Properties

Label 177.8.a.d
Level $177$
Weight $8$
Character orbit 177.a
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + 27 q^{3} + ( 75 + \beta_{1} + \beta_{2} ) q^{4} + ( 37 + \beta_{1} + \beta_{4} ) q^{5} + ( 27 + 27 \beta_{1} ) q^{6} + ( 170 + 5 \beta_{1} + \beta_{7} ) q^{7} + ( 207 + 61 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{8} + 729 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + 27 q^{3} + ( 75 + \beta_{1} + \beta_{2} ) q^{4} + ( 37 + \beta_{1} + \beta_{4} ) q^{5} + ( 27 + 27 \beta_{1} ) q^{6} + ( 170 + 5 \beta_{1} + \beta_{7} ) q^{7} + ( 207 + 61 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{8} + 729 q^{9} + ( 179 + 62 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{10} + ( 815 + 69 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{11} + ( 2025 + 27 \beta_{1} + 27 \beta_{2} ) q^{12} + ( 743 + 53 \beta_{1} - 13 \beta_{2} + \beta_{4} + 2 \beta_{7} + \beta_{15} ) q^{13} + ( 1085 + 223 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{14} + ( 999 + 27 \beta_{1} + 27 \beta_{4} ) q^{15} + ( 3207 + 424 \beta_{1} + 67 \beta_{2} + 3 \beta_{3} + 14 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{16} + ( 3967 + 77 \beta_{1} + 19 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{17} + ( 729 + 729 \beta_{1} ) q^{18} + ( 3168 - 128 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 4 \beta_{12} - 3 \beta_{13} - \beta_{15} + \beta_{17} ) q^{19} + ( 7840 + 213 \beta_{1} + 102 \beta_{2} - 6 \beta_{3} + 80 \beta_{4} - \beta_{5} - \beta_{6} - 9 \beta_{7} - \beta_{8} + 3 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} + \beta_{16} - 2 \beta_{17} ) q^{20} + ( 4590 + 135 \beta_{1} + 27 \beta_{7} ) q^{21} + ( 14833 + 1082 \beta_{1} + 118 \beta_{2} - 6 \beta_{3} + 42 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 22 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} + 6 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} - 3 \beta_{16} - 2 \beta_{17} ) q^{22} + ( 8737 - 1172 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} - 19 \beta_{4} + 9 \beta_{5} + \beta_{6} + 5 \beta_{7} - 3 \beta_{9} - 5 \beta_{10} - 2 \beta_{11} - 7 \beta_{12} + 2 \beta_{13} + \beta_{14} + 5 \beta_{15} - 2 \beta_{16} + 7 \beta_{17} ) q^{23} + ( 5589 + 1647 \beta_{1} + 108 \beta_{2} + 27 \beta_{3} ) q^{24} + ( 22114 + 134 \beta_{1} + 25 \beta_{2} - 14 \beta_{3} + 89 \beta_{4} - 9 \beta_{5} + 5 \beta_{6} - 29 \beta_{7} - 2 \beta_{8} + \beta_{9} + 4 \beta_{10} + 4 \beta_{11} + \beta_{12} + 5 \beta_{14} + \beta_{15} + 7 \beta_{16} - 4 \beta_{17} ) q^{25} + ( 10397 - 656 \beta_{1} + 66 \beta_{2} - 27 \beta_{3} - 73 \beta_{4} + 4 \beta_{5} - 15 \beta_{6} - 20 \beta_{7} + 7 \beta_{9} + 9 \beta_{10} - 7 \beta_{11} + 8 \beta_{12} - 5 \beta_{13} + 9 \beta_{14} + 5 \beta_{15} + \beta_{16} - 4 \beta_{17} ) q^{26} + 19683 q^{27} + ( 23998 + 370 \beta_{1} + 166 \beta_{2} + 3 \beta_{3} - 66 \beta_{4} + 12 \beta_{6} + 10 \beta_{7} + 7 \beta_{9} - 13 \beta_{10} + 3 \beta_{11} - 14 \beta_{12} - \beta_{13} - \beta_{14} - 3 \beta_{15} + 7 \beta_{16} + 8 \beta_{17} ) q^{28} + ( 32864 - 63 \beta_{1} + 45 \beta_{2} - 31 \beta_{3} + \beta_{4} - 9 \beta_{5} - 7 \beta_{6} - 11 \beta_{7} - \beta_{9} + 2 \beta_{10} + 8 \beta_{11} + \beta_{12} + 2 \beta_{13} - 7 \beta_{14} + 3 \beta_{15} - 7 \beta_{16} - 5 \beta_{17} ) q^{29} + ( 4833 + 1674 \beta_{1} + 27 \beta_{2} + 27 \beta_{3} + 27 \beta_{4} - 27 \beta_{6} ) q^{30} + ( 23286 + 1209 \beta_{1} + 120 \beta_{2} - 4 \beta_{3} - 7 \beta_{4} + 11 \beta_{5} + \beta_{6} - 43 \beta_{7} + 24 \beta_{8} - 4 \beta_{9} + 14 \beta_{11} + 17 \beta_{12} - 3 \beta_{13} - 7 \beta_{14} + \beta_{15} - 13 \beta_{16} + \beta_{17} ) q^{31} + ( 66038 + 2776 \beta_{1} + 534 \beta_{2} + 29 \beta_{3} - 137 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} + 9 \beta_{8} - 24 \beta_{9} - 9 \beta_{10} - 10 \beta_{12} + 14 \beta_{13} - 16 \beta_{14} - 8 \beta_{15} - 5 \beta_{16} + 6 \beta_{17} ) q^{32} + ( 22005 + 1863 \beta_{1} + 54 \beta_{2} - 27 \beta_{4} + 27 \beta_{7} + 27 \beta_{8} + 27 \beta_{11} ) q^{33} + ( 20326 + 6231 \beta_{1} - 22 \beta_{2} + 20 \beta_{3} - 93 \beta_{4} - 20 \beta_{5} - 6 \beta_{6} - 119 \beta_{7} + 15 \beta_{8} - 10 \beta_{9} + 43 \beta_{10} - 16 \beta_{11} - \beta_{12} - \beta_{13} - 8 \beta_{15} - 11 \beta_{16} - 2 \beta_{17} ) q^{34} + ( 50379 + 1310 \beta_{1} - 388 \beta_{2} - 39 \beta_{3} - 9 \beta_{4} - 26 \beta_{5} - 12 \beta_{6} + 65 \beta_{7} + 17 \beta_{8} + 20 \beta_{9} + 5 \beta_{10} - 6 \beta_{11} + 16 \beta_{12} - 10 \beta_{13} + 4 \beta_{14} - 7 \beta_{15} + 29 \beta_{16} - 22 \beta_{17} ) q^{35} + ( 54675 + 729 \beta_{1} + 729 \beta_{2} ) q^{36} + ( -931 + 4251 \beta_{1} - 423 \beta_{2} + 41 \beta_{3} - 157 \beta_{4} + 52 \beta_{5} + 22 \beta_{6} - 34 \beta_{7} - 46 \beta_{8} - 3 \beta_{9} - 25 \beta_{10} - 6 \beta_{11} - 20 \beta_{12} + 9 \beta_{13} - 10 \beta_{14} + 5 \beta_{15} - \beta_{16} + 11 \beta_{17} ) q^{37} + ( -24125 + 3304 \beta_{1} - 498 \beta_{2} + 31 \beta_{3} - 588 \beta_{4} + 16 \beta_{5} + 12 \beta_{6} + 61 \beta_{7} - 146 \beta_{8} + 5 \beta_{9} - 33 \beta_{10} - 23 \beta_{11} - 37 \beta_{12} - 28 \beta_{13} + 5 \beta_{14} + \beta_{15} + 5 \beta_{16} + 40 \beta_{17} ) q^{38} + ( 20061 + 1431 \beta_{1} - 351 \beta_{2} + 27 \beta_{4} + 54 \beta_{7} + 27 \beta_{15} ) q^{39} + ( 29692 + 13657 \beta_{1} - 149 \beta_{2} + 173 \beta_{3} - 12 \beta_{4} + 19 \beta_{5} - 53 \beta_{6} - 153 \beta_{7} + 69 \beta_{8} - 8 \beta_{9} - \beta_{10} - 36 \beta_{11} + 6 \beta_{12} - 10 \beta_{13} - 4 \beta_{14} + 14 \beta_{15} + 17 \beta_{16} - 52 \beta_{17} ) q^{40} + ( 42224 + 2592 \beta_{1} - 492 \beta_{2} - 28 \beta_{3} - 350 \beta_{4} + 17 \beta_{5} + 23 \beta_{6} + 105 \beta_{7} - 21 \beta_{8} + 9 \beta_{9} + 8 \beta_{10} - 31 \beta_{11} + 39 \beta_{12} - 8 \beta_{13} + 15 \beta_{14} - 12 \beta_{15} - \beta_{16} + 6 \beta_{17} ) q^{41} + ( 29295 + 6021 \beta_{1} - 54 \beta_{2} + 27 \beta_{3} + 54 \beta_{4} - 27 \beta_{5} + 81 \beta_{7} - 27 \beta_{8} + 27 \beta_{9} - 27 \beta_{11} + 27 \beta_{14} - 27 \beta_{15} ) q^{42} + ( 40018 + 12264 \beta_{1} - 716 \beta_{2} - 8 \beta_{3} + 188 \beta_{4} - 17 \beta_{5} + 53 \beta_{6} + 22 \beta_{7} + 5 \beta_{8} - 18 \beta_{9} + 14 \beta_{10} + 24 \beta_{11} + 37 \beta_{12} + 31 \beta_{13} + 7 \beta_{14} - \beta_{15} - 11 \beta_{16} - 3 \beta_{17} ) q^{43} + ( 137778 + 19365 \beta_{1} + 387 \beta_{2} + 35 \beta_{3} - 691 \beta_{4} - 66 \beta_{5} + 34 \beta_{6} + 127 \beta_{7} + 165 \beta_{8} - 20 \beta_{9} + 66 \beta_{11} - 17 \beta_{12} + 34 \beta_{13} + 18 \beta_{14} - 14 \beta_{15} - 16 \beta_{16} - 16 \beta_{17} ) q^{44} + ( 26973 + 729 \beta_{1} + 729 \beta_{4} ) q^{45} + ( -228793 + 9092 \beta_{1} - 2051 \beta_{2} - 28 \beta_{3} - 141 \beta_{4} + 66 \beta_{5} + 42 \beta_{6} + 59 \beta_{7} - 244 \beta_{8} + 7 \beta_{9} + 14 \beta_{10} - 39 \beta_{11} - 19 \beta_{12} - 10 \beta_{13} - 3 \beta_{14} + 51 \beta_{15} + 4 \beta_{16} + 104 \beta_{17} ) q^{46} + ( 74720 + 11897 \beta_{1} - 458 \beta_{2} - 117 \beta_{3} - 442 \beta_{4} - 37 \beta_{5} + 51 \beta_{6} + 224 \beta_{7} - 93 \beta_{8} + 16 \beta_{9} + 41 \beta_{10} + 13 \beta_{11} - 15 \beta_{12} + 39 \beta_{13} + 13 \beta_{14} - 44 \beta_{15} - 12 \beta_{16} - 21 \beta_{17} ) q^{47} + ( 86589 + 11448 \beta_{1} + 1809 \beta_{2} + 81 \beta_{3} + 378 \beta_{4} + 54 \beta_{5} + 54 \beta_{6} - 27 \beta_{7} - 27 \beta_{8} - 27 \beta_{9} - 27 \beta_{11} + 27 \beta_{12} + 27 \beta_{13} - 27 \beta_{14} - 27 \beta_{15} ) q^{48} + ( 87148 + 12408 \beta_{1} - 2551 \beta_{2} + 18 \beta_{3} + 422 \beta_{4} - 66 \beta_{5} + 8 \beta_{6} + 284 \beta_{7} + 147 \beta_{8} + 15 \beta_{9} - 89 \beta_{10} - 9 \beta_{11} - 18 \beta_{12} + 31 \beta_{13} - 20 \beta_{14} - 49 \beta_{15} - 21 \beta_{16} + 2 \beta_{17} ) q^{49} + ( 48453 + 25994 \beta_{1} - 812 \beta_{2} + 7 \beta_{3} - 959 \beta_{4} - 7 \beta_{5} - 160 \beta_{6} - 24 \beta_{7} + 183 \beta_{8} + 57 \beta_{9} - 87 \beta_{10} + 99 \beta_{11} + 21 \beta_{12} - 37 \beta_{13} + 21 \beta_{14} + 11 \beta_{15} - 29 \beta_{16} - 110 \beta_{17} ) q^{50} + ( 107109 + 2079 \beta_{1} + 513 \beta_{2} - 54 \beta_{3} + 54 \beta_{4} + 27 \beta_{5} + 27 \beta_{6} + 54 \beta_{7} - 54 \beta_{8} - 54 \beta_{9} - 27 \beta_{10} - 27 \beta_{12} - 27 \beta_{13} - 27 \beta_{14} + 27 \beta_{15} ) q^{51} + ( -212219 + 13098 \beta_{1} - 3397 \beta_{2} - 176 \beta_{3} + 176 \beta_{4} - 95 \beta_{5} - 87 \beta_{6} + 71 \beta_{7} + 204 \beta_{8} + 53 \beta_{9} + 100 \beta_{10} - 9 \beta_{11} + 54 \beta_{12} - 95 \beta_{13} - 21 \beta_{14} + 145 \beta_{15} + 30 \beta_{16} + 18 \beta_{17} ) q^{52} + ( 60624 - 7172 \beta_{1} - 796 \beta_{2} - 334 \beta_{3} - 421 \beta_{4} - 36 \beta_{5} - 20 \beta_{6} + 78 \beta_{7} + 205 \beta_{8} + 10 \beta_{9} + 214 \beta_{10} - 86 \beta_{11} + 88 \beta_{12} - 28 \beta_{13} + 20 \beta_{14} + 20 \beta_{15} - 40 \beta_{16} + 28 \beta_{17} ) q^{53} + ( 19683 + 19683 \beta_{1} ) q^{54} + ( 11656 + 18942 \beta_{1} - 2662 \beta_{2} - 33 \beta_{3} + 1684 \beta_{4} + 3 \beta_{5} - 55 \beta_{6} - 80 \beta_{7} - 369 \beta_{8} - 15 \beta_{9} - 118 \beta_{10} + 36 \beta_{11} - 71 \beta_{12} + 50 \beta_{13} - 39 \beta_{14} - 48 \beta_{15} + 75 \beta_{16} - 63 \beta_{17} ) q^{55} + ( -25387 + 12140 \beta_{1} + 1212 \beta_{2} + 20 \beta_{3} - 1860 \beta_{4} + 222 \beta_{5} + 190 \beta_{6} + 135 \beta_{7} - 260 \beta_{8} - 161 \beta_{10} + 94 \beta_{11} - 191 \beta_{12} - 12 \beta_{13} - 46 \beta_{14} + 12 \beta_{15} + 79 \beta_{16} + 156 \beta_{17} ) q^{56} + ( 85536 - 3456 \beta_{1} - 162 \beta_{2} - 162 \beta_{3} + 162 \beta_{4} - 54 \beta_{5} + 54 \beta_{6} + 27 \beta_{7} + 108 \beta_{8} + 27 \beta_{9} + 54 \beta_{10} - 27 \beta_{11} - 108 \beta_{12} - 81 \beta_{13} - 27 \beta_{15} + 27 \beta_{17} ) q^{57} + ( 22384 + 40938 \beta_{1} - 2374 \beta_{2} - 139 \beta_{3} + 702 \beta_{4} - 52 \beta_{5} - 169 \beta_{6} - 739 \beta_{7} + 72 \beta_{8} - \beta_{9} + 64 \beta_{10} + 57 \beta_{11} + 57 \beta_{12} - 8 \beta_{13} + 85 \beta_{14} - 17 \beta_{15} + 2 \beta_{16} - 152 \beta_{17} ) q^{58} + 205379 q^{59} + ( 211680 + 5751 \beta_{1} + 2754 \beta_{2} - 162 \beta_{3} + 2160 \beta_{4} - 27 \beta_{5} - 27 \beta_{6} - 243 \beta_{7} - 27 \beta_{8} + 81 \beta_{10} - 108 \beta_{11} + 108 \beta_{12} + 54 \beta_{13} - 108 \beta_{14} + 27 \beta_{16} - 54 \beta_{17} ) q^{60} + ( -75752 - 1612 \beta_{1} - 3001 \beta_{2} - 79 \beta_{3} + 943 \beta_{4} + 198 \beta_{5} - 98 \beta_{6} - 145 \beta_{7} + 123 \beta_{8} - 50 \beta_{9} - 48 \beta_{10} + 91 \beta_{11} - 22 \beta_{12} - 4 \beta_{13} + 112 \beta_{14} - 7 \beta_{15} + 34 \beta_{16} + 77 \beta_{17} ) q^{61} + ( 279783 + 35075 \beta_{1} + 2586 \beta_{2} + 77 \beta_{3} - 692 \beta_{4} - 62 \beta_{5} + 19 \beta_{6} - 789 \beta_{7} + 433 \beta_{8} - 183 \beta_{9} + 188 \beta_{10} + 193 \beta_{11} + 153 \beta_{12} + 32 \beta_{13} + 21 \beta_{14} + 91 \beta_{15} - 94 \beta_{16} + 44 \beta_{17} ) q^{62} + ( 123930 + 3645 \beta_{1} + 729 \beta_{7} ) q^{63} + ( 258409 + 66129 \beta_{1} + 66 \beta_{2} + 459 \beta_{3} + 2360 \beta_{4} - 168 \beta_{5} + 80 \beta_{6} - 1002 \beta_{7} - 252 \beta_{8} - 124 \beta_{9} - 42 \beta_{10} + 80 \beta_{11} - 58 \beta_{12} + 88 \beta_{13} - 88 \beta_{14} - 184 \beta_{15} - 94 \beta_{16} - 88 \beta_{17} ) q^{64} + ( 192071 - 35109 \beta_{1} + 28 \beta_{2} + 405 \beta_{3} + 1721 \beta_{4} + 111 \beta_{5} - 69 \beta_{6} + 128 \beta_{7} - 2 \beta_{8} + 80 \beta_{9} - 67 \beta_{10} - 152 \beta_{11} - 123 \beta_{12} - 9 \beta_{13} + 3 \beta_{14} - 20 \beta_{15} + 156 \beta_{16} + 89 \beta_{17} ) q^{65} + ( 400491 + 29214 \beta_{1} + 3186 \beta_{2} - 162 \beta_{3} + 1134 \beta_{4} - 81 \beta_{5} + 108 \beta_{6} - 594 \beta_{7} - 108 \beta_{8} + 54 \beta_{9} - 135 \beta_{10} + 162 \beta_{11} + 135 \beta_{12} + 54 \beta_{13} + 108 \beta_{14} - 108 \beta_{15} - 81 \beta_{16} - 54 \beta_{17} ) q^{66} + ( -126833 - 24940 \beta_{1} - 4602 \beta_{2} + 253 \beta_{3} + 1124 \beta_{4} + 209 \beta_{5} - 187 \beta_{6} + 193 \beta_{7} + 14 \beta_{8} - 117 \beta_{9} + 17 \beta_{10} - 106 \beta_{11} + 19 \beta_{12} + 49 \beta_{14} + 81 \beta_{15} - 136 \beta_{16} + 208 \beta_{17} ) q^{67} + ( 783223 - 1802 \beta_{1} + 10138 \beta_{2} + 436 \beta_{3} + 495 \beta_{4} + 278 \beta_{5} - 18 \beta_{6} - 851 \beta_{7} - 114 \beta_{8} - 114 \beta_{9} - 190 \beta_{10} + 2 \beta_{11} + 217 \beta_{12} + 82 \beta_{13} - 222 \beta_{14} + 22 \beta_{15} - 118 \beta_{16} - 96 \beta_{17} ) q^{68} + ( 235899 - 31644 \beta_{1} - 108 \beta_{2} - 162 \beta_{3} - 513 \beta_{4} + 243 \beta_{5} + 27 \beta_{6} + 135 \beta_{7} - 81 \beta_{9} - 135 \beta_{10} - 54 \beta_{11} - 189 \beta_{12} + 54 \beta_{13} + 27 \beta_{14} + 135 \beta_{15} - 54 \beta_{16} + 189 \beta_{17} ) q^{69} + ( 284333 + 14553 \beta_{1} - 7653 \beta_{2} - 355 \beta_{3} + 810 \beta_{4} - 435 \beta_{5} - 372 \beta_{6} + 605 \beta_{7} + 583 \beta_{8} + 404 \beta_{9} + 39 \beta_{10} - 136 \beta_{11} + 76 \beta_{12} - 123 \beta_{13} + 178 \beta_{14} - 92 \beta_{15} + 131 \beta_{16} - 320 \beta_{17} ) q^{70} + ( 804005 - 35555 \beta_{1} + 3169 \beta_{2} + 576 \beta_{3} + 2514 \beta_{4} - 163 \beta_{5} - 25 \beta_{6} - 29 \beta_{7} - 523 \beta_{8} + 98 \beta_{9} - 132 \beta_{10} - 195 \beta_{11} + 101 \beta_{12} - 9 \beta_{13} - 59 \beta_{14} - 4 \beta_{15} + 99 \beta_{16} - 159 \beta_{17} ) q^{71} + ( 150903 + 44469 \beta_{1} + 2916 \beta_{2} + 729 \beta_{3} ) q^{72} + ( 317454 - 43276 \beta_{1} + 673 \beta_{2} - 258 \beta_{3} + 3927 \beta_{4} - 164 \beta_{5} - 228 \beta_{6} - 350 \beta_{7} - 232 \beta_{8} + 143 \beta_{9} + 158 \beta_{10} - 52 \beta_{11} + 230 \beta_{12} - 133 \beta_{13} - 140 \beta_{14} - 222 \beta_{15} - 368 \beta_{16} - 85 \beta_{17} ) q^{73} + ( 850231 - 61996 \beta_{1} + 7687 \beta_{2} - 218 \beta_{3} - 2575 \beta_{4} + 191 \beta_{5} + 471 \beta_{6} - 83 \beta_{7} - 734 \beta_{8} - 171 \beta_{9} + 4 \beta_{10} - 147 \beta_{11} - 380 \beta_{12} + 158 \beta_{13} - 117 \beta_{14} + 75 \beta_{15} + 104 \beta_{16} + 410 \beta_{17} ) q^{74} + ( 597078 + 3618 \beta_{1} + 675 \beta_{2} - 378 \beta_{3} + 2403 \beta_{4} - 243 \beta_{5} + 135 \beta_{6} - 783 \beta_{7} - 54 \beta_{8} + 27 \beta_{9} + 108 \beta_{10} + 108 \beta_{11} + 27 \beta_{12} + 135 \beta_{14} + 27 \beta_{15} + 189 \beta_{16} - 108 \beta_{17} ) q^{75} + ( 243032 - 82641 \beta_{1} - \beta_{2} - 562 \beta_{3} - 4539 \beta_{4} + 478 \beta_{5} + 774 \beta_{6} + 2497 \beta_{7} - 451 \beta_{8} + 29 \beta_{9} - 37 \beta_{10} - 329 \beta_{11} - 515 \beta_{12} + 195 \beta_{13} + 3 \beta_{14} + 105 \beta_{15} + 127 \beta_{16} + 696 \beta_{17} ) q^{76} + ( 1300920 - 45872 \beta_{1} + 1839 \beta_{2} + 739 \beta_{3} + 1052 \beta_{4} - 590 \beta_{5} - 100 \beta_{6} + 398 \beta_{7} + 432 \beta_{8} + 491 \beta_{9} - 215 \beta_{10} + 56 \beta_{11} - 170 \beta_{12} - 123 \beta_{13} + 136 \beta_{14} - 203 \beta_{15} + 139 \beta_{16} - 377 \beta_{17} ) q^{77} + ( 280719 - 17712 \beta_{1} + 1782 \beta_{2} - 729 \beta_{3} - 1971 \beta_{4} + 108 \beta_{5} - 405 \beta_{6} - 540 \beta_{7} + 189 \beta_{9} + 243 \beta_{10} - 189 \beta_{11} + 216 \beta_{12} - 135 \beta_{13} + 243 \beta_{14} + 135 \beta_{15} + 27 \beta_{16} - 108 \beta_{17} ) q^{78} + ( 1108943 - 30359 \beta_{1} + 190 \beta_{2} - 284 \beta_{3} + 600 \beta_{4} - 320 \beta_{5} + 70 \beta_{6} - 145 \beta_{7} - 297 \beta_{8} + 203 \beta_{9} + 155 \beta_{10} - 17 \beta_{11} + 104 \beta_{12} - 221 \beta_{13} + 88 \beta_{14} - 182 \beta_{15} + 191 \beta_{16} - 396 \beta_{17} ) q^{79} + ( 1785184 - 35396 \beta_{1} + 18680 \beta_{2} + 207 \beta_{3} + 792 \beta_{4} - 263 \beta_{5} + 9 \beta_{6} - 1870 \beta_{7} - 60 \beta_{8} - 411 \beta_{9} + 703 \beta_{10} - 83 \beta_{11} + 443 \beta_{12} + 37 \beta_{13} - 139 \beta_{14} + 199 \beta_{15} + 17 \beta_{16} - 316 \beta_{17} ) q^{80} + 531441 q^{81} + ( 552020 - 16110 \beta_{1} - 5781 \beta_{2} - 46 \beta_{3} - 7979 \beta_{4} - 205 \beta_{5} + 400 \beta_{6} + 2903 \beta_{7} + 1128 \beta_{8} + 25 \beta_{9} + 93 \beta_{10} + 51 \beta_{11} - 258 \beta_{12} - 156 \beta_{13} + 61 \beta_{14} + 249 \beta_{15} - 31 \beta_{16} + 376 \beta_{17} ) q^{82} + ( 1717560 - 112454 \beta_{1} + 4832 \beta_{2} + 408 \beta_{3} - 3444 \beta_{4} - 325 \beta_{5} - 411 \beta_{6} + 23 \beta_{7} + 1137 \beta_{8} - 141 \beta_{9} + 472 \beta_{10} - 177 \beta_{11} + 237 \beta_{12} - 176 \beta_{13} + 57 \beta_{14} + 392 \beta_{15} - 155 \beta_{16} - 250 \beta_{17} ) q^{83} + ( 647946 + 9990 \beta_{1} + 4482 \beta_{2} + 81 \beta_{3} - 1782 \beta_{4} + 324 \beta_{6} + 270 \beta_{7} + 189 \beta_{9} - 351 \beta_{10} + 81 \beta_{11} - 378 \beta_{12} - 27 \beta_{13} - 27 \beta_{14} - 81 \beta_{15} + 189 \beta_{16} + 216 \beta_{17} ) q^{84} + ( 574985 - 69391 \beta_{1} + 3132 \beta_{2} + 1188 \beta_{3} + 1463 \beta_{4} + 690 \beta_{5} + 212 \beta_{6} - 239 \beta_{7} + 501 \beta_{8} - 694 \beta_{9} + 231 \beta_{10} + 171 \beta_{11} + 164 \beta_{12} - 10 \beta_{13} - 158 \beta_{14} + 205 \beta_{15} + 13 \beta_{16} + 551 \beta_{17} ) q^{85} + ( 2476451 - 42990 \beta_{1} + 15341 \beta_{2} - 199 \beta_{3} - 7783 \beta_{4} + 189 \beta_{5} - 224 \beta_{6} + 46 \beta_{7} + 1128 \beta_{8} - 124 \beta_{9} - 438 \beta_{10} + 718 \beta_{11} + 261 \beta_{12} + 94 \beta_{13} + 212 \beta_{14} - 188 \beta_{15} - 328 \beta_{16} - 380 \beta_{17} ) q^{86} + ( 887328 - 1701 \beta_{1} + 1215 \beta_{2} - 837 \beta_{3} + 27 \beta_{4} - 243 \beta_{5} - 189 \beta_{6} - 297 \beta_{7} - 27 \beta_{9} + 54 \beta_{10} + 216 \beta_{11} + 27 \beta_{12} + 54 \beta_{13} - 189 \beta_{14} + 81 \beta_{15} - 189 \beta_{16} - 135 \beta_{17} ) q^{87} + ( 2211860 + 27875 \beta_{1} + 18960 \beta_{2} - 4 \beta_{3} - 5323 \beta_{4} + 350 \beta_{5} + 698 \beta_{6} + 1615 \beta_{7} - 591 \beta_{8} - 218 \beta_{9} - 114 \beta_{10} + 116 \beta_{11} + 313 \beta_{12} + 108 \beta_{13} - 248 \beta_{14} - 390 \beta_{15} + 42 \beta_{16} - 498 \beta_{17} ) q^{88} + ( 1758244 - 106919 \beta_{1} + 5477 \beta_{2} + 1658 \beta_{3} + 446 \beta_{4} + 434 \beta_{5} + 414 \beta_{6} - 1265 \beta_{7} - 2196 \beta_{8} - 418 \beta_{9} - 708 \beta_{10} + 124 \beta_{11} - 696 \beta_{12} + 460 \beta_{13} - 62 \beta_{14} - 245 \beta_{15} - 394 \beta_{16} + 296 \beta_{17} ) q^{89} + ( 130491 + 45198 \beta_{1} + 729 \beta_{2} + 729 \beta_{3} + 729 \beta_{4} - 729 \beta_{6} ) q^{90} + ( 1500987 - 101746 \beta_{1} - 4819 \beta_{2} + 61 \beta_{3} - 262 \beta_{4} + 841 \beta_{5} - 243 \beta_{6} + 1558 \beta_{7} + 454 \beta_{8} - 330 \beta_{9} - 703 \beta_{10} + 359 \beta_{11} + 191 \beta_{12} + 441 \beta_{13} - 515 \beta_{14} + 737 \beta_{15} + 212 \beta_{16} - \beta_{17} ) q^{91} + ( 374164 - 318619 \beta_{1} - 2231 \beta_{2} - 892 \beta_{3} - 11671 \beta_{4} + 522 \beta_{5} - 390 \beta_{6} + 3059 \beta_{7} + 11 \beta_{8} + 785 \beta_{9} + 13 \beta_{10} - 545 \beta_{11} - 685 \beta_{12} - 1017 \beta_{13} + 219 \beta_{14} + 241 \beta_{15} + 333 \beta_{16} + 616 \beta_{17} ) q^{92} + ( 628722 + 32643 \beta_{1} + 3240 \beta_{2} - 108 \beta_{3} - 189 \beta_{4} + 297 \beta_{5} + 27 \beta_{6} - 1161 \beta_{7} + 648 \beta_{8} - 108 \beta_{9} + 378 \beta_{11} + 459 \beta_{12} - 81 \beta_{13} - 189 \beta_{14} + 27 \beta_{15} - 351 \beta_{16} + 27 \beta_{17} ) q^{93} + ( 2465235 + 27695 \beta_{1} + 1258 \beta_{2} + 128 \beta_{3} - 8997 \beta_{4} - 57 \beta_{5} + 788 \beta_{6} + 2737 \beta_{7} - 995 \beta_{8} + 168 \beta_{9} - 1622 \beta_{10} + 590 \beta_{11} - 898 \beta_{12} + 141 \beta_{13} + 288 \beta_{14} - 474 \beta_{15} + 2 \beta_{16} - 214 \beta_{17} ) q^{94} + ( 964285 - 305339 \beta_{1} - 13293 \beta_{2} - 301 \beta_{3} + 4069 \beta_{4} + 83 \beta_{5} + 433 \beta_{6} - 758 \beta_{7} - 1696 \beta_{8} - 248 \beta_{9} + 196 \beta_{10} - 404 \beta_{11} + 599 \beta_{12} - 55 \beta_{13} + 73 \beta_{14} + 408 \beta_{15} - 317 \beta_{16} + 1106 \beta_{17} ) q^{95} + ( 1783026 + 74952 \beta_{1} + 14418 \beta_{2} + 783 \beta_{3} - 3699 \beta_{4} + 162 \beta_{5} + 54 \beta_{6} - 216 \beta_{7} + 243 \beta_{8} - 648 \beta_{9} - 243 \beta_{10} - 270 \beta_{12} + 378 \beta_{13} - 432 \beta_{14} - 216 \beta_{15} - 135 \beta_{16} + 162 \beta_{17} ) q^{96} + ( 690307 - 57648 \beta_{1} - 10366 \beta_{2} - 1143 \beta_{3} + 4464 \beta_{4} + 212 \beta_{5} - 384 \beta_{6} - 404 \beta_{7} - 1181 \beta_{8} + 316 \beta_{9} + 610 \beta_{10} + 333 \beta_{11} - 276 \beta_{12} + 366 \beta_{13} - 130 \beta_{14} - 106 \beta_{15} + 580 \beta_{16} - 447 \beta_{17} ) q^{97} + ( 2406828 - 173615 \beta_{1} + 2701 \beta_{2} - 1681 \beta_{3} + 4390 \beta_{4} - 406 \beta_{5} + 102 \beta_{6} + 2403 \beta_{7} - 1010 \beta_{8} + 386 \beta_{9} + 114 \beta_{10} + 108 \beta_{11} + 217 \beta_{12} + 393 \beta_{13} + 448 \beta_{14} - 1370 \beta_{15} + 424 \beta_{16} - 162 \beta_{17} ) q^{98} + ( 594135 + 50301 \beta_{1} + 1458 \beta_{2} - 729 \beta_{4} + 729 \beta_{7} + 729 \beta_{8} + 729 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 202 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 318 \nu + 346 \)
\(\beta_{4}\)\(=\)\((\)\(\)\(42\!\cdots\!09\)\( \nu^{17} + \)\(33\!\cdots\!95\)\( \nu^{16} - \)\(71\!\cdots\!47\)\( \nu^{15} - \)\(56\!\cdots\!16\)\( \nu^{14} + \)\(47\!\cdots\!53\)\( \nu^{13} + \)\(39\!\cdots\!29\)\( \nu^{12} - \)\(16\!\cdots\!75\)\( \nu^{11} - \)\(14\!\cdots\!10\)\( \nu^{10} + \)\(28\!\cdots\!03\)\( \nu^{9} + \)\(28\!\cdots\!41\)\( \nu^{8} - \)\(21\!\cdots\!69\)\( \nu^{7} - \)\(32\!\cdots\!48\)\( \nu^{6} + \)\(79\!\cdots\!59\)\( \nu^{5} + \)\(19\!\cdots\!31\)\( \nu^{4} + \)\(67\!\cdots\!87\)\( \nu^{3} - \)\(60\!\cdots\!02\)\( \nu^{2} - \)\(20\!\cdots\!28\)\( \nu + \)\(78\!\cdots\!20\)\(\)\()/ \)\(14\!\cdots\!52\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(63\!\cdots\!95\)\( \nu^{17} + \)\(16\!\cdots\!75\)\( \nu^{16} + \)\(11\!\cdots\!65\)\( \nu^{15} - \)\(30\!\cdots\!92\)\( \nu^{14} - \)\(91\!\cdots\!35\)\( \nu^{13} + \)\(21\!\cdots\!09\)\( \nu^{12} + \)\(37\!\cdots\!01\)\( \nu^{11} - \)\(83\!\cdots\!66\)\( \nu^{10} - \)\(88\!\cdots\!25\)\( \nu^{9} + \)\(17\!\cdots\!17\)\( \nu^{8} + \)\(12\!\cdots\!87\)\( \nu^{7} - \)\(21\!\cdots\!76\)\( \nu^{6} - \)\(95\!\cdots\!17\)\( \nu^{5} + \)\(13\!\cdots\!03\)\( \nu^{4} + \)\(37\!\cdots\!39\)\( \nu^{3} - \)\(37\!\cdots\!26\)\( \nu^{2} - \)\(61\!\cdots\!48\)\( \nu + \)\(34\!\cdots\!64\)\(\)\()/ \)\(51\!\cdots\!84\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(59\!\cdots\!49\)\( \nu^{17} - \)\(54\!\cdots\!35\)\( \nu^{16} + \)\(10\!\cdots\!99\)\( \nu^{15} + \)\(90\!\cdots\!32\)\( \nu^{14} - \)\(75\!\cdots\!09\)\( \nu^{13} - \)\(61\!\cdots\!77\)\( \nu^{12} + \)\(28\!\cdots\!99\)\( \nu^{11} + \)\(21\!\cdots\!94\)\( \nu^{10} - \)\(61\!\cdots\!67\)\( \nu^{9} - \)\(40\!\cdots\!25\)\( \nu^{8} + \)\(76\!\cdots\!77\)\( \nu^{7} + \)\(41\!\cdots\!12\)\( \nu^{6} - \)\(51\!\cdots\!75\)\( \nu^{5} - \)\(19\!\cdots\!67\)\( \nu^{4} + \)\(17\!\cdots\!29\)\( \nu^{3} + \)\(28\!\cdots\!78\)\( \nu^{2} - \)\(23\!\cdots\!56\)\( \nu + \)\(22\!\cdots\!04\)\(\)\()/ \)\(14\!\cdots\!52\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(69\!\cdots\!03\)\( \nu^{17} + \)\(17\!\cdots\!55\)\( \nu^{16} + \)\(12\!\cdots\!65\)\( \nu^{15} - \)\(33\!\cdots\!84\)\( \nu^{14} - \)\(92\!\cdots\!99\)\( \nu^{13} + \)\(26\!\cdots\!05\)\( \nu^{12} + \)\(36\!\cdots\!65\)\( \nu^{11} - \)\(10\!\cdots\!22\)\( \nu^{10} - \)\(82\!\cdots\!97\)\( \nu^{9} + \)\(25\!\cdots\!41\)\( \nu^{8} + \)\(10\!\cdots\!99\)\( \nu^{7} - \)\(33\!\cdots\!44\)\( \nu^{6} - \)\(77\!\cdots\!13\)\( \nu^{5} + \)\(24\!\cdots\!11\)\( \nu^{4} + \)\(27\!\cdots\!59\)\( \nu^{3} - \)\(85\!\cdots\!66\)\( \nu^{2} - \)\(37\!\cdots\!72\)\( \nu + \)\(11\!\cdots\!40\)\(\)\()/ \)\(10\!\cdots\!68\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(87\!\cdots\!53\)\( \nu^{17} - \)\(23\!\cdots\!45\)\( \nu^{16} - \)\(15\!\cdots\!47\)\( \nu^{15} + \)\(44\!\cdots\!44\)\( \nu^{14} + \)\(11\!\cdots\!61\)\( \nu^{13} - \)\(34\!\cdots\!87\)\( \nu^{12} - \)\(45\!\cdots\!43\)\( \nu^{11} + \)\(14\!\cdots\!22\)\( \nu^{10} + \)\(10\!\cdots\!35\)\( \nu^{9} - \)\(32\!\cdots\!43\)\( \nu^{8} - \)\(12\!\cdots\!93\)\( \nu^{7} + \)\(43\!\cdots\!88\)\( \nu^{6} + \)\(91\!\cdots\!07\)\( \nu^{5} - \)\(30\!\cdots\!09\)\( \nu^{4} - \)\(31\!\cdots\!81\)\( \nu^{3} + \)\(10\!\cdots\!78\)\( \nu^{2} + \)\(42\!\cdots\!64\)\( \nu - \)\(12\!\cdots\!04\)\(\)\()/ \)\(90\!\cdots\!72\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(19\!\cdots\!75\)\( \nu^{17} + \)\(29\!\cdots\!53\)\( \nu^{16} - \)\(36\!\cdots\!57\)\( \nu^{15} - \)\(43\!\cdots\!80\)\( \nu^{14} + \)\(27\!\cdots\!15\)\( \nu^{13} + \)\(24\!\cdots\!39\)\( \nu^{12} - \)\(10\!\cdots\!01\)\( \nu^{11} - \)\(61\!\cdots\!50\)\( \nu^{10} + \)\(25\!\cdots\!21\)\( \nu^{9} + \)\(40\!\cdots\!67\)\( \nu^{8} - \)\(35\!\cdots\!63\)\( \nu^{7} + \)\(11\!\cdots\!88\)\( \nu^{6} + \)\(27\!\cdots\!93\)\( \nu^{5} - \)\(25\!\cdots\!27\)\( \nu^{4} - \)\(11\!\cdots\!67\)\( \nu^{3} + \)\(18\!\cdots\!94\)\( \nu^{2} + \)\(17\!\cdots\!28\)\( \nu - \)\(41\!\cdots\!84\)\(\)\()/ \)\(14\!\cdots\!52\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(21\!\cdots\!43\)\( \nu^{17} + \)\(78\!\cdots\!19\)\( \nu^{16} + \)\(39\!\cdots\!21\)\( \nu^{15} - \)\(14\!\cdots\!92\)\( \nu^{14} - \)\(28\!\cdots\!43\)\( \nu^{13} + \)\(10\!\cdots\!81\)\( \nu^{12} + \)\(11\!\cdots\!13\)\( \nu^{11} - \)\(43\!\cdots\!66\)\( \nu^{10} - \)\(24\!\cdots\!77\)\( \nu^{9} + \)\(98\!\cdots\!85\)\( \nu^{8} + \)\(31\!\cdots\!47\)\( \nu^{7} - \)\(12\!\cdots\!80\)\( \nu^{6} - \)\(21\!\cdots\!57\)\( \nu^{5} + \)\(85\!\cdots\!35\)\( \nu^{4} + \)\(71\!\cdots\!31\)\( \nu^{3} - \)\(26\!\cdots\!06\)\( \nu^{2} - \)\(92\!\cdots\!32\)\( \nu + \)\(29\!\cdots\!28\)\(\)\()/ \)\(14\!\cdots\!52\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(22\!\cdots\!61\)\( \nu^{17} + \)\(34\!\cdots\!87\)\( \nu^{16} - \)\(40\!\cdots\!71\)\( \nu^{15} - \)\(51\!\cdots\!32\)\( \nu^{14} + \)\(29\!\cdots\!21\)\( \nu^{13} + \)\(29\!\cdots\!33\)\( \nu^{12} - \)\(11\!\cdots\!59\)\( \nu^{11} - \)\(75\!\cdots\!34\)\( \nu^{10} + \)\(26\!\cdots\!07\)\( \nu^{9} + \)\(63\!\cdots\!49\)\( \nu^{8} - \)\(33\!\cdots\!45\)\( \nu^{7} + \)\(89\!\cdots\!64\)\( \nu^{6} + \)\(23\!\cdots\!87\)\( \nu^{5} - \)\(21\!\cdots\!29\)\( \nu^{4} - \)\(82\!\cdots\!77\)\( \nu^{3} + \)\(13\!\cdots\!54\)\( \nu^{2} + \)\(10\!\cdots\!84\)\( \nu - \)\(26\!\cdots\!52\)\(\)\()/ \)\(14\!\cdots\!52\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(37\!\cdots\!75\)\( \nu^{17} - \)\(47\!\cdots\!71\)\( \nu^{16} - \)\(66\!\cdots\!61\)\( \nu^{15} + \)\(98\!\cdots\!48\)\( \nu^{14} + \)\(49\!\cdots\!23\)\( \nu^{13} - \)\(83\!\cdots\!33\)\( \nu^{12} - \)\(19\!\cdots\!69\)\( \nu^{11} + \)\(37\!\cdots\!66\)\( \nu^{10} + \)\(43\!\cdots\!45\)\( \nu^{9} - \)\(94\!\cdots\!93\)\( \nu^{8} - \)\(56\!\cdots\!67\)\( \nu^{7} + \)\(13\!\cdots\!28\)\( \nu^{6} + \)\(40\!\cdots\!73\)\( \nu^{5} - \)\(10\!\cdots\!59\)\( \nu^{4} - \)\(14\!\cdots\!99\)\( \nu^{3} + \)\(40\!\cdots\!34\)\( \nu^{2} + \)\(19\!\cdots\!20\)\( \nu - \)\(56\!\cdots\!64\)\(\)\()/ \)\(20\!\cdots\!36\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(29\!\cdots\!37\)\( \nu^{17} + \)\(15\!\cdots\!05\)\( \nu^{16} + \)\(54\!\cdots\!31\)\( \nu^{15} - \)\(29\!\cdots\!32\)\( \nu^{14} - \)\(40\!\cdots\!77\)\( \nu^{13} + \)\(21\!\cdots\!55\)\( \nu^{12} + \)\(15\!\cdots\!07\)\( \nu^{11} - \)\(85\!\cdots\!10\)\( \nu^{10} - \)\(36\!\cdots\!43\)\( \nu^{9} + \)\(19\!\cdots\!87\)\( \nu^{8} + \)\(47\!\cdots\!17\)\( \nu^{7} - \)\(24\!\cdots\!04\)\( \nu^{6} - \)\(34\!\cdots\!27\)\( \nu^{5} + \)\(16\!\cdots\!33\)\( \nu^{4} + \)\(12\!\cdots\!33\)\( \nu^{3} - \)\(54\!\cdots\!18\)\( \nu^{2} - \)\(17\!\cdots\!84\)\( \nu + \)\(63\!\cdots\!16\)\(\)\()/ \)\(14\!\cdots\!52\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(37\!\cdots\!43\)\( \nu^{17} + \)\(80\!\cdots\!91\)\( \nu^{16} + \)\(68\!\cdots\!85\)\( \nu^{15} - \)\(15\!\cdots\!20\)\( \nu^{14} - \)\(51\!\cdots\!55\)\( \nu^{13} + \)\(12\!\cdots\!45\)\( \nu^{12} + \)\(20\!\cdots\!97\)\( \nu^{11} - \)\(52\!\cdots\!42\)\( \nu^{10} - \)\(49\!\cdots\!13\)\( \nu^{9} + \)\(12\!\cdots\!29\)\( \nu^{8} + \)\(67\!\cdots\!71\)\( \nu^{7} - \)\(18\!\cdots\!16\)\( \nu^{6} - \)\(52\!\cdots\!09\)\( \nu^{5} + \)\(14\!\cdots\!71\)\( \nu^{4} + \)\(21\!\cdots\!31\)\( \nu^{3} - \)\(59\!\cdots\!30\)\( \nu^{2} - \)\(32\!\cdots\!00\)\( \nu + \)\(93\!\cdots\!96\)\(\)\()/ \)\(18\!\cdots\!44\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(15\!\cdots\!51\)\( \nu^{17} + \)\(21\!\cdots\!75\)\( \nu^{16} + \)\(27\!\cdots\!37\)\( \nu^{15} - \)\(45\!\cdots\!32\)\( \nu^{14} - \)\(20\!\cdots\!19\)\( \nu^{13} + \)\(38\!\cdots\!85\)\( \nu^{12} + \)\(79\!\cdots\!45\)\( \nu^{11} - \)\(17\!\cdots\!10\)\( \nu^{10} - \)\(18\!\cdots\!73\)\( \nu^{9} + \)\(43\!\cdots\!89\)\( \nu^{8} + \)\(23\!\cdots\!75\)\( \nu^{7} - \)\(63\!\cdots\!64\)\( \nu^{6} - \)\(17\!\cdots\!65\)\( \nu^{5} + \)\(50\!\cdots\!07\)\( \nu^{4} + \)\(63\!\cdots\!07\)\( \nu^{3} - \)\(19\!\cdots\!78\)\( \nu^{2} - \)\(90\!\cdots\!56\)\( \nu + \)\(26\!\cdots\!16\)\(\)\()/ \)\(72\!\cdots\!76\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(28\!\cdots\!57\)\( \nu^{17} - \)\(11\!\cdots\!17\)\( \nu^{16} - \)\(52\!\cdots\!79\)\( \nu^{15} + \)\(20\!\cdots\!32\)\( \nu^{14} + \)\(38\!\cdots\!89\)\( \nu^{13} - \)\(15\!\cdots\!87\)\( \nu^{12} - \)\(15\!\cdots\!43\)\( \nu^{11} + \)\(63\!\cdots\!82\)\( \nu^{10} + \)\(34\!\cdots\!59\)\( \nu^{9} - \)\(14\!\cdots\!55\)\( \nu^{8} - \)\(44\!\cdots\!37\)\( \nu^{7} + \)\(18\!\cdots\!40\)\( \nu^{6} + \)\(31\!\cdots\!63\)\( \nu^{5} - \)\(13\!\cdots\!97\)\( \nu^{4} - \)\(11\!\cdots\!21\)\( \nu^{3} + \)\(41\!\cdots\!10\)\( \nu^{2} + \)\(14\!\cdots\!32\)\( \nu - \)\(48\!\cdots\!00\)\(\)\()/ \)\(72\!\cdots\!76\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(67\!\cdots\!09\)\( \nu^{17} - \)\(83\!\cdots\!49\)\( \nu^{16} - \)\(12\!\cdots\!35\)\( \nu^{15} + \)\(43\!\cdots\!44\)\( \nu^{14} + \)\(88\!\cdots\!49\)\( \nu^{13} - \)\(54\!\cdots\!27\)\( \nu^{12} - \)\(34\!\cdots\!83\)\( \nu^{11} + \)\(30\!\cdots\!58\)\( \nu^{10} + \)\(76\!\cdots\!27\)\( \nu^{9} - \)\(92\!\cdots\!79\)\( \nu^{8} - \)\(97\!\cdots\!57\)\( \nu^{7} + \)\(15\!\cdots\!84\)\( \nu^{6} + \)\(68\!\cdots\!87\)\( \nu^{5} - \)\(13\!\cdots\!05\)\( \nu^{4} - \)\(23\!\cdots\!97\)\( \nu^{3} + \)\(55\!\cdots\!58\)\( \nu^{2} + \)\(31\!\cdots\!60\)\( \nu - \)\(83\!\cdots\!44\)\(\)\()/ \)\(14\!\cdots\!52\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 202\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 317 \beta_{1} - 144\)
\(\nu^{4}\)\(=\)\(-\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 14 \beta_{4} - \beta_{3} + 441 \beta_{2} - 458 \beta_{1} + 64138\)
\(\nu^{5}\)\(=\)\(6 \beta_{17} - 5 \beta_{16} - 3 \beta_{15} - 11 \beta_{14} + 9 \beta_{13} - 15 \beta_{12} + 5 \beta_{11} - 9 \beta_{10} - 19 \beta_{9} + 14 \beta_{8} - 3 \beta_{7} - 8 \beta_{6} - 4 \beta_{5} - 207 \beta_{4} + 536 \beta_{3} + 357 \beta_{2} + 115053 \beta_{1} - 67329\)
\(\nu^{6}\)\(=\)\(-124 \beta_{17} - 64 \beta_{16} - 791 \beta_{15} - 647 \beta_{14} + 659 \beta_{13} + 657 \beta_{12} - 575 \beta_{11} + 12 \beta_{10} - 635 \beta_{9} - 961 \beta_{8} - 1609 \beta_{7} + 1378 \beta_{6} + 1106 \beta_{5} + 12352 \beta_{4} - 842 \beta_{3} + 181610 \beta_{2} - 204834 \beta_{1} + 23297602\)
\(\nu^{7}\)\(=\)\(2474 \beta_{17} - 3415 \beta_{16} - 2716 \beta_{15} - 7840 \beta_{14} + 6558 \beta_{13} - 11534 \beta_{12} + 6480 \beta_{11} - 6591 \beta_{10} - 13976 \beta_{9} + 13659 \beta_{8} - 2036 \beta_{7} - 7246 \beta_{6} - 3346 \beta_{5} - 149567 \beta_{4} + 245025 \beta_{3} + 106923 \beta_{2} + 44351590 \beta_{1} - 31157507\)
\(\nu^{8}\)\(=\)\(-116554 \beta_{17} - 52891 \beta_{16} - 447118 \beta_{15} - 328802 \beta_{14} + 351956 \beta_{13} + 359890 \beta_{12} - 257626 \beta_{11} + 7433 \beta_{10} - 318758 \beta_{9} - 584593 \beta_{8} - 1155836 \beta_{7} + 726386 \beta_{6} + 478486 \beta_{5} + 7414971 \beta_{4} - 511113 \beta_{3} + 74858936 \beta_{2} - 92966196 \beta_{1} + 8984566067\)
\(\nu^{9}\)\(=\)\(388322 \beta_{17} - 1825573 \beta_{16} - 1726918 \beta_{15} - 4078574 \beta_{14} + 3596304 \beta_{13} - 6216366 \beta_{12} + 4630522 \beta_{11} - 3534565 \beta_{10} - 7571986 \beta_{9} + 9340381 \beta_{8} - 1459420 \beta_{7} - 4706518 \beta_{6} - 2040274 \beta_{5} - 80925323 \beta_{4} + 107536138 \beta_{3} + 26320798 \beta_{2} + 17695645476 \beta_{1} - 14615002473\)
\(\nu^{10}\)\(=\)\(-75945018 \beta_{17} - 31120411 \beta_{16} - 221976003 \beta_{15} - 155030463 \beta_{14} + 173120393 \beta_{13} + 184817587 \beta_{12} - 105545031 \beta_{11} + 5840577 \beta_{10} - 149035083 \beta_{9} - 298451098 \beta_{8} - 653102317 \beta_{7} + 348164004 \beta_{6} + 191384400 \beta_{5} + 3871033813 \beta_{4} - 274666134 \beta_{3} + 31103588912 \beta_{2} - 42481059061 \beta_{1} + 3585271738667\)
\(\nu^{11}\)\(=\)\(-228023992 \beta_{17} - 908704302 \beta_{16} - 934849297 \beta_{15} - 1888558801 \beta_{14} + 1763596785 \beta_{13} - 2924352141 \beta_{12} + 2662864647 \beta_{11} - 1687719610 \beta_{10} - 3676537693 \beta_{9} + 5403763567 \beta_{8} - 1099095263 \beta_{7} - 2684247214 \beta_{6} - 1096900542 \beta_{5} - 39512490878 \beta_{4} + 46539072077 \beta_{3} + 3620702686 \beta_{2} + 7217503389736 \beta_{1} - 6867264837542\)
\(\nu^{12}\)\(=\)\(-42522848694 \beta_{17} - 16097755799 \beta_{16} - 103292091881 \beta_{15} - 70929671893 \beta_{14} + 81470489051 \beta_{13} + 91236803507 \beta_{12} - 41560911765 \beta_{11} + 4833963913 \beta_{10} - 67764754597 \beta_{9} - 139673788574 \beta_{8} - 332561073237 \beta_{7} + 159538319508 \beta_{6} + 74084469752 \beta_{5} + 1892276922611 \beta_{4} - 139084523891 \beta_{3} + 13019721695685 \beta_{2} - 19407684291361 \beta_{1} + 1462353467116559\)
\(\nu^{13}\)\(=\)\(-278185379156 \beta_{17} - 438056649436 \beta_{16} - 461373418794 \beta_{15} - 829096386854 \beta_{14} + 815874882166 \beta_{13} - 1286465288348 \beta_{12} + 1376841652370 \beta_{11} - 764106901460 \beta_{10} - 1697787635154 \beta_{9} + 2842708287848 \beta_{8} - 760179295568 \beta_{7} - 1429603814404 \beta_{6} - 547639490740 \beta_{5} - 18393885757330 \beta_{4} + 20039534104018 \beta_{3} - 1399164315288 \beta_{2} + 2987921417051933 \beta_{1} - 3208968238534860\)
\(\nu^{14}\)\(=\)\(-21971913318404 \beta_{17} - 7808170309630 \beta_{16} - 46346257405376 \beta_{15} - 31997668747184 \beta_{14} + 37340874144940 \beta_{13} + 43804923964164 \beta_{12} - 16102368794736 \beta_{11} + 3587997827890 \beta_{10} - 30414027868608 \beta_{9} - 62225710177282 \beta_{8} - 160553964576648 \beta_{7} + 71328150285044 \beta_{6} + 28208023274900 \beta_{5} + 892785008830970 \beta_{4} - 68022824702854 \beta_{3} + 5482729379558087 \beta_{2} - 8835268582974167 \beta_{1} + 605359660364174204\)
\(\nu^{15}\)\(=\)\(-195154928700636 \beta_{17} - 207010529679718 \beta_{16} - 214807477197556 \beta_{15} - 354539077892948 \beta_{14} + 364571231003256 \beta_{13} - 544925313069452 \beta_{12} + 670325261909340 \beta_{11} - 337269456117118 \beta_{10} - 763866031252508 \beta_{9} + 1410547932773462 \beta_{8} - 475420428838520 \beta_{7} - 730023483614300 \beta_{6} - 260104122906588 \beta_{5} - 8346143868388354 \beta_{4} + 8616600750540887 \beta_{3} - 1731756944620897 \beta_{2} + 1249879656684639703 \beta_{1} - 1487545602461313878\)
\(\nu^{16}\)\(=\)\(-10820295662465012 \beta_{17} - 3647373336892490 \beta_{16} - 20345717899537013 \beta_{15} - 14326817836278765 \beta_{14} + 16833510154701713 \beta_{13} + 20611792230126869 \beta_{12} - 6218813200910205 \beta_{11} + 2380055527875478 \beta_{10} - 13553989836150965 \beta_{9} - 26899584202261919 \beta_{8} - 75195607829850649 \beta_{7} + 31427820835606902 \beta_{6} + 10624888758742502 \beta_{5} + 412538050487200416 \beta_{4} - 32512479641415623 \beta_{3} + 2319772929304213875 \beta_{2} - 4003291868441954812 \beta_{1} + 253206914713408641168\)
\(\nu^{17}\)\(=\)\(-114569915468905206 \beta_{17} - 96417570520028491 \beta_{16} - 96218585116703551 \beta_{15} - 149630903286342967 \beta_{14} + 159319592809759881 \beta_{13} - 225507317743521243 \beta_{12} + 314473261651701225 \beta_{11} - 147214330736520247 \beta_{10} - 338669365944039255 \beta_{9} + 673994326505990708 \beta_{8} - 274350611142830427 \beta_{7} - 362307627738937412 \beta_{6} - 119222169681113520 \beta_{5} - 3730378140311332025 \beta_{4} + 3705509146726947542 \beta_{3} - 1176850859131287005 \beta_{2} + 526759179851060168247 \beta_{1} - 683909960273644887927\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−20.9501
−20.5204
−17.7235
−16.0158
−13.7979
−10.5486
−6.66616
−6.28523
2.80127
2.93322
6.43791
8.74396
9.20800
13.6971
15.9283
17.9457
19.9328
20.8794
−19.9501 27.0000 270.005 505.865 −538.652 222.970 −2833.01 729.000 −10092.0
1.2 −19.5204 27.0000 253.046 −49.6011 −527.051 −119.919 −2440.94 729.000 968.233
1.3 −16.7235 27.0000 151.676 −298.819 −451.535 200.095 −395.941 729.000 4997.30
1.4 −15.0158 27.0000 97.4757 6.55078 −405.428 1466.16 458.348 729.000 −98.3656
1.5 −12.7979 27.0000 35.7870 −405.222 −345.544 −375.715 1180.14 729.000 5186.00
1.6 −9.54862 27.0000 −36.8240 248.981 −257.813 −1453.31 1573.84 729.000 −2377.42
1.7 −5.66616 27.0000 −95.8947 −15.0142 −152.986 499.350 1268.62 729.000 85.0729
1.8 −5.28523 27.0000 −100.066 498.961 −142.701 341.176 1205.38 729.000 −2637.13
1.9 3.80127 27.0000 −113.550 −87.8737 102.634 1306.06 −918.197 729.000 −334.031
1.10 3.93322 27.0000 −112.530 −305.648 106.197 −1406.85 −946.057 729.000 −1202.18
1.11 7.43791 27.0000 −72.6775 171.827 200.824 1529.87 −1492.62 729.000 1278.03
1.12 9.74396 27.0000 −33.0553 321.882 263.087 −939.921 −1569.32 729.000 3136.41
1.13 10.2080 27.0000 −23.7967 −397.678 275.616 −505.577 −1549.54 729.000 −4059.50
1.14 14.6971 27.0000 88.0043 457.882 396.821 1576.42 −587.821 729.000 6729.53
1.15 16.9283 27.0000 158.568 −421.077 457.065 −62.1753 517.469 729.000 −7128.13
1.16 18.9457 27.0000 230.941 260.561 511.535 254.948 1950.29 729.000 4936.51
1.17 20.9328 27.0000 310.183 −149.882 565.186 1201.09 3813.60 729.000 −3137.45
1.18 21.8794 27.0000 350.709 336.305 590.745 −653.674 4872.75 729.000 7358.16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.8.a.d 18
3.b odd 2 1 531.8.a.e 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.d 18 1.a even 1 1 trivial
531.8.a.e 18 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!84\)\( T_{2}^{9} - \)\(32\!\cdots\!32\)\( T_{2}^{8} + \)\(17\!\cdots\!04\)\( T_{2}^{7} - \)\(48\!\cdots\!48\)\( T_{2}^{6} - \)\(10\!\cdots\!76\)\( T_{2}^{5} + \)\(17\!\cdots\!84\)\( T_{2}^{4} + \)\(29\!\cdots\!24\)\( T_{2}^{3} - \)\(69\!\cdots\!36\)\( T_{2}^{2} - \)\(29\!\cdots\!56\)\( T_{2} + \)\(85\!\cdots\!24\)\( \)">\(T_{2}^{18} - \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(177))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8546066599350042624 - 2986166801834541056 T - 693942416741339136 T^{2} + 291940745415092224 T^{3} + 17094064606518784 T^{4} - 10471805881131776 T^{5} - 48249714830848 T^{6} + 178756197263104 T^{7} - 3235536461632 T^{8} - 1636933261984 T^{9} + 47652762640 T^{10} + 8480054376 T^{11} - 296611506 T^{12} - 24830263 T^{13} + 951840 T^{14} + 38223 T^{15} - 1543 T^{16} - 24 T^{17} + T^{18} \)
$3$ \( ( -27 + T )^{18} \)
$5$ \( -\)\(55\!\cdots\!00\)\( + \)\(33\!\cdots\!00\)\( T + \)\(69\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} - \)\(22\!\cdots\!00\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} - \)\(63\!\cdots\!50\)\( T^{6} + \)\(83\!\cdots\!50\)\( T^{7} - \)\(27\!\cdots\!75\)\( T^{8} - \)\(18\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!85\)\( T^{10} + 22281867330443888452 T^{11} - 22478246164794430 T^{12} - 146078131940412 T^{13} + 176937790588 T^{14} + 495010858 T^{15} - 673119 T^{16} - 678 T^{17} + T^{18} \)
$7$ \( \)\(19\!\cdots\!40\)\( + \)\(21\!\cdots\!88\)\( T - \)\(28\!\cdots\!04\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!48\)\( T^{4} - \)\(61\!\cdots\!40\)\( T^{5} - \)\(19\!\cdots\!27\)\( T^{6} + \)\(24\!\cdots\!85\)\( T^{7} + \)\(94\!\cdots\!96\)\( T^{8} - \)\(14\!\cdots\!27\)\( T^{9} - \)\(20\!\cdots\!96\)\( T^{10} + \)\(34\!\cdots\!53\)\( T^{11} + 16499868697159016378 T^{12} - 37615099704794105 T^{13} - 1263517610260 T^{14} + 17929894423 T^{15} - 3484320 T^{16} - 3081 T^{17} + T^{18} \)
$11$ \( \)\(41\!\cdots\!36\)\( - \)\(10\!\cdots\!40\)\( T - \)\(10\!\cdots\!48\)\( T^{2} + \)\(81\!\cdots\!24\)\( T^{3} + \)\(58\!\cdots\!04\)\( T^{4} - \)\(27\!\cdots\!52\)\( T^{5} - \)\(13\!\cdots\!33\)\( T^{6} + \)\(52\!\cdots\!86\)\( T^{7} + \)\(14\!\cdots\!52\)\( T^{8} - \)\(56\!\cdots\!70\)\( T^{9} - \)\(81\!\cdots\!81\)\( T^{10} + \)\(34\!\cdots\!52\)\( T^{11} + \)\(19\!\cdots\!44\)\( T^{12} - \)\(11\!\cdots\!40\)\( T^{13} + 291928190138493 T^{14} + 2122286941002 T^{15} - 85979552 T^{16} - 15070 T^{17} + T^{18} \)
$13$ \( -\)\(22\!\cdots\!20\)\( + \)\(38\!\cdots\!24\)\( T - \)\(17\!\cdots\!32\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!38\)\( T^{4} - \)\(89\!\cdots\!74\)\( T^{5} - \)\(65\!\cdots\!39\)\( T^{6} + \)\(89\!\cdots\!96\)\( T^{7} - \)\(25\!\cdots\!88\)\( T^{8} - \)\(38\!\cdots\!78\)\( T^{9} + \)\(22\!\cdots\!07\)\( T^{10} + \)\(85\!\cdots\!80\)\( T^{11} - \)\(59\!\cdots\!90\)\( T^{12} - \)\(10\!\cdots\!90\)\( T^{13} + 74554157511247471 T^{14} + 5993828370396 T^{15} - 444025036 T^{16} - 13662 T^{17} + T^{18} \)
$17$ \( -\)\(15\!\cdots\!40\)\( + \)\(37\!\cdots\!08\)\( T - \)\(23\!\cdots\!12\)\( T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!28\)\( T^{4} - \)\(25\!\cdots\!08\)\( T^{5} - \)\(52\!\cdots\!17\)\( T^{6} + \)\(54\!\cdots\!77\)\( T^{7} + \)\(45\!\cdots\!52\)\( T^{8} - \)\(40\!\cdots\!17\)\( T^{9} - \)\(16\!\cdots\!22\)\( T^{10} + \)\(14\!\cdots\!13\)\( T^{11} + \)\(22\!\cdots\!22\)\( T^{12} - \)\(25\!\cdots\!47\)\( T^{13} - 224803045408108142 T^{14} + 221440443915091 T^{15} - 1827757584 T^{16} - 71919 T^{17} + T^{18} \)
$19$ \( \)\(18\!\cdots\!00\)\( + \)\(32\!\cdots\!44\)\( T - \)\(10\!\cdots\!24\)\( T^{2} - \)\(41\!\cdots\!28\)\( T^{3} + \)\(26\!\cdots\!36\)\( T^{4} - \)\(53\!\cdots\!92\)\( T^{5} - \)\(19\!\cdots\!72\)\( T^{6} + \)\(81\!\cdots\!08\)\( T^{7} + \)\(50\!\cdots\!15\)\( T^{8} - \)\(30\!\cdots\!67\)\( T^{9} - \)\(31\!\cdots\!73\)\( T^{10} + \)\(41\!\cdots\!58\)\( T^{11} - \)\(21\!\cdots\!41\)\( T^{12} - \)\(23\!\cdots\!57\)\( T^{13} + 27604467082890023677 T^{14} + 598936500449594 T^{15} - 9176680215 T^{16} - 56231 T^{17} + T^{18} \)
$23$ \( -\)\(34\!\cdots\!44\)\( - \)\(25\!\cdots\!24\)\( T - \)\(31\!\cdots\!08\)\( T^{2} + \)\(60\!\cdots\!96\)\( T^{3} + \)\(46\!\cdots\!72\)\( T^{4} - \)\(29\!\cdots\!48\)\( T^{5} - \)\(14\!\cdots\!04\)\( T^{6} + \)\(56\!\cdots\!44\)\( T^{7} + \)\(12\!\cdots\!07\)\( T^{8} - \)\(47\!\cdots\!41\)\( T^{9} - \)\(40\!\cdots\!93\)\( T^{10} + \)\(19\!\cdots\!70\)\( T^{11} + \)\(27\!\cdots\!51\)\( T^{12} - \)\(41\!\cdots\!31\)\( T^{13} + \)\(11\!\cdots\!37\)\( T^{14} + 4166842232603762 T^{15} - 21752118163 T^{16} - 150029 T^{17} + T^{18} \)
$29$ \( \)\(31\!\cdots\!00\)\( - \)\(55\!\cdots\!00\)\( T + \)\(16\!\cdots\!72\)\( T^{2} + \)\(17\!\cdots\!92\)\( T^{3} - \)\(89\!\cdots\!80\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!62\)\( T^{6} - \)\(70\!\cdots\!16\)\( T^{7} - \)\(58\!\cdots\!17\)\( T^{8} + \)\(10\!\cdots\!99\)\( T^{9} + \)\(21\!\cdots\!21\)\( T^{10} - \)\(20\!\cdots\!90\)\( T^{11} + \)\(15\!\cdots\!59\)\( T^{12} + \)\(25\!\cdots\!29\)\( T^{13} - \)\(88\!\cdots\!45\)\( T^{14} + 33539298219220330 T^{15} + 54512768469 T^{16} - 591285 T^{17} + T^{18} \)
$31$ \( -\)\(14\!\cdots\!60\)\( - \)\(10\!\cdots\!48\)\( T + \)\(22\!\cdots\!16\)\( T^{2} + \)\(22\!\cdots\!52\)\( T^{3} - \)\(19\!\cdots\!76\)\( T^{4} - \)\(11\!\cdots\!52\)\( T^{5} - \)\(15\!\cdots\!82\)\( T^{6} + \)\(23\!\cdots\!32\)\( T^{7} + \)\(26\!\cdots\!43\)\( T^{8} - \)\(25\!\cdots\!61\)\( T^{9} - \)\(70\!\cdots\!19\)\( T^{10} + \)\(14\!\cdots\!02\)\( T^{11} - \)\(89\!\cdots\!05\)\( T^{12} - \)\(45\!\cdots\!27\)\( T^{13} + \)\(63\!\cdots\!47\)\( T^{14} + 71404098235687914 T^{15} - 138399370867 T^{16} - 426733 T^{17} + T^{18} \)
$37$ \( \)\(76\!\cdots\!72\)\( - \)\(98\!\cdots\!80\)\( T - \)\(23\!\cdots\!20\)\( T^{2} + \)\(39\!\cdots\!88\)\( T^{3} + \)\(95\!\cdots\!50\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{5} + \)\(69\!\cdots\!93\)\( T^{6} + \)\(17\!\cdots\!75\)\( T^{7} - \)\(61\!\cdots\!40\)\( T^{8} - \)\(37\!\cdots\!17\)\( T^{9} + \)\(17\!\cdots\!28\)\( T^{10} + \)\(40\!\cdots\!75\)\( T^{11} - \)\(24\!\cdots\!82\)\( T^{12} - \)\(21\!\cdots\!23\)\( T^{13} + \)\(18\!\cdots\!84\)\( T^{14} + 46359733597784145 T^{15} - 675249694046 T^{16} - 7703 T^{17} + T^{18} \)
$41$ \( \)\(89\!\cdots\!00\)\( + \)\(42\!\cdots\!20\)\( T - \)\(21\!\cdots\!88\)\( T^{2} + \)\(16\!\cdots\!52\)\( T^{3} + \)\(14\!\cdots\!56\)\( T^{4} - \)\(16\!\cdots\!24\)\( T^{5} - \)\(33\!\cdots\!17\)\( T^{6} + \)\(59\!\cdots\!89\)\( T^{7} + \)\(20\!\cdots\!96\)\( T^{8} - \)\(99\!\cdots\!65\)\( T^{9} + \)\(38\!\cdots\!94\)\( T^{10} + \)\(86\!\cdots\!57\)\( T^{11} - \)\(68\!\cdots\!26\)\( T^{12} - \)\(39\!\cdots\!23\)\( T^{13} + \)\(40\!\cdots\!86\)\( T^{14} + 892121690436796915 T^{15} - 1062477031572 T^{16} - 770959 T^{17} + T^{18} \)
$43$ \( -\)\(10\!\cdots\!16\)\( + \)\(18\!\cdots\!76\)\( T - \)\(13\!\cdots\!36\)\( T^{2} + \)\(46\!\cdots\!28\)\( T^{3} - \)\(89\!\cdots\!56\)\( T^{4} + \)\(88\!\cdots\!88\)\( T^{5} - \)\(30\!\cdots\!33\)\( T^{6} - \)\(20\!\cdots\!74\)\( T^{7} + \)\(21\!\cdots\!24\)\( T^{8} - \)\(41\!\cdots\!42\)\( T^{9} - \)\(22\!\cdots\!13\)\( T^{10} + \)\(94\!\cdots\!68\)\( T^{11} + \)\(29\!\cdots\!32\)\( T^{12} - \)\(56\!\cdots\!60\)\( T^{13} + \)\(38\!\cdots\!41\)\( T^{14} + 1234259307888869374 T^{15} - 1314378952012 T^{16} - 793050 T^{17} + T^{18} \)
$47$ \( -\)\(27\!\cdots\!68\)\( + \)\(26\!\cdots\!64\)\( T + \)\(12\!\cdots\!60\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} - \)\(60\!\cdots\!44\)\( T^{4} - \)\(33\!\cdots\!80\)\( T^{5} + \)\(85\!\cdots\!32\)\( T^{6} + \)\(33\!\cdots\!56\)\( T^{7} - \)\(53\!\cdots\!03\)\( T^{8} - \)\(16\!\cdots\!17\)\( T^{9} + \)\(18\!\cdots\!03\)\( T^{10} + \)\(43\!\cdots\!50\)\( T^{11} - \)\(38\!\cdots\!99\)\( T^{12} - \)\(62\!\cdots\!51\)\( T^{13} + \)\(47\!\cdots\!09\)\( T^{14} + 4683862160543968258 T^{15} - 3355518768729 T^{16} - 1410373 T^{17} + T^{18} \)
$53$ \( \)\(75\!\cdots\!56\)\( + \)\(19\!\cdots\!12\)\( T + \)\(40\!\cdots\!92\)\( T^{2} - \)\(47\!\cdots\!72\)\( T^{3} - \)\(12\!\cdots\!36\)\( T^{4} + \)\(26\!\cdots\!28\)\( T^{5} + \)\(86\!\cdots\!38\)\( T^{6} - \)\(42\!\cdots\!42\)\( T^{7} - \)\(21\!\cdots\!31\)\( T^{8} + \)\(46\!\cdots\!74\)\( T^{9} + \)\(26\!\cdots\!33\)\( T^{10} + \)\(38\!\cdots\!84\)\( T^{11} - \)\(17\!\cdots\!18\)\( T^{12} - \)\(32\!\cdots\!24\)\( T^{13} + \)\(64\!\cdots\!24\)\( T^{14} + 9831686130049143066 T^{15} - 12552905271943 T^{16} - 1037934 T^{17} + T^{18} \)
$59$ \( ( -205379 + T )^{18} \)
$61$ \( \)\(36\!\cdots\!56\)\( + \)\(22\!\cdots\!16\)\( T - \)\(14\!\cdots\!72\)\( T^{2} - \)\(22\!\cdots\!68\)\( T^{3} - \)\(31\!\cdots\!60\)\( T^{4} + \)\(73\!\cdots\!08\)\( T^{5} + \)\(17\!\cdots\!62\)\( T^{6} - \)\(11\!\cdots\!36\)\( T^{7} - \)\(44\!\cdots\!25\)\( T^{8} + \)\(85\!\cdots\!47\)\( T^{9} + \)\(43\!\cdots\!65\)\( T^{10} - \)\(29\!\cdots\!46\)\( T^{11} - \)\(17\!\cdots\!01\)\( T^{12} + \)\(52\!\cdots\!93\)\( T^{13} + \)\(32\!\cdots\!91\)\( T^{14} - 43364517261107475158 T^{15} - 29409868811463 T^{16} + 1374623 T^{17} + T^{18} \)
$67$ \( -\)\(90\!\cdots\!24\)\( - \)\(25\!\cdots\!72\)\( T + \)\(12\!\cdots\!92\)\( T^{2} + \)\(27\!\cdots\!44\)\( T^{3} - \)\(38\!\cdots\!88\)\( T^{4} + \)\(35\!\cdots\!88\)\( T^{5} + \)\(52\!\cdots\!72\)\( T^{6} - \)\(83\!\cdots\!16\)\( T^{7} - \)\(38\!\cdots\!97\)\( T^{8} + \)\(71\!\cdots\!16\)\( T^{9} + \)\(15\!\cdots\!01\)\( T^{10} - \)\(29\!\cdots\!56\)\( T^{11} - \)\(37\!\cdots\!94\)\( T^{12} + \)\(61\!\cdots\!80\)\( T^{13} + \)\(50\!\cdots\!02\)\( T^{14} - 62518256259238817720 T^{15} - 35485606292597 T^{16} + 2436904 T^{17} + T^{18} \)
$71$ \( \)\(29\!\cdots\!64\)\( - \)\(40\!\cdots\!60\)\( T - \)\(95\!\cdots\!12\)\( T^{2} + \)\(40\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!66\)\( T^{4} - \)\(12\!\cdots\!10\)\( T^{5} + \)\(85\!\cdots\!67\)\( T^{6} + \)\(70\!\cdots\!74\)\( T^{7} - \)\(12\!\cdots\!50\)\( T^{8} + \)\(54\!\cdots\!84\)\( T^{9} + \)\(48\!\cdots\!83\)\( T^{10} - \)\(11\!\cdots\!92\)\( T^{11} + \)\(32\!\cdots\!62\)\( T^{12} + \)\(30\!\cdots\!74\)\( T^{13} - \)\(30\!\cdots\!19\)\( T^{14} + \)\(42\!\cdots\!42\)\( T^{15} + 37152599964818 T^{16} - 14289172 T^{17} + T^{18} \)
$73$ \( -\)\(12\!\cdots\!68\)\( - \)\(12\!\cdots\!48\)\( T - \)\(43\!\cdots\!76\)\( T^{2} - \)\(64\!\cdots\!96\)\( T^{3} - \)\(25\!\cdots\!40\)\( T^{4} + \)\(30\!\cdots\!20\)\( T^{5} + \)\(29\!\cdots\!28\)\( T^{6} - \)\(14\!\cdots\!24\)\( T^{7} - \)\(83\!\cdots\!45\)\( T^{8} - \)\(10\!\cdots\!59\)\( T^{9} + \)\(11\!\cdots\!23\)\( T^{10} + \)\(20\!\cdots\!02\)\( T^{11} - \)\(88\!\cdots\!21\)\( T^{12} - \)\(14\!\cdots\!29\)\( T^{13} + \)\(40\!\cdots\!81\)\( T^{14} + \)\(46\!\cdots\!34\)\( T^{15} - 98889506527283 T^{16} - 5482515 T^{17} + T^{18} \)
$79$ \( \)\(10\!\cdots\!40\)\( + \)\(44\!\cdots\!32\)\( T - \)\(43\!\cdots\!52\)\( T^{2} - \)\(22\!\cdots\!76\)\( T^{3} + \)\(51\!\cdots\!40\)\( T^{4} + \)\(32\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!35\)\( T^{6} - \)\(20\!\cdots\!42\)\( T^{7} + \)\(81\!\cdots\!24\)\( T^{8} + \)\(71\!\cdots\!90\)\( T^{9} - \)\(11\!\cdots\!09\)\( T^{10} - \)\(11\!\cdots\!08\)\( T^{11} + \)\(35\!\cdots\!84\)\( T^{12} + \)\(67\!\cdots\!32\)\( T^{13} - \)\(37\!\cdots\!95\)\( T^{14} + \)\(19\!\cdots\!62\)\( T^{15} + 110280473650448 T^{16} - 19786414 T^{17} + T^{18} \)
$83$ \( -\)\(20\!\cdots\!36\)\( + \)\(82\!\cdots\!32\)\( T + \)\(49\!\cdots\!56\)\( T^{2} - \)\(20\!\cdots\!92\)\( T^{3} + \)\(29\!\cdots\!16\)\( T^{4} - \)\(20\!\cdots\!68\)\( T^{5} + \)\(58\!\cdots\!95\)\( T^{6} + \)\(90\!\cdots\!55\)\( T^{7} - \)\(11\!\cdots\!24\)\( T^{8} + \)\(27\!\cdots\!49\)\( T^{9} + \)\(69\!\cdots\!38\)\( T^{10} - \)\(14\!\cdots\!93\)\( T^{11} + \)\(19\!\cdots\!82\)\( T^{12} + \)\(99\!\cdots\!27\)\( T^{13} - \)\(46\!\cdots\!82\)\( T^{14} + \)\(30\!\cdots\!93\)\( T^{15} + 191844106203420 T^{16} - 30227337 T^{17} + T^{18} \)
$89$ \( -\)\(37\!\cdots\!00\)\( + \)\(10\!\cdots\!20\)\( T + \)\(24\!\cdots\!16\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} - \)\(24\!\cdots\!92\)\( T^{4} + \)\(22\!\cdots\!92\)\( T^{5} - \)\(18\!\cdots\!92\)\( T^{6} - \)\(11\!\cdots\!88\)\( T^{7} + \)\(22\!\cdots\!51\)\( T^{8} + \)\(19\!\cdots\!51\)\( T^{9} - \)\(77\!\cdots\!93\)\( T^{10} + \)\(14\!\cdots\!98\)\( T^{11} + \)\(11\!\cdots\!79\)\( T^{12} - \)\(81\!\cdots\!99\)\( T^{13} - \)\(70\!\cdots\!47\)\( T^{14} + \)\(88\!\cdots\!30\)\( T^{15} + 6694208664213 T^{16} - 31061677 T^{17} + T^{18} \)
$97$ \( -\)\(17\!\cdots\!84\)\( - \)\(39\!\cdots\!44\)\( T + \)\(30\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} - \)\(74\!\cdots\!28\)\( T^{4} - \)\(21\!\cdots\!72\)\( T^{5} + \)\(67\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!76\)\( T^{7} - \)\(26\!\cdots\!63\)\( T^{8} - \)\(32\!\cdots\!66\)\( T^{9} + \)\(50\!\cdots\!01\)\( T^{10} + \)\(42\!\cdots\!16\)\( T^{11} - \)\(52\!\cdots\!58\)\( T^{12} - \)\(29\!\cdots\!76\)\( T^{13} + \)\(29\!\cdots\!90\)\( T^{14} + \)\(95\!\cdots\!44\)\( T^{15} - 864962403425135 T^{16} - 12084118 T^{17} + T^{18} \)
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