# 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 $2$

# Related objects

## 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
 0.5 + 105.104i 0.5 − 187.698i 0.5 + 19.6223i 0.5 + 4.33815i 0.5 + 62.0735i 0.5 − 22.2192i 0.5 − 3.30996i 0.5 + 100.713i 0.5 + 3.46508i 0.5 − 77.8364i 0.5 + 22.2192i 0.5 + 3.30996i 0.5 − 100.713i 0.5 − 3.46508i 0.5 + 77.8364i 0.5 − 105.104i 0.5 + 187.698i 0.5 − 19.6223i 0.5 − 4.33815i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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