539154 \beta_{5} + 129164714401 \beta_{4} - 378777286367 \beta_{3} + 300792669181 \beta_{2} + 233164867477 \beta_{1} - 33167410765564\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(103942207979079 \beta_{19} + 125492989445627 \beta_{18} + 145648312616615 \beta_{17} + 18571609580247 \beta_{16} + 36697714996121 \beta_{15} + 5221463255880 \beta_{14} + 88465036998142 \beta_{13} + 92356861416540 \beta_{12} + 5759167770612 \beta_{11} - 21698816139786 \beta_{10} + 122730176685805 \beta_{9} + 150535200379242 \beta_{8} + 3857308208861117 \beta_{7} + 4905981253609563 \beta_{6} + 1185992000082601 \beta_{5} + 11305830263071 \beta_{4} + 84345394775071 \beta_{3} - 35117823208673 \beta_{2} - 63621675264433 \beta_{1} + 1813130220444526\)\()/5\)
\(\nu^{8}\)\(=\)\((\)\(-13842091634976246 \beta_{19} + 502024747818216 \beta_{18} - 19863187805628261 \beta_{17} - 9272202344431941 \beta_{16} + 3305364833424614 \beta_{15} + 3179449857469132 \beta_{14} + 20736794945362629 \beta_{13} - 20013428633019519 \beta_{12} - 3135524116262934 \beta_{11} - 86806806822574 \beta_{10} + 14748872341150028 \beta_{9} + 602204626769700 \beta_{8} + 15430755830019455 \beta_{7} + 710128405362732351 \beta_{6} - 685758226699016458 \beta_{5} - 5146121823564637 \beta_{4} + 14393453173070906 \beta_{3} - 9869463026340296 \beta_{2} - 8759533318090158 \beta_{1} + 1136450780209231125\)\()/5\)
\(\nu^{9}\)\(=\)\((\)\(-3695513010394813661 \beta_{19} - 4343930651521299044 \beta_{18} - 5224591443921971412 \beta_{17} - 623422536314347559 \beta_{16} - 1374929497068435096 \beta_{15} - 188996745689043328 \beta_{14} - 2998460386016271955 \beta_{13} - 3181859743346373811 \beta_{12} - 217417354819860553 \beta_{11} + 693475569813778383 \beta_{10} - 4260702407590637309 \beta_{9} - 5203542622590426787 \beta_{8} - 135921643415500440007 \beta_{7} - 174000381131149096091 \beta_{6} - 44291613084053573371 \beta_{5} - 370093386982350489 \beta_{4} - 2987970392247062398 \beta_{3} + 1188493266280253673 \beta_{2} + 2216770492395775727 \beta_{1} - 62880819243245144623\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(479836068175989206254 \beta_{19} - 21723418522400568448 \beta_{18} + 683539070185226970327 \beta_{17} + 324592822553215183066 \beta_{16} - 123236595491872756846 \beta_{15} - 117305987357702616654 \beta_{14} - 741080286579040534855 \beta_{13} + 710173260566806098259 \beta_{12} + 115273587407273416344 \beta_{11} + 3468028917433596230 \beta_{10} - 519623946257964610251 \beta_{9} - 26022229743391263024 \beta_{8} - 679723950030727780980 \beta_{7} - 25171903133021231919285 \beta_{6} + 24080260382176756162245 \beta_{5} + 187356604894633221727 \beta_{4} - 519984676026358624398 \beta_{3} + 341144705608932788058 \beta_{2} + 315899511700433993017 \beta_{1} - 39751180186230190798940\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(130865106710435947710514 \beta_{19} + 152188142473817684131306 \beta_{18} + 185531260562466637476268 \beta_{17} + 21888455589211535053793 \beta_{16} + 49241171243796784489259 \beta_{15} + 6582149837139629871372 \beta_{14} + 103828055870162952223994 \beta_{13} + 111811631583051438858874 \beta_{12} + 7861598026512170987014 \beta_{11} - 23710164828296866314008 \beta_{10} + 149096139629789588752717 \beta_{9} + 182296993979596052720774 \beta_{8} + 4787903496107920783154726 \beta_{7} + 6125198308898388831145278 \beta_{6} + 1604500183266970659937939 \beta_{5} + 12895294918631582218717 \beta_{4} + 105068226134381838040547 \beta_{3} - 41066674170434819799714 \beta_{2} - 77617640560328495624683 \beta_{1} + 2177142929932229668719775\)\()/5\)
\(\nu^{12}\)\(=\)\((\)\(-16731238997072047594581246 \beta_{19} + 913367820730309718757025 \beta_{18} - 23776489534328747013103407 \beta_{17} - 11398309967357291615689305 \beta_{16} + 4451866083406179956896328 \beta_{15} + 4195846716199117057606979 \beta_{14} + 26288888631409624442422429 \beta_{13} - 24994710517468288258863271 \beta_{12} - 4109161871892961635079722 \beta_{11} - 142299138720239694055486 \beta_{10} + 18411444156734613986488200 \beta_{9} + 1094068218341430854145007 \beta_{8} + 28734898194712513762840630 \beta_{7} + 892567303376877065258064198 \beta_{6} - 846177103951506895883973241 \beta_{5} - 6657851053825384856507286 \beta_{4} + 18526240972950589339308795 \beta_{3} - 11980237238127709239502603 \beta_{2} - 11259305183309247648527418 \beta_{1} + 1398930716721958535863563030\)\()/5\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(46\!\cdots\!81\)\( \beta_{19} - \)\(53\!\cdots\!12\)\( \beta_{18} - \)\(65\!\cdots\!91\)\( \beta_{17} - \)\(77\!\cdots\!01\)\( \beta_{16} - \)\(17\!\cdots\!02\)\( \beta_{15} - \)\(22\!\cdots\!86\)\( \beta_{14} - \)\(36\!\cdots\!89\)\( \beta_{13} - \)\(39\!\cdots\!20\)\( \beta_{12} - \)\(28\!\cdots\!41\)\( \beta_{11} + \)\(82\!\cdots\!46\)\( \beta_{10} - \)\(52\!\cdots\!22\)\( \beta_{9} - \)\(64\!\cdots\!00\)\( \beta_{8} - \)\(16\!\cdots\!89\)\( \beta_{7} - \)\(21\!\cdots\!64\)\( \beta_{6} - \)\(57\!\cdots\!10\)\( \beta_{5} - \)\(45\!\cdots\!53\)\( \beta_{4} - \)\(36\!\cdots\!08\)\( \beta_{3} + \)\(14\!\cdots\!69\)\( \beta_{2} + \)\(27\!\cdots\!92\)\( \beta_{1} - \)\(75\!\cdots\!77\)\(\)\()/5\)
\(\nu^{14}\)\(=\)\((\)\(\)\(58\!\cdots\!39\)\( \beta_{19} - \)\(37\!\cdots\!28\)\( \beta_{18} + \)\(82\!\cdots\!07\)\( \beta_{17} + \)\(40\!\cdots\!29\)\( \beta_{16} - \)\(15\!\cdots\!59\)\( \beta_{15} - \)\(14\!\cdots\!22\)\( \beta_{14} - \)\(93\!\cdots\!53\)\( \beta_{13} + \)\(87\!\cdots\!38\)\( \beta_{12} + \)\(14\!\cdots\!01\)\( \beta_{11} + \)\(57\!\cdots\!73\)\( \beta_{10} - \)\(65\!\cdots\!74\)\( \beta_{9} - \)\(44\!\cdots\!02\)\( \beta_{8} - \)\(11\!\cdots\!58\)\( \beta_{7} - \)\(31\!\cdots\!17\)\( \beta_{6} + \)\(29\!\cdots\!08\)\( \beta_{5} + \)\(23\!\cdots\!83\)\( \beta_{4} - \)\(65\!\cdots\!50\)\( \beta_{3} + \)\(42\!\cdots\!41\)\( \beta_{2} + \)\(39\!\cdots\!47\)\( \beta_{1} - \)\(49\!\cdots\!97\)\(\)\()/5\)
\(\nu^{15}\)\(=\)\((\)\(\)\(16\!\cdots\!29\)\( \beta_{19} + \)\(18\!\cdots\!20\)\( \beta_{18} + \)\(23\!\cdots\!19\)\( \beta_{17} + \)\(27\!\cdots\!27\)\( \beta_{16} + \)\(61\!\cdots\!77\)\( \beta_{15} + \)\(78\!\cdots\!19\)\( \beta_{14} + \)\(12\!\cdots\!19\)\( \beta_{13} + \)\(14\!\cdots\!65\)\( \beta_{12} + \)\(10\!\cdots\!36\)\( \beta_{11} - \)\(29\!\cdots\!35\)\( \beta_{10} + \)\(18\!\cdots\!14\)\( \beta_{9} + \)\(22\!\cdots\!16\)\( \beta_{8} + \)\(59\!\cdots\!80\)\( \beta_{7} + \)\(75\!\cdots\!80\)\( \beta_{6} + \)\(20\!\cdots\!60\)\( \beta_{5} + \)\(16\!\cdots\!73\)\( \beta_{4} + \)\(12\!\cdots\!22\)\( \beta_{3} - \)\(49\!\cdots\!61\)\( \beta_{2} - \)\(95\!\cdots\!48\)\( \beta_{1} + \)\(26\!\cdots\!25\)\(\)\()/5\)
\(\nu^{16}\)\(=\)\((\)\(-\)\(20\!\cdots\!41\)\( \beta_{19} + \)\(15\!\cdots\!34\)\( \beta_{18} - \)\(28\!\cdots\!84\)\( \beta_{17} - \)\(14\!\cdots\!20\)\( \beta_{16} + \)\(56\!\cdots\!97\)\( \beta_{15} + \)\(52\!\cdots\!01\)\( \beta_{14} + \)\(32\!\cdots\!93\)\( \beta_{13} - \)\(30\!\cdots\!07\)\( \beta_{12} - \)\(51\!\cdots\!25\)\( \beta_{11} - \)\(23\!\cdots\!78\)\( \beta_{10} + \)\(23\!\cdots\!27\)\( \beta_{9} + \)\(18\!\cdots\!14\)\( \beta_{8} + \)\(47\!\cdots\!65\)\( \beta_{7} + \)\(11\!\cdots\!86\)\( \beta_{6} - \)\(10\!\cdots\!65\)\( \beta_{5} - \)\(82\!\cdots\!95\)\( \beta_{4} + \)\(23\!\cdots\!81\)\( \beta_{3} - \)\(14\!\cdots\!28\)\( \beta_{2} - \)\(14\!\cdots\!08\)\( \beta_{1} + \)\(17\!\cdots\!43\)\(\)\()/5\)
\(\nu^{17}\)\(=\)\((\)\(-\)\(57\!\cdots\!41\)\( \beta_{19} - \)\(66\!\cdots\!78\)\( \beta_{18} - \)\(82\!\cdots\!99\)\( \beta_{17} - \)\(99\!\cdots\!79\)\( \beta_{16} - \)\(21\!\cdots\!72\)\( \beta_{15} - \)\(27\!\cdots\!94\)\( \beta_{14} - \)\(44\!\cdots\!80\)\( \beta_{13} - \)\(49\!\cdots\!10\)\( \beta_{12} - \)\(36\!\cdots\!54\)\( \beta_{11} + \)\(10\!\cdots\!39\)\( \beta_{10} - \)\(64\!\cdots\!90\)\( \beta_{9} - \)\(79\!\cdots\!13\)\( \beta_{8} - \)\(20\!\cdots\!45\)\( \beta_{7} - \)\(26\!\cdots\!97\)\( \beta_{6} - \)\(73\!\cdots\!16\)\( \beta_{5} - \)\(58\!\cdots\!72\)\( \beta_{4} - \)\(45\!\cdots\!59\)\( \beta_{3} + \)\(17\!\cdots\!43\)\( \beta_{2} + \)\(33\!\cdots\!77\)\( \beta_{1} - \)\(90\!\cdots\!64\)\(\)\()/5\)
\(\nu^{18}\)\(=\)\((\)\(\)\(71\!\cdots\!69\)\( \beta_{19} - \)\(59\!\cdots\!26\)\( \beta_{18} + \)\(10\!\cdots\!36\)\( \beta_{17} + \)\(49\!\cdots\!35\)\( \beta_{16} - \)\(20\!\cdots\!74\)\( \beta_{15} - \)\(18\!\cdots\!10\)\( \beta_{14} - \)\(11\!\cdots\!02\)\( \beta_{13} + \)\(10\!\cdots\!82\)\( \beta_{12} + \)\(17\!\cdots\!23\)\( \beta_{11} + \)\(92\!\cdots\!66\)\( \beta_{10} - \)\(82\!\cdots\!17\)\( \beta_{9} - \)\(71\!\cdots\!20\)\( \beta_{8} - \)\(18\!\cdots\!02\)\( \beta_{7} - \)\(39\!\cdots\!73\)\( \beta_{6} + \)\(36\!\cdots\!92\)\( \beta_{5} + \)\(29\!\cdots\!64\)\( \beta_{4} - \)\(82\!\cdots\!02\)\( \beta_{3} + \)\(52\!\cdots\!48\)\( \beta_{2} + \)\(50\!\cdots\!87\)\( \beta_{1} - \)\(61\!\cdots\!68\)\(\)\()/5\)
\(\nu^{19}\)\(=\)\((\)\(\)\(20\!\cdots\!04\)\( \beta_{19} + \)\(23\!\cdots\!05\)\( \beta_{18} + \)\(29\!\cdots\!86\)\( \beta_{17} + \)\(35\!\cdots\!53\)\( \beta_{16} + \)\(75\!\cdots\!97\)\( \beta_{15} + \)\(93\!\cdots\!61\)\( \beta_{14} + \)\(15\!\cdots\!72\)\( \beta_{13} + \)\(17\!\cdots\!58\)\( \beta_{12} + \)\(12\!\cdots\!68\)\( \beta_{11} - \)\(36\!\cdots\!62\)\( \beta_{10} + \)\(22\!\cdots\!51\)\( \beta_{9} + \)\(27\!\cdots\!68\)\( \beta_{8} + \)\(73\!\cdots\!43\)\( \beta_{7} + \)\(92\!\cdots\!22\)\( \beta_{6} + \)\(26\!\cdots\!43\)\( \beta_{5} + \)\(20\!\cdots\!49\)\( \beta_{4} + \)\(15\!\cdots\!52\)\( \beta_{3} - \)\(60\!\cdots\!56\)\( \beta_{2} - \)\(11\!\cdots\!98\)\( \beta_{1} + \)\(31\!\cdots\!88\)\(\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 105.104i
0.500000 187.698i
0.500000 + 19.6223i
0.500000 + 4.33815i
0.500000 + 62.0735i
0.500000 22.2192i
0.500000 3.30996i
0.500000 + 100.713i
0.500000 + 3.46508i
0.500000 77.8364i
0.500000 + 22.2192i
0.500000 + 3.30996i
0.500000 100.713i
0.500000 3.46508i
0.500000 + 77.8364i
0.500000 105.104i
0.500000 + 187.698i
0.500000 19.6223i
0.500000 4.33815i
0.500000 62.0735i
−1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −55.7220 + 4.47850i −29.1246 21.1603i −89.1519 51.7771 + 37.6183i 25.0304 77.0356i 85.9134 + 206.443i
31.2 −1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −49.2840 26.3835i −29.1246 21.1603i 255.057 51.7771 + 37.6183i 25.0304 77.0356i −39.4504 + 220.099i
31.3 −1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −13.4674 + 54.2552i −29.1246 21.1603i 11.3383 51.7771 + 37.6183i 25.0304 77.0356i 223.046 15.8303i
31.4 −1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i 11.2821 54.7514i −29.1246 21.1603i 29.3060 51.7771 + 37.6183i 25.0304 77.0356i −222.232 + 24.7567i
31.5 −1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i 51.5150 + 21.7074i −29.1246 21.1603i −38.5661 51.7771 + 37.6183i 25.0304 77.0356i 18.9037 222.806i
61.1 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i −41.6409 37.2966i 11.1246 + 34.2380i −58.1692 −19.7771 60.8676i −65.5304 47.6106i −222.442 22.7907i
61.2 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i −38.1312 + 40.8780i 11.1246 + 34.2380i −22.2017 −19.7771 60.8676i −65.5304 47.6106i −27.2854 + 221.936i
61.3 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 22.2851 51.2677i 11.1246 + 34.2380i 175.661 −19.7771 60.8676i −65.5304 47.6106i −48.4213 218.301i
61.4 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 32.6761 + 45.3572i 11.1246 + 34.2380i −9.31479 −19.7771 60.8676i −65.5304 47.6106i 212.383 + 69.9528i
61.5 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 52.9872 17.8146i 11.1246 + 34.2380i −163.959 −19.7771 60.8676i −65.5304 47.6106i 129.585 182.230i
91.1 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i −41.6409 + 37.2966i 11.1246 34.2380i −58.1692 −19.7771 + 60.8676i −65.5304 + 47.6106i −222.442 + 22.7907i
91.2 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i −38.1312 40.8780i 11.1246 34.2380i −22.2017 −19.7771 + 60.8676i −65.5304 + 47.6106i −27.2854 221.936i
91.3 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i 22.2851 + 51.2677i 11.1246 34.2380i 175.661 −19.7771 + 60.8676i −65.5304 + 47.6106i −48.4213 + 218.301i
91.4 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i 32.6761 45.3572i 11.1246 34.2380i −9.31479 −19.7771 + 60.8676i −65.5304 + 47.6106i 212.383 69.9528i
91.5 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i 52.9872 + 17.8146i 11.1246 34.2380i −163.959 −19.7771 + 60.8676i −65.5304 + 47.6106i 129.585 + 182.230i
121.1 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i −55.7220 4.47850i −29.1246 + 21.1603i −89.1519 51.7771 37.6183i 25.0304 + 77.0356i 85.9134 206.443i
121.2 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i −49.2840 + 26.3835i −29.1246 + 21.1603i 255.057 51.7771 37.6183i 25.0304 + 77.0356i −39.4504 220.099i
121.3 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i −13.4674 54.2552i −29.1246 + 21.1603i 11.3383 51.7771 37.6183i 25.0304 + 77.0356i 223.046 + 15.8303i
121.4 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i 11.2821 + 54.7514i −29.1246 + 21.1603i 29.3060 51.7771 37.6183i 25.0304 + 77.0356i −222.232 24.7567i
121.5 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i 51.5150 21.7074i −29.1246 + 21.1603i −38.5661 51.7771 37.6183i 25.0304 + 77.0356i 18.9037 + 222.806i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.6.g.a 20
25.d even 5 1 inner 150.6.g.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.g.a 20 1.a even 1 1 trivial
150.6.g.a 20 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(66\!\cdots\!80\)\( T_{7}^{3} - \)\(86\!\cdots\!95\)\( T_{7}^{2} + \)\(81\!\cdots\!20\)\( T_{7} + \)\(10\!\cdots\!84\)\( \)">\(T_{7}^{10} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} )^{5} \)
$3$ \( ( 6561 - 729 T + 81 T^{2} - 9 T^{3} + T^{4} )^{5} \)
$5$ \( \)\(88\!\cdots\!25\)\( + \)\(15\!\cdots\!75\)\( T + \)\(59\!\cdots\!50\)\( T^{2} + \)\(14\!\cdots\!50\)\( T^{3} + \)\(64\!\cdots\!00\)\( T^{4} - \)\(32\!\cdots\!25\)\( T^{5} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(34\!\cdots\!75\)\( T^{7} + \)\(81\!\cdots\!00\)\( T^{8} + 17950515747070312500 T^{9} + 357291107177734375 T^{10} + 5744165039062500 T^{11} + 83565234375000 T^{12} + 1127412109375 T^{13} - 14637500000 T^{14} - 109140625 T^{15} + 6972500 T^{16} + 48250 T^{17} + 650 T^{18} + 55 T^{19} + T^{20} \)
$7$ \( ( 100958609795235984 + 8167718788477620 T - 864743821499495 T^{2} - 66684308302480 T^{3} + 17548207095 T^{4} + 65226626294 T^{5} + 1080778560 T^{6} + 499140 T^{7} - 64540 T^{8} - 90 T^{9} + T^{10} )^{2} \)
$11$ \( \)\(10\!\cdots\!56\)\( + \)\(61\!\cdots\!80\)\( T + \)\(23\!\cdots\!20\)\( T^{2} + \)\(52\!\cdots\!80\)\( T^{3} + \)\(71\!\cdots\!65\)\( T^{4} + \)\(42\!\cdots\!08\)\( T^{5} + \)\(66\!\cdots\!05\)\( T^{6} + \)\(21\!\cdots\!70\)\( T^{7} + \)\(10\!\cdots\!55\)\( T^{8} + \)\(47\!\cdots\!90\)\( T^{9} + \)\(30\!\cdots\!29\)\( T^{10} + \)\(70\!\cdots\!60\)\( T^{11} + \)\(35\!\cdots\!90\)\( T^{12} + 2425673059295238160 T^{13} + 37318282988186005 T^{14} - 77399363864188 T^{15} + 300096878715 T^{16} - 399757090 T^{17} + 746590 T^{18} - 380 T^{19} + T^{20} \)
$13$ \( \)\(69\!\cdots\!41\)\( + \)\(27\!\cdots\!85\)\( T + \)\(92\!\cdots\!05\)\( T^{2} - \)\(25\!\cdots\!85\)\( T^{3} + \)\(10\!\cdots\!80\)\( T^{4} - \)\(13\!\cdots\!51\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!65\)\( T^{7} + \)\(57\!\cdots\!45\)\( T^{8} + \)\(37\!\cdots\!40\)\( T^{9} + \)\(19\!\cdots\!76\)\( T^{10} + \)\(49\!\cdots\!05\)\( T^{11} + \)\(14\!\cdots\!15\)\( T^{12} + \)\(24\!\cdots\!95\)\( T^{13} + 471564893841021915 T^{14} + 456440101067944 T^{15} + 405074128110 T^{16} - 217493195 T^{17} + 795090 T^{18} + 230 T^{19} + T^{20} \)
$17$ \( \)\(43\!\cdots\!21\)\( - \)\(49\!\cdots\!05\)\( T + \)\(12\!\cdots\!90\)\( T^{2} - \)\(24\!\cdots\!90\)\( T^{3} + \)\(26\!\cdots\!90\)\( T^{4} - \)\(14\!\cdots\!23\)\( T^{5} + \)\(48\!\cdots\!55\)\( T^{6} - \)\(11\!\cdots\!65\)\( T^{7} + \)\(20\!\cdots\!65\)\( T^{8} - \)\(26\!\cdots\!40\)\( T^{9} + \)\(26\!\cdots\!34\)\( T^{10} - \)\(20\!\cdots\!35\)\( T^{11} + \)\(15\!\cdots\!55\)\( T^{12} - \)\(10\!\cdots\!55\)\( T^{13} + 61384778983845356380 T^{14} - 36904386480344857 T^{15} + 28413228022940 T^{16} - 15841600245 T^{17} + 10406845 T^{18} - 2870 T^{19} + T^{20} \)
$19$ \( \)\(58\!\cdots\!00\)\( + \)\(59\!\cdots\!00\)\( T + \)\(44\!\cdots\!00\)\( T^{2} - \)\(33\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!25\)\( T^{4} - \)\(79\!\cdots\!50\)\( T^{5} + \)\(16\!\cdots\!75\)\( T^{6} + \)\(65\!\cdots\!50\)\( T^{7} + \)\(12\!\cdots\!25\)\( T^{8} - \)\(17\!\cdots\!50\)\( T^{9} + \)\(14\!\cdots\!75\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(22\!\cdots\!00\)\( T^{12} + \)\(11\!\cdots\!50\)\( T^{13} + 79180586215281062625 T^{14} + 4778425582348650 T^{15} + 15375391191945 T^{16} - 1408355830 T^{17} + 6111865 T^{18} - 1560 T^{19} + T^{20} \)
$23$ \( \)\(51\!\cdots\!76\)\( - \)\(18\!\cdots\!40\)\( T + \)\(80\!\cdots\!60\)\( T^{2} + \)\(42\!\cdots\!40\)\( T^{3} + \)\(91\!\cdots\!65\)\( T^{4} + \)\(23\!\cdots\!54\)\( T^{5} + \)\(40\!\cdots\!05\)\( T^{6} - \)\(17\!\cdots\!70\)\( T^{7} + \)\(27\!\cdots\!20\)\( T^{8} + \)\(17\!\cdots\!20\)\( T^{9} + \)\(12\!\cdots\!31\)\( T^{10} + \)\(29\!\cdots\!30\)\( T^{11} + \)\(16\!\cdots\!05\)\( T^{12} + \)\(24\!\cdots\!20\)\( T^{13} + \)\(21\!\cdots\!45\)\( T^{14} + 224536466679690854 T^{15} + 181758909627710 T^{16} + 29477547810 T^{17} + 20063815 T^{18} - 760 T^{19} + T^{20} \)
$29$ \( \)\(23\!\cdots\!25\)\( - \)\(25\!\cdots\!75\)\( T + \)\(26\!\cdots\!25\)\( T^{2} - \)\(10\!\cdots\!75\)\( T^{3} + \)\(34\!\cdots\!75\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} - \)\(33\!\cdots\!75\)\( T^{7} + \)\(17\!\cdots\!75\)\( T^{8} - \)\(90\!\cdots\!75\)\( T^{9} + \)\(45\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!75\)\( T^{11} + \)\(24\!\cdots\!75\)\( T^{12} - \)\(30\!\cdots\!50\)\( T^{13} + \)\(20\!\cdots\!00\)\( T^{14} - 34729695877602166775 T^{15} + 7961321110505845 T^{16} - 1057703140435 T^{17} + 154161315 T^{18} - 12000 T^{19} + T^{20} \)
$31$ \( \)\(16\!\cdots\!56\)\( + \)\(39\!\cdots\!60\)\( T + \)\(42\!\cdots\!60\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(51\!\cdots\!45\)\( T^{4} + \)\(12\!\cdots\!08\)\( T^{5} + \)\(97\!\cdots\!85\)\( T^{6} + \)\(14\!\cdots\!10\)\( T^{7} + \)\(16\!\cdots\!80\)\( T^{8} + \)\(12\!\cdots\!70\)\( T^{9} + \)\(13\!\cdots\!79\)\( T^{10} + \)\(90\!\cdots\!20\)\( T^{11} + \)\(89\!\cdots\!45\)\( T^{12} + \)\(30\!\cdots\!10\)\( T^{13} + \)\(53\!\cdots\!65\)\( T^{14} + 1437590491843589862 T^{15} + 2614053871983730 T^{16} + 60672169380 T^{17} + 76015265 T^{18} + 1710 T^{19} + T^{20} \)
$37$ \( \)\(94\!\cdots\!21\)\( - \)\(10\!\cdots\!10\)\( T + \)\(28\!\cdots\!90\)\( T^{2} + \)\(17\!\cdots\!95\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} + \)\(17\!\cdots\!52\)\( T^{5} + \)\(42\!\cdots\!35\)\( T^{6} + \)\(32\!\cdots\!85\)\( T^{7} + \)\(13\!\cdots\!55\)\( T^{8} + \)\(27\!\cdots\!15\)\( T^{9} + \)\(13\!\cdots\!84\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{11} + \)\(31\!\cdots\!05\)\( T^{12} + \)\(12\!\cdots\!15\)\( T^{13} + \)\(34\!\cdots\!20\)\( T^{14} + \)\(13\!\cdots\!43\)\( T^{15} + 17003225240131880 T^{16} - 110978219595 T^{17} + 140706065 T^{18} + 5595 T^{19} + T^{20} \)
$41$ \( \)\(56\!\cdots\!21\)\( + \)\(99\!\cdots\!05\)\( T + \)\(68\!\cdots\!40\)\( T^{2} + \)\(59\!\cdots\!40\)\( T^{3} + \)\(74\!\cdots\!90\)\( T^{4} + \)\(12\!\cdots\!23\)\( T^{5} + \)\(21\!\cdots\!30\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(54\!\cdots\!40\)\( T^{8} + \)\(17\!\cdots\!15\)\( T^{9} + \)\(56\!\cdots\!84\)\( T^{10} + \)\(13\!\cdots\!10\)\( T^{11} + \)\(28\!\cdots\!30\)\( T^{12} + \)\(44\!\cdots\!55\)\( T^{13} + \)\(60\!\cdots\!80\)\( T^{14} + \)\(50\!\cdots\!32\)\( T^{15} + 32784405067571640 T^{16} + 297703238395 T^{17} - 12346730 T^{18} + 2820 T^{19} + T^{20} \)
$43$ \( ( -\)\(92\!\cdots\!24\)\( + \)\(47\!\cdots\!60\)\( T + \)\(47\!\cdots\!85\)\( T^{2} - \)\(28\!\cdots\!10\)\( T^{3} + \)\(39\!\cdots\!40\)\( T^{4} + \)\(35\!\cdots\!52\)\( T^{5} - 132517226218625515 T^{6} + 9903014454760 T^{7} + 49439740 T^{8} - 33010 T^{9} + T^{10} )^{2} \)
$47$ \( \)\(80\!\cdots\!16\)\( + \)\(18\!\cdots\!00\)\( T + \)\(72\!\cdots\!80\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!65\)\( T^{4} + \)\(81\!\cdots\!32\)\( T^{5} + \)\(14\!\cdots\!25\)\( T^{6} + \)\(31\!\cdots\!70\)\( T^{7} + \)\(42\!\cdots\!50\)\( T^{8} + \)\(17\!\cdots\!10\)\( T^{9} + \)\(55\!\cdots\!49\)\( T^{10} + \)\(15\!\cdots\!50\)\( T^{11} - \)\(44\!\cdots\!15\)\( T^{12} - \)\(25\!\cdots\!50\)\( T^{13} + \)\(13\!\cdots\!55\)\( T^{14} + \)\(32\!\cdots\!08\)\( T^{15} + 103106705445656250 T^{16} - 5294491639690 T^{17} + 189375325 T^{18} + 19630 T^{19} + T^{20} \)
$53$ \( \)\(37\!\cdots\!41\)\( - \)\(16\!\cdots\!90\)\( T + \)\(20\!\cdots\!90\)\( T^{2} - \)\(14\!\cdots\!65\)\( T^{3} + \)\(16\!\cdots\!40\)\( T^{4} + \)\(70\!\cdots\!24\)\( T^{5} + \)\(23\!\cdots\!05\)\( T^{6} - \)\(95\!\cdots\!05\)\( T^{7} + \)\(23\!\cdots\!05\)\( T^{8} - \)\(35\!\cdots\!55\)\( T^{9} + \)\(58\!\cdots\!26\)\( T^{10} - \)\(61\!\cdots\!70\)\( T^{11} + \)\(60\!\cdots\!45\)\( T^{12} - \)\(38\!\cdots\!95\)\( T^{13} + \)\(18\!\cdots\!20\)\( T^{14} - \)\(49\!\cdots\!81\)\( T^{15} + 429179158140756210 T^{16} + 4067586593365 T^{17} + 1372346685 T^{18} - 62735 T^{19} + T^{20} \)
$59$ \( \)\(27\!\cdots\!00\)\( - \)\(11\!\cdots\!00\)\( T + \)\(20\!\cdots\!00\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!25\)\( T^{4} - \)\(40\!\cdots\!50\)\( T^{5} + \)\(28\!\cdots\!50\)\( T^{6} - \)\(74\!\cdots\!50\)\( T^{7} + \)\(10\!\cdots\!75\)\( T^{8} - \)\(72\!\cdots\!50\)\( T^{9} + \)\(45\!\cdots\!50\)\( T^{10} - \)\(16\!\cdots\!50\)\( T^{11} + \)\(67\!\cdots\!75\)\( T^{12} - \)\(20\!\cdots\!50\)\( T^{13} + \)\(61\!\cdots\!50\)\( T^{14} - \)\(15\!\cdots\!00\)\( T^{15} + 5562973455077805425 T^{16} - 183050941753400 T^{17} + 5763165240 T^{18} - 105100 T^{19} + T^{20} \)
$61$ \( \)\(50\!\cdots\!81\)\( - \)\(10\!\cdots\!25\)\( T + \)\(89\!\cdots\!70\)\( T^{2} - \)\(15\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!20\)\( T^{4} - \)\(16\!\cdots\!67\)\( T^{5} + \)\(11\!\cdots\!50\)\( T^{6} - \)\(64\!\cdots\!30\)\( T^{7} + \)\(34\!\cdots\!40\)\( T^{8} - \)\(14\!\cdots\!05\)\( T^{9} + \)\(51\!\cdots\!04\)\( T^{10} - \)\(16\!\cdots\!50\)\( T^{11} + \)\(47\!\cdots\!40\)\( T^{12} - \)\(70\!\cdots\!95\)\( T^{13} + \)\(19\!\cdots\!90\)\( T^{14} - \)\(33\!\cdots\!88\)\( T^{15} + 9860006355067315300 T^{16} - 138053835759665 T^{17} + 4003247170 T^{18} - 34790 T^{19} + T^{20} \)
$67$ \( \)\(50\!\cdots\!96\)\( - \)\(26\!\cdots\!00\)\( T + \)\(55\!\cdots\!00\)\( T^{2} - \)\(60\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!85\)\( T^{4} - \)\(11\!\cdots\!48\)\( T^{5} + \)\(70\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!50\)\( T^{7} + \)\(82\!\cdots\!40\)\( T^{8} + \)\(84\!\cdots\!40\)\( T^{9} + \)\(16\!\cdots\!84\)\( T^{10} + \)\(24\!\cdots\!50\)\( T^{11} + \)\(11\!\cdots\!50\)\( T^{12} + \)\(67\!\cdots\!70\)\( T^{13} + \)\(37\!\cdots\!20\)\( T^{14} + \)\(27\!\cdots\!18\)\( T^{15} + 18461092522588703575 T^{16} + 125920926239450 T^{17} + 5843161720 T^{18} + 19470 T^{19} + T^{20} \)
$71$ \( \)\(32\!\cdots\!56\)\( + \)\(19\!\cdots\!00\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!65\)\( T^{4} + \)\(10\!\cdots\!08\)\( T^{5} + \)\(91\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!35\)\( T^{8} + \)\(49\!\cdots\!40\)\( T^{9} + \)\(16\!\cdots\!29\)\( T^{10} + \)\(29\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!50\)\( T^{12} + \)\(33\!\cdots\!70\)\( T^{13} + \)\(11\!\cdots\!55\)\( T^{14} + \)\(24\!\cdots\!62\)\( T^{15} + 53387090117325127625 T^{16} + 824344294987050 T^{17} + 12634619005 T^{18} + 111720 T^{19} + T^{20} \)
$73$ \( \)\(32\!\cdots\!81\)\( - \)\(44\!\cdots\!25\)\( T + \)\(16\!\cdots\!40\)\( T^{2} + \)\(28\!\cdots\!50\)\( T^{3} + \)\(93\!\cdots\!05\)\( T^{4} + \)\(12\!\cdots\!44\)\( T^{5} + \)\(20\!\cdots\!50\)\( T^{6} + \)\(22\!\cdots\!70\)\( T^{7} + \)\(15\!\cdots\!50\)\( T^{8} + \)\(63\!\cdots\!90\)\( T^{9} + \)\(16\!\cdots\!46\)\( T^{10} + \)\(29\!\cdots\!00\)\( T^{11} + \)\(48\!\cdots\!95\)\( T^{12} + \)\(12\!\cdots\!75\)\( T^{13} + \)\(40\!\cdots\!90\)\( T^{14} + \)\(96\!\cdots\!84\)\( T^{15} + \)\(15\!\cdots\!50\)\( T^{16} + 1951457312319015 T^{17} + 22447906750 T^{18} + 198830 T^{19} + T^{20} \)
$79$ \( \)\(48\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T + \)\(17\!\cdots\!00\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!25\)\( T^{4} - \)\(20\!\cdots\!50\)\( T^{5} + \)\(18\!\cdots\!25\)\( T^{6} - \)\(87\!\cdots\!00\)\( T^{7} + \)\(34\!\cdots\!75\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!75\)\( T^{10} - \)\(54\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!00\)\( T^{12} - \)\(78\!\cdots\!00\)\( T^{13} + \)\(12\!\cdots\!75\)\( T^{14} - \)\(78\!\cdots\!50\)\( T^{15} + \)\(11\!\cdots\!95\)\( T^{16} - 569642176593880 T^{17} + 15065532390 T^{18} - 71210 T^{19} + T^{20} \)
$83$ \( \)\(78\!\cdots\!36\)\( - \)\(20\!\cdots\!80\)\( T + \)\(43\!\cdots\!00\)\( T^{2} - \)\(41\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!65\)\( T^{4} + \)\(75\!\cdots\!34\)\( T^{5} + \)\(28\!\cdots\!10\)\( T^{6} + \)\(73\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!65\)\( T^{8} + \)\(36\!\cdots\!70\)\( T^{9} + \)\(63\!\cdots\!86\)\( T^{10} + \)\(88\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!75\)\( T^{12} + \)\(20\!\cdots\!40\)\( T^{13} + \)\(33\!\cdots\!70\)\( T^{14} + \)\(43\!\cdots\!14\)\( T^{15} + \)\(57\!\cdots\!45\)\( T^{16} + 5846362196462300 T^{17} + 50548325830 T^{18} + 288690 T^{19} + T^{20} \)
$89$ \( \)\(12\!\cdots\!25\)\( + \)\(49\!\cdots\!00\)\( T + \)\(93\!\cdots\!25\)\( T^{2} + \)\(15\!\cdots\!25\)\( T^{3} + \)\(17\!\cdots\!50\)\( T^{4} - \)\(41\!\cdots\!25\)\( T^{5} + \)\(33\!\cdots\!00\)\( T^{6} - \)\(23\!\cdots\!25\)\( T^{7} + \)\(46\!\cdots\!25\)\( T^{8} - \)\(12\!\cdots\!25\)\( T^{9} + \)\(82\!\cdots\!00\)\( T^{10} - \)\(18\!\cdots\!00\)\( T^{11} + \)\(44\!\cdots\!25\)\( T^{12} - \)\(65\!\cdots\!25\)\( T^{13} + \)\(10\!\cdots\!75\)\( T^{14} - \)\(93\!\cdots\!75\)\( T^{15} + 94958196907162770800 T^{16} + 113411088119750 T^{17} + 4193667360 T^{18} - 114225 T^{19} + T^{20} \)
$97$ \( \)\(63\!\cdots\!21\)\( + \)\(17\!\cdots\!95\)\( T + \)\(26\!\cdots\!70\)\( T^{2} + \)\(28\!\cdots\!40\)\( T^{3} + \)\(70\!\cdots\!60\)\( T^{4} + \)\(16\!\cdots\!77\)\( T^{5} + \)\(40\!\cdots\!30\)\( T^{6} + \)\(66\!\cdots\!30\)\( T^{7} + \)\(10\!\cdots\!60\)\( T^{8} + \)\(12\!\cdots\!15\)\( T^{9} + \)\(13\!\cdots\!84\)\( T^{10} + \)\(12\!\cdots\!90\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} + \)\(73\!\cdots\!05\)\( T^{13} + \)\(48\!\cdots\!70\)\( T^{14} + \)\(29\!\cdots\!68\)\( T^{15} + \)\(20\!\cdots\!40\)\( T^{16} + 15530025285591315 T^{17} + 104614357130 T^{18} + 446970 T^{19} + T^{20} \)
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