# Properties

 Label 76.7.h.a Level $76$ Weight $7$ Character orbit 76.h Analytic conductor $17.484$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 76.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.4841103551$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + 64290781142478432 x^{10} + 26741969359067432592 x^{8} + 6092313787108272225600 x^{6} + 632563499034568748983104 x^{4} + 15152233769669432690883072 x^{2} + 103372598905826219735270400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{22}\cdot 3^{8}\cdot 19^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -6 + 6 \beta_{3} + \beta_{6} ) q^{5} + ( 23 - \beta_{9} + \beta_{10} ) q^{7} + ( -6 \beta_{1} - 3 \beta_{2} + 221 \beta_{3} - 2 \beta_{5} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -6 + 6 \beta_{3} + \beta_{6} ) q^{5} + ( 23 - \beta_{9} + \beta_{10} ) q^{7} + ( -6 \beta_{1} - 3 \beta_{2} + 221 \beta_{3} - 2 \beta_{5} + \beta_{8} ) q^{9} + ( -183 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - \beta_{12} + \beta_{13} ) q^{11} + ( -477 - \beta_{1} + 2 \beta_{2} + 238 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + \beta_{14} - \beta_{17} ) q^{13} + ( 614 + \beta_{1} - 42 \beta_{2} - 307 \beta_{3} - \beta_{4} - 3 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{11} ) q^{15} + ( 113 - 7 \beta_{1} + 6 \beta_{2} - 113 \beta_{3} + 2 \beta_{6} - \beta_{7} - \beta_{16} ) q^{17} + ( 249 + 45 \beta_{1} + 35 \beta_{2} - 285 \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{19} + ( -290 - 8 \beta_{1} - 8 \beta_{2} - 291 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{14} + \beta_{18} + \beta_{19} ) q^{21} + ( 80 \beta_{1} + 41 \beta_{2} + 83 \beta_{3} + \beta_{5} - 4 \beta_{8} + \beta_{10} + 4 \beta_{12} - \beta_{13} + \beta_{16} + \beta_{18} + 2 \beta_{19} ) q^{23} + ( 2 + 98 \beta_{1} + 49 \beta_{2} - 2778 \beta_{3} + \beta_{4} + 22 \beta_{5} - 2 \beta_{8} - 5 \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{25} + ( 2842 + 249 \beta_{1} + 2 \beta_{2} - 5681 \beta_{3} + 24 \beta_{5} - 24 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} - 9 \beta_{9} - 4 \beta_{10} - 5 \beta_{11} - \beta_{12} - 3 \beta_{13} - \beta_{15} + \beta_{16} + 3 \beta_{17} + 2 \beta_{19} ) q^{27} + ( -726 + \beta_{1} - 91 \beta_{2} + 362 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 5 \beta_{11} - 2 \beta_{12} + 5 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 5 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{29} + ( -3465 - 33 \beta_{1} + 3 \beta_{2} + 6926 \beta_{3} - 36 \beta_{5} + 36 \beta_{6} + \beta_{7} + 10 \beta_{9} + 9 \beta_{10} + \beta_{11} + 8 \beta_{12} + 5 \beta_{13} - 2 \beta_{15} + \beta_{16} - 4 \beta_{17} + 3 \beta_{19} ) q^{31} + ( -997 - 378 \beta_{1} - 375 \beta_{2} - 1007 \beta_{3} - 6 \beta_{4} - 30 \beta_{5} - 20 \beta_{6} - 8 \beta_{7} + 10 \beta_{8} - 6 \beta_{10} - 16 \beta_{12} + 8 \beta_{13} + 10 \beta_{14} - 6 \beta_{15} + 3 \beta_{16} + 2 \beta_{18} + 2 \beta_{19} ) q^{33} + ( 398 + 37 \beta_{1} - 56 \beta_{2} - 405 \beta_{3} + 7 \beta_{5} + 36 \beta_{6} - 24 \beta_{7} - 28 \beta_{9} - 5 \beta_{13} + 7 \beta_{14} - 5 \beta_{16} - 14 \beta_{17} - \beta_{18} + \beta_{19} ) q^{35} + ( -5433 + 51 \beta_{1} + 2 \beta_{2} + 10851 \beta_{3} - 46 \beta_{5} + 46 \beta_{6} + 18 \beta_{7} + 26 \beta_{8} + 8 \beta_{9} + 16 \beta_{10} - 8 \beta_{11} - 10 \beta_{12} - 12 \beta_{13} + 2 \beta_{16} - 15 \beta_{17} + 2 \beta_{19} ) q^{37} + ( 4086 - 212 \beta_{1} - 433 \beta_{2} - 8 \beta_{4} - 53 \beta_{5} - 53 \beta_{6} - 32 \beta_{7} + 28 \beta_{8} - 8 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - 24 \beta_{12} + 25 \beta_{13} + 4 \beta_{15} - 5 \beta_{16} + 2 \beta_{18} + \beta_{19} ) q^{39} + ( 3660 - 101 \beta_{1} - 99 \beta_{2} + 3640 \beta_{3} - 48 \beta_{5} - 34 \beta_{6} - 27 \beta_{7} + 54 \beta_{8} - 36 \beta_{9} + 72 \beta_{10} + 16 \beta_{12} - 8 \beta_{13} + 20 \beta_{14} - 4 \beta_{15} + 2 \beta_{16} + 2 \beta_{18} + 2 \beta_{19} ) q^{41} + ( 11046 + 488 \beta_{1} - 445 \beta_{2} - 11035 \beta_{3} - 9 \beta_{4} - 11 \beta_{5} + 100 \beta_{6} - 33 \beta_{7} + 9 \beta_{8} - 7 \beta_{9} + 9 \beta_{10} - 18 \beta_{11} + 22 \beta_{13} - 11 \beta_{14} + 3 \beta_{16} + 22 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{43} + ( -39450 + 659 \beta_{1} + 1351 \beta_{2} - 8 \beta_{3} + 18 \beta_{4} + 372 \beta_{5} + 356 \beta_{6} + 111 \beta_{7} - 102 \beta_{8} - 77 \beta_{9} + 86 \beta_{10} + 9 \beta_{11} - 48 \beta_{12} + 49 \beta_{13} + 16 \beta_{14} - 2 \beta_{15} + \beta_{16} - 8 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{45} + ( 2 + 33 \beta_{1} + 18 \beta_{2} + 10803 \beta_{3} - 5 \beta_{4} - 125 \beta_{5} + 35 \beta_{8} - 174 \beta_{10} + 5 \beta_{11} + 28 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} + 4 \beta_{19} ) q^{47} + ( 6662 + 104 \beta_{1} + 165 \beta_{2} + 22 \beta_{3} + 2 \beta_{4} + 184 \beta_{5} + 228 \beta_{6} + 62 \beta_{7} - 61 \beta_{8} - 179 \beta_{9} + 180 \beta_{10} + \beta_{11} + 4 \beta_{12} - 3 \beta_{13} - 44 \beta_{14} - \beta_{16} + 22 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{49} + ( 13504 - 17 \beta_{1} + 862 \beta_{2} - 6760 \beta_{3} + \beta_{4} - 238 \beta_{5} - 508 \beta_{6} - 2 \beta_{7} + \beta_{8} - 102 \beta_{9} + 51 \beta_{10} + \beta_{11} + 49 \beta_{12} - 99 \beta_{13} + 16 \beta_{14} + \beta_{16} - 16 \beta_{17} - \beta_{18} ) q^{51} + ( 16981 + 2 \beta_{1} - 49 \beta_{2} - 8486 \beta_{3} + 11 \beta_{4} + 110 \beta_{5} + 238 \beta_{6} - 94 \beta_{7} + 47 \beta_{8} + 358 \beta_{9} - 179 \beta_{10} + 11 \beta_{11} + 10 \beta_{12} - 19 \beta_{13} - 9 \beta_{14} - 4 \beta_{15} + 7 \beta_{16} + 9 \beta_{17} + \beta_{18} ) q^{53} + ( 35743 + 306 \beta_{1} - 300 \beta_{2} - 35733 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} - 518 \beta_{6} - 140 \beta_{7} - 10 \beta_{8} + 95 \beta_{9} - 10 \beta_{10} + 20 \beta_{11} - 4 \beta_{13} - 10 \beta_{14} + 6 \beta_{16} + 20 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{55} + ( -9097 - 565 \beta_{1} + 150 \beta_{2} + 39833 \beta_{3} + 7 \beta_{4} - 732 \beta_{5} - 154 \beta_{6} + 119 \beta_{7} + 54 \beta_{8} + 279 \beta_{9} - 143 \beta_{10} + 8 \beta_{11} + 38 \beta_{12} - 64 \beta_{13} - 32 \beta_{14} + 3 \beta_{15} - 10 \beta_{16} + 4 \beta_{17} + \beta_{18} - 11 \beta_{19} ) q^{57} + ( -20249 - 992 \beta_{1} - 993 \beta_{2} - 20246 \beta_{3} - 11 \beta_{4} - 547 \beta_{5} - 272 \beta_{6} + 101 \beta_{7} - 213 \beta_{8} + 109 \beta_{9} - 229 \beta_{10} - 14 \beta_{12} + 7 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} - \beta_{16} - 10 \beta_{18} - 10 \beta_{19} ) q^{59} + ( -20 + 1490 \beta_{1} + 742 \beta_{2} + 4396 \beta_{3} + 793 \beta_{5} - 124 \beta_{8} + 234 \beta_{10} + 122 \beta_{12} + 10 \beta_{13} - 20 \beta_{14} + 6 \beta_{15} - 10 \beta_{16} - 20 \beta_{17} - 10 \beta_{18} - 20 \beta_{19} ) q^{61} + ( 14 - 2682 \beta_{1} - 1362 \beta_{2} - 25758 \beta_{3} - 2 \beta_{4} + 890 \beta_{5} + 36 \beta_{8} - 260 \beta_{10} + 2 \beta_{11} - 60 \beta_{12} + 10 \beta_{13} + 14 \beta_{14} + 10 \beta_{15} - 10 \beta_{16} + 14 \beta_{17} - 10 \beta_{18} - 20 \beta_{19} ) q^{63} + ( 25252 + 308 \beta_{1} - 23 \beta_{2} - 50500 \beta_{3} + 660 \beta_{5} - 660 \beta_{6} - 39 \beta_{7} - 54 \beta_{8} - 303 \beta_{9} - 318 \beta_{10} + 15 \beta_{11} - 28 \beta_{12} - 5 \beta_{13} + 11 \beta_{15} - 12 \beta_{16} + 4 \beta_{17} - 23 \beta_{19} ) q^{65} + ( -55935 + 10 \beta_{1} + 2822 \beta_{2} + 27981 \beta_{3} + 32 \beta_{4} - 81 \beta_{5} - 108 \beta_{6} + 192 \beta_{7} - 96 \beta_{8} + 248 \beta_{9} - 124 \beta_{10} + 32 \beta_{11} - 27 \beta_{12} + 39 \beta_{13} - 27 \beta_{14} - 15 \beta_{15} + 45 \beta_{16} + 27 \beta_{17} - 15 \beta_{18} ) q^{67} + ( -36172 - 2614 \beta_{1} - 36 \beta_{2} + 72332 \beta_{3} - 87 \beta_{5} + 87 \beta_{6} + 88 \beta_{7} + 84 \beta_{8} + 366 \beta_{9} + 362 \beta_{10} + 4 \beta_{11} + 42 \beta_{12} + 78 \beta_{13} + 18 \beta_{15} - 18 \beta_{16} - 12 \beta_{17} - 36 \beta_{19} ) q^{69} + ( 12286 - 1240 \beta_{1} - 1266 \beta_{2} + 12314 \beta_{3} + 26 \beta_{4} - 374 \beta_{5} - 173 \beta_{6} - 62 \beta_{7} + 150 \beta_{8} - 42 \beta_{9} + 110 \beta_{10} - 40 \beta_{12} + 20 \beta_{13} - 28 \beta_{14} + 52 \beta_{15} - 26 \beta_{16} - 21 \beta_{18} - 21 \beta_{19} ) q^{71} + ( 85346 + 3143 \beta_{1} - 3105 \beta_{2} - 85340 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} + 78 \beta_{6} - 208 \beta_{7} - 9 \beta_{8} - 521 \beta_{9} - 9 \beta_{10} + 18 \beta_{11} - 54 \beta_{13} - 6 \beta_{14} + 44 \beta_{16} + 12 \beta_{17} + 11 \beta_{18} - 11 \beta_{19} ) q^{73} + ( -56465 - 5180 \beta_{1} - 20 \beta_{2} + 112942 \beta_{3} - 1022 \beta_{5} + 1022 \beta_{6} + 284 \beta_{7} + 218 \beta_{8} + 226 \beta_{9} + 160 \beta_{10} + 66 \beta_{11} - 59 \beta_{12} - 39 \beta_{13} + 6 \beta_{15} - 14 \beta_{16} + 12 \beta_{17} - 20 \beta_{19} ) q^{75} + ( 38481 + 578 \beta_{1} + 1191 \beta_{2} + 27 \beta_{3} + 78 \beta_{4} - 954 \beta_{5} - 900 \beta_{6} - 67 \beta_{7} + 106 \beta_{8} - 61 \beta_{9} + 100 \beta_{10} + 39 \beta_{11} - 6 \beta_{12} - 5 \beta_{13} - 54 \beta_{14} - 40 \beta_{15} + 51 \beta_{16} + 27 \beta_{17} - 22 \beta_{18} - 11 \beta_{19} ) q^{77} + ( 57160 + 1008 \beta_{1} + 986 \beta_{2} + 57218 \beta_{3} - 8 \beta_{4} + 200 \beta_{5} + 129 \beta_{6} - 170 \beta_{7} + 332 \beta_{8} - 430 \beta_{9} + 852 \beta_{10} + 48 \beta_{12} - 24 \beta_{13} - 58 \beta_{14} + 44 \beta_{15} - 22 \beta_{16} - 21 \beta_{18} - 21 \beta_{19} ) q^{79} + ( -106862 - 4447 \beta_{1} + 4219 \beta_{2} + 106830 \beta_{3} + 68 \beta_{4} + 32 \beta_{5} + 3614 \beta_{6} + 189 \beta_{7} - 68 \beta_{8} + 202 \beta_{9} - 68 \beta_{10} + 136 \beta_{11} + 60 \beta_{13} + 32 \beta_{14} - 28 \beta_{16} - 64 \beta_{17} + 24 \beta_{18} - 24 \beta_{19} ) q^{81} + ( 21750 + 1652 \beta_{1} + 3092 \beta_{2} + 53 \beta_{3} - 118 \beta_{4} - 198 \beta_{5} - 92 \beta_{6} + 235 \beta_{7} - 294 \beta_{8} - 1011 \beta_{9} + 952 \beta_{10} - 59 \beta_{11} + 44 \beta_{12} - 56 \beta_{13} - 106 \beta_{14} + 17 \beta_{15} - 5 \beta_{16} + 53 \beta_{17} - 24 \beta_{18} - 12 \beta_{19} ) q^{83} + ( -15 + 7985 \beta_{1} + 3982 \beta_{2} - 39890 \beta_{3} + 24 \beta_{4} + 292 \beta_{5} + 16 \beta_{8} - 1402 \beta_{10} - 24 \beta_{11} + 20 \beta_{12} + 22 \beta_{13} - 15 \beta_{14} - 32 \beta_{15} - 22 \beta_{16} - 15 \beta_{17} - 22 \beta_{18} - 44 \beta_{19} ) q^{85} + ( -81797 - 1882 \beta_{1} - 3577 \beta_{2} - 86 \beta_{3} + 4 \beta_{4} + 3373 \beta_{5} + 3201 \beta_{6} + 216 \beta_{7} - 214 \beta_{8} - 1203 \beta_{9} + 1205 \beta_{10} + 2 \beta_{11} - 18 \beta_{12} + 7 \beta_{13} + 172 \beta_{14} + 11 \beta_{16} - 86 \beta_{17} - 22 \beta_{18} - 11 \beta_{19} ) q^{87} + ( 33486 + 81 \beta_{1} + 9550 \beta_{2} - 16693 \beta_{3} + 24 \beta_{4} - 2172 \beta_{5} - 4144 \beta_{6} - 14 \beta_{7} + 7 \beta_{8} - 352 \beta_{9} + 176 \beta_{10} + 24 \beta_{11} - 66 \beta_{12} + 146 \beta_{13} - 100 \beta_{14} - 5 \beta_{15} - 4 \beta_{16} + 100 \beta_{17} + 14 \beta_{18} ) q^{89} + ( 38840 + 12 \beta_{1} - 1608 \beta_{2} - 19416 \beta_{3} - 44 \beta_{4} + 2048 \beta_{5} + 4112 \beta_{6} - 444 \beta_{7} + 222 \beta_{8} + 1924 \beta_{9} - 962 \beta_{10} - 44 \beta_{11} + 18 \beta_{12} - 52 \beta_{13} - 8 \beta_{14} + 40 \beta_{15} - 64 \beta_{16} + 8 \beta_{17} - 16 \beta_{18} ) q^{91} + ( 84406 + 4892 \beta_{1} - 4844 \beta_{2} - 84455 \beta_{3} - 70 \beta_{4} + 49 \beta_{5} - 3288 \beta_{6} - 276 \beta_{7} + 70 \beta_{8} + 648 \beta_{9} + 70 \beta_{10} - 140 \beta_{11} + 6 \beta_{13} + 49 \beta_{14} + 6 \beta_{16} - 98 \beta_{17} + 24 \beta_{18} - 24 \beta_{19} ) q^{93} + ( -50639 + 4795 \beta_{1} - 10087 \beta_{2} - 1610 \beta_{3} - 84 \beta_{4} - 3606 \beta_{5} - 1619 \beta_{6} - 213 \beta_{7} + 178 \beta_{8} + 1172 \beta_{9} - 851 \beta_{10} - 65 \beta_{11} - 62 \beta_{12} + 127 \beta_{13} + 184 \beta_{14} + 34 \beta_{15} + 39 \beta_{16} - 12 \beta_{17} - 11 \beta_{18} + 44 \beta_{19} ) q^{95} + ( -5027 - 8998 \beta_{1} - 8992 \beta_{2} - 5011 \beta_{3} + 133 \beta_{4} - 6552 \beta_{5} - 3268 \beta_{6} + 237 \beta_{7} - 341 \beta_{8} + 425 \beta_{9} - 717 \beta_{10} + 96 \beta_{12} - 48 \beta_{13} - 16 \beta_{14} - 12 \beta_{15} + 6 \beta_{16} + 33 \beta_{18} + 33 \beta_{19} ) q^{97} + ( 121 + 6258 \beta_{1} + 3129 \beta_{2} - 236602 \beta_{3} - 4 \beta_{4} + 6857 \beta_{5} - 164 \beta_{8} + 602 \beta_{10} + 4 \beta_{11} - 435 \beta_{12} - 31 \beta_{13} + 121 \beta_{14} - 55 \beta_{15} + 31 \beta_{16} + 121 \beta_{17} + 31 \beta_{18} + 62 \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 30q^{3} - 56q^{5} + 464q^{7} + 2200q^{9} + O(q^{10})$$ $$20q - 30q^{3} - 56q^{5} + 464q^{7} + 2200q^{9} - 3644q^{11} - 7140q^{13} + 9168q^{15} + 1132q^{17} + 2110q^{19} - 8748q^{21} + 832q^{23} - 27698q^{25} - 10920q^{29} - 30306q^{33} + 4172q^{35} + 81144q^{39} + 109206q^{41} + 110740q^{43} - 785440q^{45} + 107080q^{47} + 136092q^{49} + 199872q^{51} + 254796q^{53} + 354840q^{55} + 212268q^{57} - 610638q^{59} + 47864q^{61} - 254476q^{63} - 839562q^{67} + 366660q^{71} + 854482q^{73} + 763088q^{77} + 1718592q^{79} - 1054142q^{81} + 439612q^{83} - 400236q^{85} - 1604736q^{87} + 478032q^{89} + 599856q^{91} + 829380q^{93} - 1055660q^{95} - 191286q^{97} - 2336728q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + 64290781142478432 x^{10} + 26741969359067432592 x^{8} + 6092313787108272225600 x^{6} + 632563499034568748983104 x^{4} + 15152233769669432690883072 x^{2} + 103372598905826219735270400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$59\!\cdots\!99$$$$\nu^{18} -$$$$53\!\cdots\!99$$$$\nu^{16} -$$$$19\!\cdots\!06$$$$\nu^{14} -$$$$36\!\cdots\!84$$$$\nu^{12} -$$$$37\!\cdots\!60$$$$\nu^{10} -$$$$20\!\cdots\!68$$$$\nu^{8} -$$$$61\!\cdots\!96$$$$\nu^{6} -$$$$76\!\cdots\!56$$$$\nu^{4} -$$$$19\!\cdots\!72$$$$\nu^{2} -$$$$32\!\cdots\!60$$$$\nu -$$$$12\!\cdots\!00$$$$)/$$$$65\!\cdots\!20$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$38\!\cdots\!87$$$$\nu^{19} -$$$$34\!\cdots\!36$$$$\nu^{17} -$$$$12\!\cdots\!12$$$$\nu^{15} -$$$$22\!\cdots\!48$$$$\nu^{13} -$$$$23\!\cdots\!84$$$$\nu^{11} -$$$$12\!\cdots\!24$$$$\nu^{9} -$$$$34\!\cdots\!16$$$$\nu^{7} -$$$$33\!\cdots\!64$$$$\nu^{5} +$$$$83\!\cdots\!48$$$$\nu^{3} +$$$$67\!\cdots\!88$$$$\nu +$$$$10\!\cdots\!60$$$$)/$$$$21\!\cdots\!20$$ $$\beta_{4}$$ $$=$$ $$($$$$15\!\cdots\!65$$$$\nu^{19} +$$$$13\!\cdots\!32$$$$\nu^{18} +$$$$72\!\cdots\!52$$$$\nu^{17} +$$$$11\!\cdots\!02$$$$\nu^{16} -$$$$11\!\cdots\!68$$$$\nu^{15} +$$$$42\!\cdots\!28$$$$\nu^{14} -$$$$13\!\cdots\!72$$$$\nu^{13} +$$$$79\!\cdots\!52$$$$\nu^{12} -$$$$32\!\cdots\!08$$$$\nu^{11} +$$$$80\!\cdots\!60$$$$\nu^{10} -$$$$37\!\cdots\!80$$$$\nu^{9} +$$$$45\!\cdots\!84$$$$\nu^{8} -$$$$22\!\cdots\!36$$$$\nu^{7} +$$$$12\!\cdots\!88$$$$\nu^{6} -$$$$65\!\cdots\!52$$$$\nu^{5} +$$$$15\!\cdots\!28$$$$\nu^{4} -$$$$76\!\cdots\!92$$$$\nu^{3} +$$$$22\!\cdots\!16$$$$\nu^{2} -$$$$86\!\cdots\!24$$$$\nu +$$$$30\!\cdots\!60$$$$)/$$$$10\!\cdots\!60$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$43\!\cdots\!61$$$$\nu^{19} -$$$$27\!\cdots\!56$$$$\nu^{18} -$$$$39\!\cdots\!80$$$$\nu^{17} -$$$$24\!\cdots\!56$$$$\nu^{16} -$$$$14\!\cdots\!68$$$$\nu^{15} -$$$$89\!\cdots\!64$$$$\nu^{14} -$$$$27\!\cdots\!52$$$$\nu^{13} -$$$$16\!\cdots\!96$$$$\nu^{12} -$$$$28\!\cdots\!24$$$$\nu^{11} -$$$$17\!\cdots\!20$$$$\nu^{10} -$$$$17\!\cdots\!12$$$$\nu^{9} -$$$$96\!\cdots\!32$$$$\nu^{8} -$$$$56\!\cdots\!12$$$$\nu^{7} -$$$$28\!\cdots\!44$$$$\nu^{6} -$$$$91\!\cdots\!20$$$$\nu^{5} -$$$$36\!\cdots\!04$$$$\nu^{4} -$$$$61\!\cdots\!44$$$$\nu^{3} -$$$$12\!\cdots\!48$$$$\nu^{2} -$$$$13\!\cdots\!32$$$$\nu -$$$$25\!\cdots\!40$$$$)/$$$$20\!\cdots\!40$$ $$\beta_{6}$$ $$=$$ $$($$$$43\!\cdots\!61$$$$\nu^{19} -$$$$27\!\cdots\!56$$$$\nu^{18} +$$$$39\!\cdots\!80$$$$\nu^{17} -$$$$24\!\cdots\!56$$$$\nu^{16} +$$$$14\!\cdots\!68$$$$\nu^{15} -$$$$89\!\cdots\!64$$$$\nu^{14} +$$$$27\!\cdots\!52$$$$\nu^{13} -$$$$16\!\cdots\!96$$$$\nu^{12} +$$$$28\!\cdots\!24$$$$\nu^{11} -$$$$17\!\cdots\!20$$$$\nu^{10} +$$$$17\!\cdots\!12$$$$\nu^{9} -$$$$96\!\cdots\!32$$$$\nu^{8} +$$$$56\!\cdots\!12$$$$\nu^{7} -$$$$28\!\cdots\!44$$$$\nu^{6} +$$$$91\!\cdots\!20$$$$\nu^{5} -$$$$36\!\cdots\!04$$$$\nu^{4} +$$$$61\!\cdots\!44$$$$\nu^{3} -$$$$12\!\cdots\!48$$$$\nu^{2} +$$$$13\!\cdots\!32$$$$\nu -$$$$25\!\cdots\!40$$$$)/$$$$20\!\cdots\!40$$ $$\beta_{7}$$ $$=$$ $$($$$$16\!\cdots\!73$$$$\nu^{19} +$$$$15\!\cdots\!80$$$$\nu^{18} +$$$$13\!\cdots\!76$$$$\nu^{17} +$$$$13\!\cdots\!80$$$$\nu^{16} +$$$$48\!\cdots\!00$$$$\nu^{15} +$$$$50\!\cdots\!20$$$$\nu^{14} +$$$$82\!\cdots\!20$$$$\nu^{13} +$$$$93\!\cdots\!80$$$$\nu^{12} +$$$$73\!\cdots\!88$$$$\nu^{11} +$$$$96\!\cdots\!60$$$$\nu^{10} +$$$$29\!\cdots\!76$$$$\nu^{9} +$$$$54\!\cdots\!40$$$$\nu^{8} +$$$$97\!\cdots\!68$$$$\nu^{7} +$$$$16\!\cdots\!60$$$$\nu^{6} -$$$$25\!\cdots\!36$$$$\nu^{5} +$$$$20\!\cdots\!00$$$$\nu^{4} -$$$$54\!\cdots\!84$$$$\nu^{3} +$$$$69\!\cdots\!20$$$$\nu^{2} -$$$$19\!\cdots\!96$$$$\nu +$$$$32\!\cdots\!20$$$$)/$$$$43\!\cdots\!40$$ $$\beta_{8}$$ $$=$$ $$($$$$16\!\cdots\!73$$$$\nu^{19} -$$$$15\!\cdots\!80$$$$\nu^{18} +$$$$13\!\cdots\!76$$$$\nu^{17} -$$$$13\!\cdots\!80$$$$\nu^{16} +$$$$48\!\cdots\!00$$$$\nu^{15} -$$$$50\!\cdots\!20$$$$\nu^{14} +$$$$82\!\cdots\!20$$$$\nu^{13} -$$$$93\!\cdots\!80$$$$\nu^{12} +$$$$73\!\cdots\!88$$$$\nu^{11} -$$$$96\!\cdots\!60$$$$\nu^{10} +$$$$29\!\cdots\!76$$$$\nu^{9} -$$$$54\!\cdots\!40$$$$\nu^{8} +$$$$97\!\cdots\!68$$$$\nu^{7} -$$$$16\!\cdots\!60$$$$\nu^{6} -$$$$25\!\cdots\!36$$$$\nu^{5} -$$$$20\!\cdots\!00$$$$\nu^{4} -$$$$54\!\cdots\!84$$$$\nu^{3} -$$$$69\!\cdots\!20$$$$\nu^{2} -$$$$19\!\cdots\!96$$$$\nu -$$$$32\!\cdots\!20$$$$)/$$$$43\!\cdots\!40$$ $$\beta_{9}$$ $$=$$ $$($$$$46\!\cdots\!49$$$$\nu^{19} +$$$$18\!\cdots\!44$$$$\nu^{18} +$$$$44\!\cdots\!16$$$$\nu^{17} +$$$$16\!\cdots\!84$$$$\nu^{16} +$$$$17\!\cdots\!68$$$$\nu^{15} +$$$$55\!\cdots\!56$$$$\nu^{14} +$$$$37\!\cdots\!72$$$$\nu^{13} +$$$$94\!\cdots\!84$$$$\nu^{12} +$$$$46\!\cdots\!32$$$$\nu^{11} +$$$$81\!\cdots\!80$$$$\nu^{10} +$$$$34\!\cdots\!88$$$$\nu^{9} +$$$$29\!\cdots\!28$$$$\nu^{8} +$$$$15\!\cdots\!00$$$$\nu^{7} -$$$$17\!\cdots\!84$$$$\nu^{6} +$$$$35\!\cdots\!44$$$$\nu^{5} -$$$$37\!\cdots\!44$$$$\nu^{4} +$$$$36\!\cdots\!40$$$$\nu^{3} -$$$$69\!\cdots\!28$$$$\nu^{2} +$$$$49\!\cdots\!56$$$$\nu -$$$$99\!\cdots\!20$$$$)/$$$$43\!\cdots\!40$$ $$\beta_{10}$$ $$=$$ $$($$$$46\!\cdots\!49$$$$\nu^{19} -$$$$18\!\cdots\!44$$$$\nu^{18} +$$$$44\!\cdots\!16$$$$\nu^{17} -$$$$16\!\cdots\!84$$$$\nu^{16} +$$$$17\!\cdots\!68$$$$\nu^{15} -$$$$55\!\cdots\!56$$$$\nu^{14} +$$$$37\!\cdots\!72$$$$\nu^{13} -$$$$94\!\cdots\!84$$$$\nu^{12} +$$$$46\!\cdots\!32$$$$\nu^{11} -$$$$81\!\cdots\!80$$$$\nu^{10} +$$$$34\!\cdots\!88$$$$\nu^{9} -$$$$29\!\cdots\!28$$$$\nu^{8} +$$$$15\!\cdots\!00$$$$\nu^{7} +$$$$17\!\cdots\!84$$$$\nu^{6} +$$$$35\!\cdots\!44$$$$\nu^{5} +$$$$37\!\cdots\!44$$$$\nu^{4} +$$$$36\!\cdots\!40$$$$\nu^{3} +$$$$69\!\cdots\!28$$$$\nu^{2} +$$$$49\!\cdots\!56$$$$\nu +$$$$99\!\cdots\!20$$$$)/$$$$43\!\cdots\!40$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$37\!\cdots\!71$$$$\nu^{19} -$$$$17\!\cdots\!12$$$$\nu^{18} -$$$$32\!\cdots\!04$$$$\nu^{17} -$$$$15\!\cdots\!32$$$$\nu^{16} -$$$$10\!\cdots\!12$$$$\nu^{15} -$$$$53\!\cdots\!88$$$$\nu^{14} -$$$$17\!\cdots\!08$$$$\nu^{13} -$$$$94\!\cdots\!32$$$$\nu^{12} -$$$$14\!\cdots\!28$$$$\nu^{11} -$$$$89\!\cdots\!20$$$$\nu^{10} -$$$$39\!\cdots\!12$$$$\nu^{9} -$$$$42\!\cdots\!84$$$$\nu^{8} +$$$$12\!\cdots\!60$$$$\nu^{7} -$$$$71\!\cdots\!88$$$$\nu^{6} +$$$$96\!\cdots\!04$$$$\nu^{5} +$$$$86\!\cdots\!72$$$$\nu^{4} +$$$$15\!\cdots\!40$$$$\nu^{3} +$$$$31\!\cdots\!04$$$$\nu^{2} +$$$$19\!\cdots\!36$$$$\nu +$$$$33\!\cdots\!00$$$$)/$$$$21\!\cdots\!20$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!65$$$$\nu^{19} -$$$$37\!\cdots\!24$$$$\nu^{18} -$$$$13\!\cdots\!96$$$$\nu^{17} -$$$$35\!\cdots\!24$$$$\nu^{16} -$$$$48\!\cdots\!96$$$$\nu^{15} -$$$$13\!\cdots\!36$$$$\nu^{14} -$$$$89\!\cdots\!04$$$$\nu^{13} -$$$$26\!\cdots\!64$$$$\nu^{12} -$$$$92\!\cdots\!76$$$$\nu^{11} -$$$$28\!\cdots\!00$$$$\nu^{10} -$$$$52\!\cdots\!60$$$$\nu^{9} -$$$$16\!\cdots\!08$$$$\nu^{8} -$$$$15\!\cdots\!72$$$$\nu^{7} -$$$$42\!\cdots\!96$$$$\nu^{6} -$$$$18\!\cdots\!44$$$$\nu^{5} -$$$$12\!\cdots\!36$$$$\nu^{4} -$$$$52\!\cdots\!64$$$$\nu^{3} +$$$$65\!\cdots\!68$$$$\nu^{2} -$$$$14\!\cdots\!28$$$$\nu -$$$$13\!\cdots\!60$$$$)/$$$$43\!\cdots\!40$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!65$$$$\nu^{19} +$$$$37\!\cdots\!24$$$$\nu^{18} -$$$$13\!\cdots\!96$$$$\nu^{17} +$$$$35\!\cdots\!24$$$$\nu^{16} -$$$$48\!\cdots\!96$$$$\nu^{15} +$$$$13\!\cdots\!36$$$$\nu^{14} -$$$$89\!\cdots\!04$$$$\nu^{13} +$$$$26\!\cdots\!64$$$$\nu^{12} -$$$$92\!\cdots\!76$$$$\nu^{11} +$$$$28\!\cdots\!00$$$$\nu^{10} -$$$$52\!\cdots\!60$$$$\nu^{9} +$$$$16\!\cdots\!08$$$$\nu^{8} -$$$$15\!\cdots\!72$$$$\nu^{7} +$$$$42\!\cdots\!96$$$$\nu^{6} -$$$$18\!\cdots\!44$$$$\nu^{5} +$$$$12\!\cdots\!36$$$$\nu^{4} -$$$$52\!\cdots\!64$$$$\nu^{3} -$$$$65\!\cdots\!68$$$$\nu^{2} -$$$$14\!\cdots\!28$$$$\nu +$$$$13\!\cdots\!60$$$$)/$$$$43\!\cdots\!40$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$32\!\cdots\!58$$$$\nu^{19} +$$$$69\!\cdots\!07$$$$\nu^{18} -$$$$30\!\cdots\!36$$$$\nu^{17} +$$$$54\!\cdots\!12$$$$\nu^{16} -$$$$11\!\cdots\!80$$$$\nu^{15} +$$$$15\!\cdots\!48$$$$\nu^{14} -$$$$22\!\cdots\!40$$$$\nu^{13} +$$$$17\!\cdots\!52$$$$\nu^{12} -$$$$26\!\cdots\!68$$$$\nu^{11} -$$$$21\!\cdots\!20$$$$\nu^{10} -$$$$17\!\cdots\!96$$$$\nu^{9} -$$$$21\!\cdots\!56$$$$\nu^{8} -$$$$68\!\cdots\!48$$$$\nu^{7} -$$$$17\!\cdots\!12$$$$\nu^{6} -$$$$14\!\cdots\!04$$$$\nu^{5} -$$$$59\!\cdots\!52$$$$\nu^{4} -$$$$12\!\cdots\!56$$$$\nu^{3} -$$$$75\!\cdots\!24$$$$\nu^{2} -$$$$15\!\cdots\!44$$$$\nu -$$$$10\!\cdots\!00$$$$)/$$$$36\!\cdots\!20$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!03$$$$\nu^{19} -$$$$19\!\cdots\!40$$$$\nu^{18} -$$$$13\!\cdots\!60$$$$\nu^{17} -$$$$18\!\cdots\!56$$$$\nu^{16} -$$$$52\!\cdots\!36$$$$\nu^{15} -$$$$65\!\cdots\!00$$$$\nu^{14} -$$$$10\!\cdots\!40$$$$\nu^{13} -$$$$12\!\cdots\!80$$$$\nu^{12} -$$$$12\!\cdots\!16$$$$\nu^{11} -$$$$13\!\cdots\!40$$$$\nu^{10} -$$$$87\!\cdots\!24$$$$\nu^{9} -$$$$78\!\cdots\!56$$$$\nu^{8} -$$$$35\!\cdots\!92$$$$\nu^{7} -$$$$25\!\cdots\!00$$$$\nu^{6} -$$$$75\!\cdots\!28$$$$\nu^{5} -$$$$38\!\cdots\!76$$$$\nu^{4} -$$$$70\!\cdots\!36$$$$\nu^{3} -$$$$20\!\cdots\!92$$$$\nu^{2} -$$$$90\!\cdots\!28$$$$\nu -$$$$23\!\cdots\!00$$$$)/$$$$86\!\cdots\!08$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!03$$$$\nu^{19} +$$$$19\!\cdots\!40$$$$\nu^{18} -$$$$13\!\cdots\!60$$$$\nu^{17} +$$$$18\!\cdots\!56$$$$\nu^{16} -$$$$52\!\cdots\!36$$$$\nu^{15} +$$$$65\!\cdots\!00$$$$\nu^{14} -$$$$10\!\cdots\!40$$$$\nu^{13} +$$$$12\!\cdots\!80$$$$\nu^{12} -$$$$12\!\cdots\!16$$$$\nu^{11} +$$$$13\!\cdots\!40$$$$\nu^{10} -$$$$87\!\cdots\!24$$$$\nu^{9} +$$$$78\!\cdots\!56$$$$\nu^{8} -$$$$35\!\cdots\!92$$$$\nu^{7} +$$$$25\!\cdots\!00$$$$\nu^{6} -$$$$75\!\cdots\!28$$$$\nu^{5} +$$$$38\!\cdots\!76$$$$\nu^{4} -$$$$70\!\cdots\!36$$$$\nu^{3} +$$$$20\!\cdots\!92$$$$\nu^{2} -$$$$90\!\cdots\!28$$$$\nu +$$$$23\!\cdots\!00$$$$)/$$$$86\!\cdots\!08$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$79\!\cdots\!25$$$$\nu^{19} -$$$$74\!\cdots\!12$$$$\nu^{17} -$$$$28\!\cdots\!72$$$$\nu^{15} -$$$$56\!\cdots\!28$$$$\nu^{13} -$$$$64\!\cdots\!12$$$$\nu^{11} -$$$$43\!\cdots\!00$$$$\nu^{9} -$$$$16\!\cdots\!84$$$$\nu^{7} -$$$$34\!\cdots\!48$$$$\nu^{5} -$$$$30\!\cdots\!08$$$$\nu^{3} -$$$$38\!\cdots\!36$$$$\nu -$$$$21\!\cdots\!20$$$$)/$$$$43\!\cdots\!40$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!01$$$$\nu^{19} +$$$$91\!\cdots\!76$$$$\nu^{18} -$$$$19\!\cdots\!58$$$$\nu^{17} +$$$$87\!\cdots\!66$$$$\nu^{16} -$$$$78\!\cdots\!96$$$$\nu^{15} +$$$$33\!\cdots\!84$$$$\nu^{14} -$$$$16\!\cdots\!84$$$$\nu^{13} +$$$$70\!\cdots\!56$$$$\nu^{12} -$$$$20\!\cdots\!52$$$$\nu^{11} +$$$$84\!\cdots\!00$$$$\nu^{10} -$$$$14\!\cdots\!52$$$$\nu^{9} +$$$$60\!\cdots\!92$$$$\nu^{8} -$$$$61\!\cdots\!48$$$$\nu^{7} +$$$$25\!\cdots\!24$$$$\nu^{6} -$$$$14\!\cdots\!12$$$$\nu^{5} +$$$$57\!\cdots\!84$$$$\nu^{4} -$$$$14\!\cdots\!96$$$$\nu^{3} +$$$$56\!\cdots\!68$$$$\nu^{2} -$$$$17\!\cdots\!56$$$$\nu +$$$$69\!\cdots\!40$$$$)/$$$$10\!\cdots\!60$$ $$\beta_{19}$$ $$=$$ $$($$$$54\!\cdots\!79$$$$\nu^{19} -$$$$49\!\cdots\!24$$$$\nu^{18} +$$$$52\!\cdots\!80$$$$\nu^{17} -$$$$44\!\cdots\!64$$$$\nu^{16} +$$$$20\!\cdots\!92$$$$\nu^{15} -$$$$16\!\cdots\!16$$$$\nu^{14} +$$$$43\!\cdots\!88$$$$\nu^{13} -$$$$30\!\cdots\!44$$$$\nu^{12} +$$$$53\!\cdots\!56$$$$\nu^{11} -$$$$32\!\cdots\!40$$$$\nu^{10} +$$$$39\!\cdots\!28$$$$\nu^{9} -$$$$19\!\cdots\!88$$$$\nu^{8} +$$$$16\!\cdots\!48$$$$\nu^{7} -$$$$62\!\cdots\!96$$$$\nu^{6} +$$$$38\!\cdots\!00$$$$\nu^{5} -$$$$95\!\cdots\!56$$$$\nu^{4} +$$$$38\!\cdots\!76$$$$\nu^{3} -$$$$52\!\cdots\!72$$$$\nu^{2} +$$$$48\!\cdots\!08$$$$\nu -$$$$58\!\cdots\!20$$$$)/$$$$21\!\cdots\!20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{2} + \beta_{1} - 947$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{19} - 3 \beta_{17} - \beta_{16} + \beta_{15} + 3 \beta_{13} + \beta_{12} + 5 \beta_{11} + 4 \beta_{10} + 9 \beta_{9} - \beta_{8} + 4 \beta_{7} + 18 \beta_{6} - 18 \beta_{5} + 2909 \beta_{3} - 2 \beta_{2} - 1689 \beta_{1} - 1456$$ $$\nu^{4}$$ $$=$$ $$-16 \beta_{19} - 32 \beta_{18} + 20 \beta_{17} + 8 \beta_{16} + 8 \beta_{15} - 40 \beta_{14} - 16 \beta_{13} - 48 \beta_{11} + 46 \beta_{10} - 94 \beta_{9} + 2262 \beta_{8} - 2310 \beta_{7} - 7736 \beta_{6} - 7776 \beta_{5} - 96 \beta_{4} + 20 \beta_{3} - 8116 \beta_{2} - 3998 \beta_{1} + 1618536$$ $$\nu^{5}$$ $$=$$ $$4774 \beta_{19} + 7164 \beta_{17} + 2416 \beta_{16} - 2358 \beta_{15} - 8460 \beta_{13} - 3686 \beta_{12} - 15136 \beta_{11} - 5638 \beta_{10} - 20774 \beta_{9} - 3394 \beta_{8} - 18530 \beta_{7} - 95392 \beta_{6} + 95392 \beta_{5} - 12027380 \beta_{3} + 4774 \beta_{2} + 3383638 \beta_{1} + 6017272$$ $$\nu^{6}$$ $$=$$ $$49732 \beta_{19} + 99464 \beta_{18} - 76612 \beta_{17} - 18396 \beta_{16} - 31336 \beta_{15} + 153224 \beta_{14} + 189928 \beta_{13} - 140196 \beta_{12} + 206264 \beta_{11} + 67032 \beta_{10} + 139232 \beta_{9} - 5042212 \beta_{8} + 5248476 \beta_{7} + 20899840 \beta_{6} + 21053064 \beta_{5} + 412528 \beta_{4} - 76612 \beta_{3} + 36067932 \beta_{2} + 17775956 \beta_{1} - 3293632188$$ $$\nu^{7}$$ $$=$$ $$-10308332 \beta_{19} - 15814644 \beta_{17} - 5343468 \beta_{16} + 4964864 \beta_{15} + 20469828 \beta_{13} + 10161496 \beta_{12} + 39292304 \beta_{11} + 1686552 \beta_{10} + 40978856 \beta_{9} + 23119808 \beta_{8} + 62412112 \beta_{7} + 333205936 \beta_{6} - 333205936 \beta_{5} + 46161518340 \beta_{3} - 10308332 \beta_{2} - 7335722416 \beta_{1} - 23088666492$$ $$\nu^{8}$$ $$=$$ $$-137328540 \beta_{19} - 274657080 \beta_{18} + 237782024 \beta_{17} + 26751916 \beta_{16} + 110576624 \beta_{15} - 475564048 \beta_{14} - 755927100 \beta_{13} + 618598560 \beta_{12} - 666754604 \beta_{11} - 423127672 \beta_{10} - 243626932 \beta_{9} + 11542909876 \beta_{8} - 12209664480 \beta_{7} - 53192934416 \beta_{6} - 53668498464 \beta_{5} - 1333509208 \beta_{4} + 237782024 \beta_{3} - 131191814780 \beta_{2} - 64760035032 \beta_{1} + 7244138989596$$ $$\nu^{9}$$ $$=$$ $$22852332736 \beta_{19} + 36056290812 \beta_{17} + 12160586612 \beta_{16} - 10691746124 \beta_{15} - 49086938820 \beta_{13} - 26234606084 \beta_{12} - 99017232244 \beta_{11} + 17749325248 \beta_{10} - 81267906996 \beta_{9} - 90107754988 \beta_{8} - 189124987232 \beta_{7} - 1005953144928 \beta_{6} + 1005953144928 \beta_{5} - 155218748744068 \beta_{3} + 22852332736 \beta_{2} + 16663803279612 \beta_{1} + 77627402517440$$ $$\nu^{10}$$ $$=$$ $$377013904280 \beta_{19} + 754027808560 \beta_{18} - 687239052304 \beta_{17} - 18988277608 \beta_{16} - 358025626672 \beta_{15} + 1374478104608 \beta_{14} + 2359831051016 \beta_{13} - 1982817146736 \beta_{12} + 1948906626360 \beta_{11} + 1212265964680 \beta_{10} + 736640661680 \beta_{9} - 27186069827880 \beta_{8} + 29134976454240 \beta_{7} + 133833423206320 \beta_{6} + 135207901310928 \beta_{5} + 3897813252720 \beta_{4} - 687239052304 \beta_{3} + 420574267706264 \beta_{2} + 207839495126608 \beta_{1} - 16656520830469680$$ $$\nu^{11}$$ $$=$$ $$-52844695637720 \beta_{19} - 85436071675344 \beta_{17} - 28820144485616 \beta_{16} + 24024551152104 \beta_{15} + 119702180794320 \beta_{13} + 66857485156600 \beta_{12} + 248573655148880 \beta_{11} - 80237179067368 \beta_{10} + 168336476081512 \beta_{9} + 294201104629688 \beta_{8} + 542774759778568 \beta_{7} + 2849031256506416 \beta_{6} - 2849031256506416 \beta_{5} + 477070773789830224 \beta_{3} - 52844695637720 \beta_{2} - 39111358012746536 \beta_{1} - 238578104930752784$$ $$\nu^{12}$$ $$=$$ $$-1031155818871232 \beta_{19} - 2062311637742464 \beta_{18} + 1910856229607264 \beta_{17} - 50163809970384 \beta_{16} + 1081319628841616 \beta_{15} - 3821712459214528 \beta_{14} - 6703062261480800 \beta_{13} + 5671906442609568 \beta_{12} - 5435593916986240 \beta_{11} - 2781671916339072 \beta_{10} - 2653922000647168 \beta_{9} + 65569748258569376 \beta_{8} - 71005342175555616 \beta_{7} - 336964386435565856 \beta_{6} - 340786098894780384 \beta_{5} - 10871187833972480 \beta_{4} + 1910856229607264 \beta_{3} - 1252010738358209376 \beta_{2} - 619174496941946800 \beta_{1} + 39486478198066626384$$ $$\nu^{13}$$ $$=$$ $$126598288879916080 \beta_{19} + 208594302406153104 \beta_{17} + 70501162650687120 \beta_{16} - 56097126229228960 \beta_{15} - 297418603296938640 \beta_{13} - 170820314417022560 \beta_{12} - 626012321213634016 \beta_{11} + 260545760847463488 \beta_{10} - 365466560366170528 \beta_{9} - 881272534847141344 \beta_{8} - 1507284856060775360 \beta_{7} - 7809485371775872544 \beta_{6} + 7809485371775872544 \beta_{5} - 1385757359945820915792 \beta_{3} + 126598288879916080 \beta_{2} + 94084327224592288736 \beta_{1} + 692982977124113534448$$ $$\nu^{14}$$ $$=$$ $$2795087374625698608 \beta_{19} + 5590174749251397216 \beta_{18} - 5184070173581355424 \beta_{17} + 314363072598861232 \beta_{16} - 3109450447224559840 \beta_{15} + 10368140347162710848 \beta_{14} + 18214105449309424272 \beta_{13} - 15419018074683725664 \beta_{12} + 14780158764665677840 \beta_{11} + 5713443740002446272 \beta_{10} + 9066715024663231568 \beta_{9} - 161121399202677041264 \beta_{8} + 175901557967342719104 \beta_{7} + 851973117362993565376 \beta_{6} + 862341257710156276224 \beta_{5} + 29560317529331355680 \beta_{4} - 5184070173581355424 \beta_{3} + 3561674194135952349040 \beta_{2} + 1762270411817041990560 \beta_{1} - 95755885763708034435696$$ $$\nu^{15}$$ $$=$$ $$-311122329396829014080 \beta_{19} - 520145827332612265776 \beta_{17} - 176150429869075450576 \beta_{16} + 134971899527753563504 \beta_{15} + 750240186822547374480 \beta_{13} + 439117857425718360400 \beta_{12} + 1584395256937073851280 \beta_{11} - 758338291199936382464 \beta_{10} + 826056965737137468816 \beta_{9} + 2513024929958500792688 \beta_{8} + 4097420186895574643968 \beta_{7} + 21011323396167895760832 \beta_{6} - 21011323396167895760832 \beta_{5} + 3882380587027106837330000 \beta_{3} - 311122329396829014080 \beta_{2} - 230631278109357544659120 \beta_{1} - 1941450366427219724797888$$ $$\nu^{16}$$ $$=$$ $$-7499938818204877312864 \beta_{19} - 14999877636409754625728 \beta_{18} + 13839013821221453929856 \beta_{17} - 1148862339670893234400 \beta_{16} + 8648801157875770547264 \beta_{15} - 27678027642442907859712 \beta_{14} - 48371252981800774001824 \beta_{13} + 40871314163595896688960 \beta_{12} - 39585471075485998193568 \beta_{11} - 10884517016745502983584 \beta_{10} - 28700954058740495209984 \beta_{9} + 401581107395346943357344 \beta_{8} - 441166578470832941550912 \beta_{7} - 2164563122676049425985472 \beta_{6} - 2192241150318492333845184 \beta_{5} - 79170942150971996387136 \beta_{4} + 13839013821221453929856 \beta_{3} - 9841696544187852344805088 \beta_{2} - 4871173756606321158935552 \beta_{1} + 236234184453655878284207232$$ $$\nu^{17}$$ $$=$$ $$778070523237152708553376 \beta_{19} + 1315689373646300765999808 \beta_{17} + 446045027727049426012288 \beta_{16} - 332025495510103282541088 \beta_{15} - 1913164715938618340811648 \beta_{13} - 1135094192701465632258272 \beta_{12} - 4029910744437329680591360 \beta_{11} + 2102036740127337229603232 \beta_{10} - 1927874004309992450988128 \beta_{9} - 6950597077529442144546976 \beta_{8} - 10980507821966771825138336 \beta_{7} - 55885431560610256699855168 \beta_{6} + 55885431560610256699855168 \beta_{5} - 10621571169526449000677379392 \beta_{3} + 778070523237152708553376 \beta_{2} + 573555691636724372635710880 \beta_{1} + 5311443429450047650721689600$$ $$\nu^{18}$$ $$=$$ $$19\!\cdots\!80$$$$\beta_{19} +$$$$39\!\cdots\!60$$$$\beta_{18} -$$$$36\!\cdots\!64$$$$\beta_{17} +$$$$35\!\cdots\!76$$$$\beta_{16} -$$$$23\!\cdots\!56$$$$\beta_{15} +$$$$73\!\cdots\!28$$$$\beta_{14} +$$$$12\!\cdots\!08$$$$\beta_{13} -$$$$10\!\cdots\!28$$$$\beta_{12} +$$$$10\!\cdots\!32$$$$\beta_{11} +$$$$19\!\cdots\!20$$$$\beta_{10} +$$$$85\!\cdots\!12$$$$\beta_{9} -$$$$10\!\cdots\!28$$$$\beta_{8} +$$$$11\!\cdots\!60$$$$\beta_{7} +$$$$55\!\cdots\!04$$$$\beta_{6} +$$$$55\!\cdots\!32$$$$\beta_{5} +$$$$21\!\cdots\!64$$$$\beta_{4} -$$$$36\!\cdots\!64$$$$\beta_{3} +$$$$26\!\cdots\!84$$$$\beta_{2} +$$$$13\!\cdots\!36$$$$\beta_{1} -$$$$59\!\cdots\!40$$ $$\nu^{19}$$ $$=$$ $$-1968867280672806058874672960 \beta_{19} - 3359595673001111842210118208 \beta_{17} - 1138954912647121846594226112 \beta_{16} + 829912368025684212280446848 \beta_{15} + 4914522729385307301978524352 \beta_{13} + 2945655448712501243103851392 \beta_{12} + 10294606042528271550561900032 \beta_{11} - 5682099507090967766977500288 \beta_{10} + 4612506535437303783584399744 \beta_{9} + 18845872946944072984326788096 \beta_{8} + 29140478989472344534888688128 \beta_{7} + 147536476496133664299405504640 \beta_{6} - 147536476496133664299405504640 \beta_{5} + 28597599722481369165786142878528 \beta_{3} - 1968867280672806058874672960 \beta_{2} - 1442031536696991547838165286016 \beta_{1} - 14300479659077185138814176498368$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 − 50.9288i − 35.1361i − 25.0503i − 21.8702i 3.67103i 4.04998i 17.5071i 21.3475i 42.7313i 43.6786i 50.9288i 35.1361i 25.0503i 21.8702i − 3.67103i − 4.04998i − 17.5071i − 21.3475i − 42.7313i − 43.6786i
0 −45.6057 26.3304i 0 −106.246 + 184.024i 0 −29.3098 0 1022.08 + 1770.30i 0
65.2 0 −31.9288 18.4341i 0 71.3406 123.566i 0 421.828 0 315.131 + 545.823i 0
65.3 0 −23.1942 13.3912i 0 16.0954 27.8780i 0 −416.808 0 −5.85193 10.1358i 0
65.4 0 −20.4402 11.8011i 0 15.3686 26.6192i 0 −77.2273 0 −85.9659 148.897i 0
65.5 0 1.67921 + 0.969492i 0 −72.6742 + 125.875i 0 621.925 0 −362.620 628.077i 0
65.6 0 2.00739 + 1.15896i 0 −67.5013 + 116.916i 0 −157.715 0 −361.814 626.680i 0
65.7 0 13.6616 + 7.88751i 0 93.3237 161.641i 0 −434.577 0 −240.074 415.821i 0
65.8 0 16.9875 + 9.80775i 0 72.3330 125.284i 0 479.542 0 −172.116 298.114i 0
65.9 0 35.5064 + 20.4996i 0 20.1615 34.9207i 0 53.8286 0 475.968 + 824.401i 0
65.10 0 36.3268 + 20.9733i 0 −70.2011 + 121.592i 0 −229.488 0 515.258 + 892.453i 0
69.1 0 −45.6057 + 26.3304i 0 −106.246 184.024i 0 −29.3098 0 1022.08 1770.30i 0
69.2 0 −31.9288 + 18.4341i 0 71.3406 + 123.566i 0 421.828 0 315.131 545.823i 0
69.3 0 −23.1942 + 13.3912i 0 16.0954 + 27.8780i 0 −416.808 0 −5.85193 + 10.1358i 0
69.4 0 −20.4402 + 11.8011i 0 15.3686 + 26.6192i 0 −77.2273 0 −85.9659 + 148.897i 0
69.5 0 1.67921 0.969492i 0 −72.6742 125.875i 0 621.925 0 −362.620 + 628.077i 0
69.6 0 2.00739 1.15896i 0 −67.5013 116.916i 0 −157.715 0 −361.814 + 626.680i 0
69.7 0 13.6616 7.88751i 0 93.3237 + 161.641i 0 −434.577 0 −240.074 + 415.821i 0
69.8 0 16.9875 9.80775i 0 72.3330 + 125.284i 0 479.542 0 −172.116 + 298.114i 0
69.9 0 35.5064 20.4996i 0 20.1615 + 34.9207i 0 53.8286 0 475.968 824.401i 0
69.10 0 36.3268 20.9733i 0 −70.2011 121.592i 0 −229.488 0 515.258 892.453i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 69.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.7.h.a 20
3.b odd 2 1 684.7.y.c 20
19.d odd 6 1 inner 76.7.h.a 20
57.f even 6 1 684.7.y.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.h.a 20 1.a even 1 1 trivial
76.7.h.a 20 19.d odd 6 1 inner
684.7.y.c 20 3.b odd 2 1
684.7.y.c 20 57.f even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$86\!\cdots\!25$$$$-$$$$14\!\cdots\!70$$$$T +$$$$10\!\cdots\!37$$$$T^{2} -$$$$37\!\cdots\!82$$$$T^{3} +$$$$58\!\cdots\!47$$$$T^{4} -$$$$11\!\cdots\!48$$$$T^{5} -$$$$26\!\cdots\!38$$$$T^{6} + 92548774443086047008 T^{7} + 8743209810087203073 T^{8} - 311919559815103182 T^{9} - 16860845133654057 T^{10} + 556330559992830 T^{11} + 25904858495841 T^{12} - 541657530216 T^{13} - 21602042718 T^{14} + 355544100 T^{15} + 13279603 T^{16} - 137850 T^{17} - 4295 T^{18} + 30 T^{19} + T^{20}$$
$5$ $$80\!\cdots\!00$$$$-$$$$66\!\cdots\!00$$$$T +$$$$38\!\cdots\!00$$$$T^{2} -$$$$12\!\cdots\!00$$$$T^{3} +$$$$33\!\cdots\!00$$$$T^{4} -$$$$50\!\cdots\!00$$$$T^{5} +$$$$72\!\cdots\!00$$$$T^{6} -$$$$48\!\cdots\!00$$$$T^{7} +$$$$98\!\cdots\!00$$$$T^{8} -$$$$33\!\cdots\!00$$$$T^{9} +$$$$81\!\cdots\!00$$$$T^{10} -$$$$11\!\cdots\!40$$$$T^{11} + 4831041099310077121 T^{12} - 2512748491438508 T^{13} + 192060089599318 T^{14} + 30640959724 T^{15} + 5494347739 T^{16} + 1625808 T^{17} + 93542 T^{18} + 56 T^{19} + T^{20}$$
$7$ $$($$$$10\!\cdots\!60$$$$+$$$$38\!\cdots\!36$$$$T - 16530284186312080320 T^{2} - 1238280827304680864 T^{3} - 8200248406879804 T^{4} + 4001628862056 T^{5} + 120848880692 T^{6} + 57262372 T^{7} - 595356 T^{8} - 232 T^{9} + T^{10} )^{2}$$
$11$ $$( -$$$$16\!\cdots\!64$$$$-$$$$10\!\cdots\!76$$$$T +$$$$19\!\cdots\!04$$$$T^{2} -$$$$13\!\cdots\!28$$$$T^{3} - 32328810278421817535 T^{4} + 23894369299865242 T^{5} + 23204968594775 T^{6} - 12293162312 T^{7} - 8106229 T^{8} + 1822 T^{9} + T^{10} )^{2}$$
$13$ $$17\!\cdots\!00$$$$-$$$$27\!\cdots\!60$$$$T +$$$$14\!\cdots\!92$$$$T^{2} -$$$$18\!\cdots\!72$$$$T^{3} -$$$$26\!\cdots\!64$$$$T^{4} +$$$$61\!\cdots\!48$$$$T^{5} +$$$$35\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!84$$$$T^{7} -$$$$20\!\cdots\!88$$$$T^{8} -$$$$28\!\cdots\!24$$$$T^{9} +$$$$85\!\cdots\!96$$$$T^{10} +$$$$20\!\cdots\!16$$$$T^{11} -$$$$19\!\cdots\!71$$$$T^{12} -$$$$61\!\cdots\!88$$$$T^{13} +$$$$29\!\cdots\!58$$$$T^{14} + 1357246554301874124 T^{15} - 111849222402981 T^{16} - 113125660200 T^{17} + 1149270 T^{18} + 7140 T^{19} + T^{20}$$
$17$ $$83\!\cdots\!00$$$$-$$$$72\!\cdots\!00$$$$T +$$$$37\!\cdots\!00$$$$T^{2} +$$$$56\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!00$$$$T^{4} +$$$$26\!\cdots\!00$$$$T^{5} +$$$$16\!\cdots\!64$$$$T^{6} +$$$$92\!\cdots\!36$$$$T^{7} +$$$$17\!\cdots\!44$$$$T^{8} +$$$$75\!\cdots\!96$$$$T^{9} +$$$$90\!\cdots\!96$$$$T^{10} +$$$$29\!\cdots\!52$$$$T^{11} +$$$$34\!\cdots\!33$$$$T^{12} +$$$$70\!\cdots\!24$$$$T^{13} +$$$$81\!\cdots\!30$$$$T^{14} + 6480072515640758524 T^{15} + 14221206461378515 T^{16} + 18781566024 T^{17} + 149142974 T^{18} - 1132 T^{19} + T^{20}$$
$19$ $$53\!\cdots\!01$$$$-$$$$23\!\cdots\!10$$$$T +$$$$30\!\cdots\!49$$$$T^{2} -$$$$15\!\cdots\!82$$$$T^{3} +$$$$96\!\cdots\!92$$$$T^{4} -$$$$68\!\cdots\!26$$$$T^{5} +$$$$17\!\cdots\!67$$$$T^{6} -$$$$23\!\cdots\!66$$$$T^{7} +$$$$20\!\cdots\!19$$$$T^{8} -$$$$65\!\cdots\!80$$$$T^{9} +$$$$21\!\cdots\!76$$$$T^{10} -$$$$13\!\cdots\!80$$$$T^{11} +$$$$92\!\cdots\!79$$$$T^{12} -$$$$22\!\cdots\!26$$$$T^{13} +$$$$35\!\cdots\!27$$$$T^{14} - 29596043962999887926 T^{15} + 8945958210968132 T^{16} - 305532365162 T^{17} + 125822389 T^{18} - 2110 T^{19} + T^{20}$$
$23$ $$36\!\cdots\!00$$$$-$$$$64\!\cdots\!00$$$$T +$$$$38\!\cdots\!00$$$$T^{2} -$$$$20\!\cdots\!80$$$$T^{3} +$$$$17\!\cdots\!24$$$$T^{4} -$$$$12\!\cdots\!08$$$$T^{5} +$$$$60\!\cdots\!96$$$$T^{6} +$$$$22\!\cdots\!12$$$$T^{7} +$$$$13\!\cdots\!56$$$$T^{8} +$$$$77\!\cdots\!32$$$$T^{9} +$$$$23\!\cdots\!72$$$$T^{10} +$$$$14\!\cdots\!88$$$$T^{11} +$$$$23\!\cdots\!41$$$$T^{12} +$$$$91\!\cdots\!88$$$$T^{13} +$$$$11\!\cdots\!08$$$$T^{14} +$$$$24\!\cdots\!56$$$$T^{15} + 398806009766960113 T^{16} + 4021565248416 T^{17} + 763368644 T^{18} - 832 T^{19} + T^{20}$$
$29$ $$20\!\cdots\!00$$$$-$$$$65\!\cdots\!40$$$$T +$$$$68\!\cdots\!88$$$$T^{2} +$$$$67\!\cdots\!40$$$$T^{3} -$$$$41\!\cdots\!28$$$$T^{4} -$$$$56\!\cdots\!20$$$$T^{5} +$$$$19\!\cdots\!44$$$$T^{6} +$$$$34\!\cdots\!84$$$$T^{7} -$$$$44\!\cdots\!60$$$$T^{8} -$$$$10\!\cdots\!60$$$$T^{9} -$$$$11\!\cdots\!60$$$$T^{10} +$$$$22\!\cdots\!48$$$$T^{11} +$$$$59\!\cdots\!53$$$$T^{12} -$$$$20\!\cdots\!72$$$$T^{13} -$$$$79\!\cdots\!22$$$$T^{14} +$$$$14\!\cdots\!44$$$$T^{15} + 7426736486043851907 T^{16} - 36606351927600 T^{17} - 3312481230 T^{18} + 10920 T^{19} + T^{20}$$
$31$ $$27\!\cdots\!76$$$$+$$$$33\!\cdots\!88$$$$T^{2} +$$$$11\!\cdots\!36$$$$T^{4} +$$$$16\!\cdots\!52$$$$T^{6} +$$$$10\!\cdots\!68$$$$T^{8} +$$$$34\!\cdots\!96$$$$T^{10} +$$$$60\!\cdots\!04$$$$T^{12} +$$$$58\!\cdots\!68$$$$T^{14} + 31167033942997357704 T^{16} + 8744900760 T^{18} + T^{20}$$
$37$ $$24\!\cdots\!00$$$$+$$$$14\!\cdots\!92$$$$T^{2} +$$$$38\!\cdots\!28$$$$T^{4} +$$$$46\!\cdots\!32$$$$T^{6} +$$$$30\!\cdots\!92$$$$T^{8} +$$$$12\!\cdots\!76$$$$T^{10} +$$$$34\!\cdots\!48$$$$T^{12} +$$$$58\!\cdots\!60$$$$T^{14} +$$$$62\!\cdots\!76$$$$T^{16} + 38093619336 T^{18} + T^{20}$$
$41$ $$77\!\cdots\!25$$$$-$$$$10\!\cdots\!50$$$$T +$$$$50\!\cdots\!43$$$$T^{2} -$$$$30\!\cdots\!34$$$$T^{3} -$$$$46\!\cdots\!89$$$$T^{4} +$$$$81\!\cdots\!64$$$$T^{5} +$$$$34\!\cdots\!98$$$$T^{6} -$$$$97\!\cdots\!44$$$$T^{7} -$$$$80\!\cdots\!35$$$$T^{8} +$$$$43\!\cdots\!30$$$$T^{9} +$$$$11\!\cdots\!29$$$$T^{10} -$$$$13\!\cdots\!74$$$$T^{11} +$$$$34\!\cdots\!09$$$$T^{12} +$$$$24\!\cdots\!44$$$$T^{13} -$$$$10\!\cdots\!14$$$$T^{14} -$$$$30\!\cdots\!84$$$$T^{15} +$$$$20\!\cdots\!71$$$$T^{16} + 2248272154467990 T^{17} - 16612124853 T^{18} - 109206 T^{19} + T^{20}$$
$43$ $$13\!\cdots\!00$$$$+$$$$20\!\cdots\!40$$$$T +$$$$26\!\cdots\!04$$$$T^{2} +$$$$61\!\cdots\!32$$$$T^{3} +$$$$14\!\cdots\!52$$$$T^{4} -$$$$72\!\cdots\!60$$$$T^{5} +$$$$62\!\cdots\!80$$$$T^{6} -$$$$14\!\cdots\!00$$$$T^{7} +$$$$80\!\cdots\!44$$$$T^{8} -$$$$19\!\cdots\!96$$$$T^{9} +$$$$68\!\cdots\!72$$$$T^{10} -$$$$96\!\cdots\!24$$$$T^{11} +$$$$17\!\cdots\!65$$$$T^{12} -$$$$94\!\cdots\!40$$$$T^{13} +$$$$19\!\cdots\!60$$$$T^{14} -$$$$88\!\cdots\!36$$$$T^{15} +$$$$12\!\cdots\!29$$$$T^{16} - 3362841824805912 T^{17} + 46107425668 T^{18} - 110740 T^{19} + T^{20}$$
$47$ $$51\!\cdots\!00$$$$+$$$$20\!\cdots\!00$$$$T +$$$$50\!\cdots\!64$$$$T^{2} -$$$$20\!\cdots\!16$$$$T^{3} +$$$$27\!\cdots\!36$$$$T^{4} -$$$$86\!\cdots\!28$$$$T^{5} +$$$$66\!\cdots\!56$$$$T^{6} -$$$$25\!\cdots\!92$$$$T^{7} +$$$$10\!\cdots\!12$$$$T^{8} -$$$$24\!\cdots\!40$$$$T^{9} +$$$$49\!\cdots\!72$$$$T^{10} -$$$$59\!\cdots\!84$$$$T^{11} +$$$$67\!\cdots\!29$$$$T^{12} -$$$$49\!\cdots\!56$$$$T^{13} +$$$$45\!\cdots\!96$$$$T^{14} -$$$$25\!\cdots\!80$$$$T^{15} +$$$$21\!\cdots\!53$$$$T^{16} - 7000757513468328 T^{17} + 54045046784 T^{18} - 107080 T^{19} + T^{20}$$
$53$ $$46\!\cdots\!00$$$$-$$$$12\!\cdots\!20$$$$T +$$$$14\!\cdots\!88$$$$T^{2} -$$$$76\!\cdots\!72$$$$T^{3} +$$$$22\!\cdots\!88$$$$T^{4} -$$$$37\!\cdots\!28$$$$T^{5} +$$$$30\!\cdots\!44$$$$T^{6} +$$$$37\!\cdots\!60$$$$T^{7} -$$$$25\!\cdots\!08$$$$T^{8} +$$$$56\!\cdots\!08$$$$T^{9} +$$$$22\!\cdots\!76$$$$T^{10} -$$$$17\!\cdots\!88$$$$T^{11} -$$$$29\!\cdots\!75$$$$T^{12} +$$$$70\!\cdots\!52$$$$T^{13} -$$$$50\!\cdots\!46$$$$T^{14} -$$$$22\!\cdots\!04$$$$T^{15} +$$$$53\!\cdots\!91$$$$T^{16} + 26102505221094600 T^{17} - 80804387478 T^{18} - 254796 T^{19} + T^{20}$$
$59$ $$26\!\cdots\!89$$$$+$$$$45\!\cdots\!38$$$$T +$$$$17\!\cdots\!41$$$$T^{2} -$$$$14\!\cdots\!22$$$$T^{3} -$$$$36\!\cdots\!93$$$$T^{4} +$$$$31\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!82$$$$T^{6} -$$$$86\!\cdots\!60$$$$T^{7} -$$$$13\!\cdots\!59$$$$T^{8} +$$$$90\!\cdots\!70$$$$T^{9} +$$$$12\!\cdots\!55$$$$T^{10} -$$$$25\!\cdots\!14$$$$T^{11} -$$$$60\!\cdots\!11$$$$T^{12} +$$$$39\!\cdots\!28$$$$T^{13} +$$$$22\!\cdots\!86$$$$T^{14} +$$$$81\!\cdots\!84$$$$T^{15} -$$$$49\!\cdots\!57$$$$T^{16} - 70871375752429266 T^{17} + 8231727741 T^{18} + 610638 T^{19} + T^{20}$$
$61$ $$10\!\cdots\!96$$$$-$$$$29\!\cdots\!16$$$$T +$$$$75\!\cdots\!36$$$$T^{2} -$$$$10\!\cdots\!84$$$$T^{3} +$$$$15\!\cdots\!64$$$$T^{4} -$$$$15\!\cdots\!04$$$$T^{5} +$$$$18\!\cdots\!08$$$$T^{6} -$$$$14\!\cdots\!84$$$$T^{7} +$$$$12\!\cdots\!88$$$$T^{8} -$$$$54\!\cdots\!52$$$$T^{9} +$$$$35\!\cdots\!20$$$$T^{10} -$$$$84\!\cdots\!56$$$$T^{11} +$$$$82\!\cdots\!37$$$$T^{12} -$$$$85\!\cdots\!96$$$$T^{13} +$$$$94\!\cdots\!06$$$$T^{14} -$$$$59\!\cdots\!60$$$$T^{15} +$$$$78\!\cdots\!95$$$$T^{16} - 23723336799805312 T^{17} + 335894265694 T^{18} - 47864 T^{19} + T^{20}$$
$67$ $$21\!\cdots\!25$$$$+$$$$99\!\cdots\!50$$$$T +$$$$13\!\cdots\!25$$$$T^{2} -$$$$83\!\cdots\!10$$$$T^{3} -$$$$31\!\cdots\!61$$$$T^{4} +$$$$92\!\cdots\!48$$$$T^{5} +$$$$53\!\cdots\!46$$$$T^{6} -$$$$82\!\cdots\!12$$$$T^{7} +$$$$13\!\cdots\!73$$$$T^{8} -$$$$56\!\cdots\!90$$$$T^{9} -$$$$24\!\cdots\!77$$$$T^{10} +$$$$19\!\cdots\!62$$$$T^{11} +$$$$45\!\cdots\!89$$$$T^{12} -$$$$45\!\cdots\!12$$$$T^{13} -$$$$14\!\cdots\!18$$$$T^{14} +$$$$42\!\cdots\!76$$$$T^{15} +$$$$76\!\cdots\!67$$$$T^{16} - 246919245006075030 T^{17} - 59150052867 T^{18} + 839562 T^{19} + T^{20}$$
$71$ $$65\!\cdots\!24$$$$+$$$$20\!\cdots\!04$$$$T -$$$$44\!\cdots\!72$$$$T^{2} -$$$$20\!\cdots\!00$$$$T^{3} +$$$$61\!\cdots\!36$$$$T^{4} +$$$$79\!\cdots\!52$$$$T^{5} -$$$$51\!\cdots\!44$$$$T^{6} -$$$$24\!\cdots\!36$$$$T^{7} +$$$$29\!\cdots\!52$$$$T^{8} +$$$$22\!\cdots\!24$$$$T^{9} -$$$$85\!\cdots\!16$$$$T^{10} -$$$$91\!\cdots\!24$$$$T^{11} +$$$$17\!\cdots\!21$$$$T^{12} +$$$$30\!\cdots\!52$$$$T^{13} -$$$$15\!\cdots\!48$$$$T^{14} -$$$$28\!\cdots\!36$$$$T^{15} +$$$$10\!\cdots\!89$$$$T^{16} + 148672245901107600 T^{17} - 360663948660 T^{18} - 366660 T^{19} + T^{20}$$
$73$ $$39\!\cdots\!25$$$$-$$$$15\!\cdots\!50$$$$T +$$$$67\!\cdots\!75$$$$T^{2} +$$$$20\!\cdots\!10$$$$T^{3} +$$$$58\!\cdots\!91$$$$T^{4} -$$$$72\!\cdots\!80$$$$T^{5} +$$$$19\!\cdots\!22$$$$T^{6} -$$$$39\!\cdots\!76$$$$T^{7} +$$$$47\!\cdots\!33$$$$T^{8} -$$$$12\!\cdots\!06$$$$T^{9} +$$$$68\!\cdots\!69$$$$T^{10} -$$$$15\!\cdots\!94$$$$T^{11} +$$$$58\!\cdots\!93$$$$T^{12} -$$$$11\!\cdots\!40$$$$T^{13} +$$$$31\!\cdots\!38$$$$T^{14} -$$$$45\!\cdots\!08$$$$T^{15} +$$$$83\!\cdots\!55$$$$T^{16} - 778169535671620938 T^{17} + 1269773791015 T^{18} - 854482 T^{19} + T^{20}$$
$79$ $$39\!\cdots\!04$$$$-$$$$12\!\cdots\!76$$$$T +$$$$11\!\cdots\!44$$$$T^{2} +$$$$78\!\cdots\!76$$$$T^{3} -$$$$12\!\cdots\!36$$$$T^{4} -$$$$52\!\cdots\!96$$$$T^{5} +$$$$10\!\cdots\!12$$$$T^{6} +$$$$16\!\cdots\!44$$$$T^{7} -$$$$21\!\cdots\!40$$$$T^{8} -$$$$25\!\cdots\!48$$$$T^{9} +$$$$30\!\cdots\!00$$$$T^{10} +$$$$76\!\cdots\!28$$$$T^{11} -$$$$24\!\cdots\!63$$$$T^{12} +$$$$50\!\cdots\!96$$$$T^{13} +$$$$14\!\cdots\!76$$$$T^{14} -$$$$13\!\cdots\!56$$$$T^{15} -$$$$37\!\cdots\!51$$$$T^{16} + 533131673602281984 T^{17} + 674305269336 T^{18} - 1718592 T^{19} + T^{20}$$
$83$ $$( -$$$$41\!\cdots\!00$$$$+$$$$32\!\cdots\!00$$$$T +$$$$15\!\cdots\!00$$$$T^{2} -$$$$67\!\cdots\!84$$$$T^{3} -$$$$24\!\cdots\!27$$$$T^{4} +$$$$29\!\cdots\!90$$$$T^{5} +$$$$12\!\cdots\!95$$$$T^{6} - 85757229543811628 T^{7} - 1952888590597 T^{8} - 219806 T^{9} + T^{10} )^{2}$$
$89$ $$82\!\cdots\!96$$$$-$$$$81\!\cdots\!36$$$$T +$$$$31\!\cdots\!28$$$$T^{2} -$$$$44\!\cdots\!76$$$$T^{3} +$$$$26\!\cdots\!64$$$$T^{4} -$$$$14\!\cdots\!40$$$$T^{5} -$$$$36\!\cdots\!96$$$$T^{6} +$$$$54\!\cdots\!24$$$$T^{7} +$$$$37\!\cdots\!04$$$$T^{8} -$$$$37\!\cdots\!04$$$$T^{9} -$$$$21\!\cdots\!92$$$$T^{10} +$$$$13\!\cdots\!80$$$$T^{11} +$$$$92\!\cdots\!09$$$$T^{12} +$$$$11\!\cdots\!84$$$$T^{13} -$$$$17\!\cdots\!74$$$$T^{14} -$$$$48\!\cdots\!64$$$$T^{15} +$$$$25\!\cdots\!75$$$$T^{16} + 931521728663143776 T^{17} - 1872488242110 T^{18} - 478032 T^{19} + T^{20}$$
$97$ $$63\!\cdots\!25$$$$-$$$$39\!\cdots\!90$$$$T +$$$$50\!\cdots\!23$$$$T^{2} +$$$$18\!\cdots\!54$$$$T^{3} -$$$$39\!\cdots\!45$$$$T^{4} -$$$$13\!\cdots\!92$$$$T^{5} +$$$$60\!\cdots\!58$$$$T^{6} -$$$$82\!\cdots\!28$$$$T^{7} +$$$$99\!\cdots\!45$$$$T^{8} +$$$$86\!\cdots\!98$$$$T^{9} -$$$$53\!\cdots\!75$$$$T^{10} -$$$$66\!\cdots\!54$$$$T^{11} +$$$$75\!\cdots\!93$$$$T^{12} +$$$$19\!\cdots\!68$$$$T^{13} -$$$$47\!\cdots\!38$$$$T^{14} -$$$$34\!\cdots\!16$$$$T^{15} +$$$$21\!\cdots\!75$$$$T^{16} - 1094638162300438662 T^{17} - 5710324275885 T^{18} + 191286 T^{19} + T^{20}$$