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 $2$

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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:

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

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