Properties

Label 177.6.a.b
Level $177$
Weight $6$
Character orbit 177.a
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -9 q^{3} + ( 17 + \beta_{2} ) q^{4} + ( 3 + \beta_{1} - \beta_{10} ) q^{5} + 9 \beta_{1} q^{6} + ( -35 + 2 \beta_{1} - \beta_{2} - \beta_{11} ) q^{7} + ( -21 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{8} + 81 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -9 q^{3} + ( 17 + \beta_{2} ) q^{4} + ( 3 + \beta_{1} - \beta_{10} ) q^{5} + 9 \beta_{1} q^{6} + ( -35 + 2 \beta_{1} - \beta_{2} - \beta_{11} ) q^{7} + ( -21 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{8} + 81 q^{9} + ( -76 + 2 \beta_{1} - 5 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{10} + ( 35 + 9 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{11} + ( -153 - 9 \beta_{2} ) q^{12} + ( -85 - 2 \beta_{1} - 10 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{13} + ( -101 + 57 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} + 7 \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{14} + ( -27 - 9 \beta_{1} + 9 \beta_{10} ) q^{15} + ( 524 + 28 \beta_{1} + 13 \beta_{2} + 4 \beta_{3} - 11 \beta_{4} + 8 \beta_{5} - 15 \beta_{6} - 4 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{16} + ( -136 + 31 \beta_{1} - 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 12 \beta_{6} + 9 \beta_{7} + 6 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{17} -81 \beta_{1} q^{18} + ( -280 + 43 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} + 7 \beta_{5} + 6 \beta_{6} - 8 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{19} + ( -135 + 227 \beta_{1} - 15 \beta_{3} - 17 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - 19 \beta_{11} ) q^{20} + ( 315 - 18 \beta_{1} + 9 \beta_{2} + 9 \beta_{11} ) q^{21} + ( -203 - 88 \beta_{1} + 7 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} - 4 \beta_{7} + 17 \beta_{8} + 24 \beta_{9} - 9 \beta_{10} + 7 \beta_{11} ) q^{22} + ( -280 + 238 \beta_{1} - 2 \beta_{2} + 13 \beta_{3} + 18 \beta_{4} + 17 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} + 7 \beta_{8} - 26 \beta_{9} - 7 \beta_{10} - 10 \beta_{11} ) q^{23} + ( 189 \beta_{1} + 9 \beta_{4} - 9 \beta_{5} - 9 \beta_{8} - 18 \beta_{10} - 9 \beta_{11} ) q^{24} + ( 638 + 225 \beta_{1} + 11 \beta_{2} + 26 \beta_{3} - 9 \beta_{4} + 12 \beta_{5} - 16 \beta_{6} + 9 \beta_{7} - 16 \beta_{9} - 8 \beta_{10} + 16 \beta_{11} ) q^{25} + ( 64 + 472 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} + 23 \beta_{4} - 13 \beta_{5} - 14 \beta_{6} + 3 \beta_{7} - 17 \beta_{8} + \beta_{9} - 20 \beta_{10} - 11 \beta_{11} ) q^{26} -729 q^{27} + ( -1849 + 393 \beta_{1} - 76 \beta_{2} + 4 \beta_{3} - 15 \beta_{4} - 6 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} - 33 \beta_{9} - 41 \beta_{10} - 13 \beta_{11} ) q^{28} + ( -295 + 337 \beta_{1} + 14 \beta_{2} - 3 \beta_{3} - 22 \beta_{4} - 23 \beta_{5} + 31 \beta_{6} - 8 \beta_{7} - 13 \beta_{8} + 10 \beta_{9} + 28 \beta_{10} + 14 \beta_{11} ) q^{29} + ( 684 - 18 \beta_{1} + 45 \beta_{2} + 9 \beta_{3} - 27 \beta_{4} + 18 \beta_{5} - 18 \beta_{6} - 9 \beta_{7} + 18 \beta_{8} - 27 \beta_{10} - 9 \beta_{11} ) q^{30} + ( -939 - 55 \beta_{1} + 24 \beta_{2} - 41 \beta_{3} + 55 \beta_{4} - 21 \beta_{5} - 11 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 20 \beta_{9} + 12 \beta_{11} ) q^{31} + ( -1067 - 456 \beta_{1} + 51 \beta_{2} + 24 \beta_{3} + 38 \beta_{4} + 16 \beta_{6} - 16 \beta_{7} + 4 \beta_{8} + 74 \beta_{9} + 18 \beta_{10} + 46 \beta_{11} ) q^{32} + ( -315 - 81 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} + 36 \beta_{6} + 18 \beta_{7} + 18 \beta_{8} - 9 \beta_{9} - 27 \beta_{10} - 9 \beta_{11} ) q^{33} + ( -1850 + 499 \beta_{1} - 46 \beta_{2} - 60 \beta_{3} + 5 \beta_{4} - 34 \beta_{5} - 43 \beta_{6} + 12 \beta_{7} - 17 \beta_{8} - 50 \beta_{9} - 39 \beta_{10} - 31 \beta_{11} ) q^{34} + ( 163 + 85 \beta_{1} + 105 \beta_{2} - 15 \beta_{3} + 15 \beta_{4} - 5 \beta_{5} - 25 \beta_{6} - 17 \beta_{7} - 9 \beta_{8} + 10 \beta_{9} + 65 \beta_{10} + 8 \beta_{11} ) q^{35} + ( 1377 + 81 \beta_{2} ) q^{36} + ( -2348 - 323 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} - 103 \beta_{4} - 26 \beta_{5} + 54 \beta_{6} + 20 \beta_{7} - 26 \beta_{8} + 56 \beta_{9} + 69 \beta_{10} + 43 \beta_{11} ) q^{37} + ( -2182 + 529 \beta_{1} - \beta_{2} + 42 \beta_{3} - 109 \beta_{4} + 65 \beta_{5} - 33 \beta_{6} - 20 \beta_{7} - 44 \beta_{8} - 59 \beta_{9} + 20 \beta_{10} + \beta_{11} ) q^{38} + ( 765 + 18 \beta_{1} + 90 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} - 27 \beta_{8} + 18 \beta_{9} - 9 \beta_{11} ) q^{39} + ( -8326 - 338 \beta_{1} - 191 \beta_{2} + 21 \beta_{3} + 168 \beta_{4} + \beta_{5} + 108 \beta_{6} + 27 \beta_{7} - 13 \beta_{8} - 10 \beta_{9} - 35 \beta_{10} + 10 \beta_{11} ) q^{40} + ( -549 - 47 \beta_{1} + 122 \beta_{2} - 62 \beta_{3} + 13 \beta_{4} - 44 \beta_{5} + 17 \beta_{6} + 4 \beta_{7} + 39 \beta_{8} + 27 \beta_{9} + 86 \beta_{10} - 9 \beta_{11} ) q^{41} + ( 909 - 513 \beta_{1} + 90 \beta_{2} - 18 \beta_{3} - 63 \beta_{4} + 9 \beta_{5} - 36 \beta_{6} + 9 \beta_{8} + 27 \beta_{9} + 9 \beta_{10} + 18 \beta_{11} ) q^{42} + ( -2607 - 510 \beta_{1} + 65 \beta_{2} + 79 \beta_{3} + 5 \beta_{4} + 37 \beta_{5} - 31 \beta_{6} + 34 \beta_{7} + 37 \beta_{8} - 26 \beta_{9} + 99 \beta_{10} - 5 \beta_{11} ) q^{43} + ( 2178 + 335 \beta_{1} + 168 \beta_{2} + 90 \beta_{3} + 50 \beta_{4} + 92 \beta_{5} - 229 \beta_{6} + 12 \beta_{7} + 45 \beta_{8} - 106 \beta_{9} + 7 \beta_{10} + 13 \beta_{11} ) q^{44} + ( 243 + 81 \beta_{1} - 81 \beta_{10} ) q^{45} + ( -12114 + 273 \beta_{1} - 259 \beta_{2} + 18 \beta_{3} - 65 \beta_{4} + 29 \beta_{5} + 115 \beta_{6} - 16 \beta_{7} + 36 \beta_{8} - 9 \beta_{9} - 14 \beta_{10} - 65 \beta_{11} ) q^{46} + ( -6599 - 122 \beta_{1} - 18 \beta_{2} - 23 \beta_{3} - 89 \beta_{4} + 23 \beta_{5} + 46 \beta_{6} - 25 \beta_{7} + 42 \beta_{8} - 5 \beta_{9} + 115 \beta_{10} - 34 \beta_{11} ) q^{47} + ( -4716 - 252 \beta_{1} - 117 \beta_{2} - 36 \beta_{3} + 99 \beta_{4} - 72 \beta_{5} + 135 \beta_{6} + 36 \beta_{7} - 9 \beta_{8} - 9 \beta_{9} + 18 \beta_{10} - 18 \beta_{11} ) q^{48} + ( -2315 - 1002 \beta_{1} + 83 \beta_{2} + 64 \beta_{3} + 91 \beta_{4} + 40 \beta_{5} - 107 \beta_{6} - 15 \beta_{7} + 57 \beta_{8} + 7 \beta_{9} + 21 \beta_{11} ) q^{49} + ( -11119 - 911 \beta_{1} - 122 \beta_{2} - 92 \beta_{3} - 42 \beta_{4} - 29 \beta_{5} + 67 \beta_{6} - 18 \beta_{7} + 70 \beta_{8} + 127 \beta_{9} + 16 \beta_{10} + 33 \beta_{11} ) q^{50} + ( 1224 - 279 \beta_{1} + 54 \beta_{3} - 36 \beta_{4} + 18 \beta_{5} - 108 \beta_{6} - 81 \beta_{7} - 54 \beta_{9} - 36 \beta_{10} + 9 \beta_{11} ) q^{51} + ( -20543 - 947 \beta_{1} - 397 \beta_{2} - 129 \beta_{3} + 139 \beta_{4} - 95 \beta_{5} + 190 \beta_{6} + 27 \beta_{7} - 9 \beta_{8} + 157 \beta_{9} + 8 \beta_{10} - 37 \beta_{11} ) q^{52} + ( -1061 + 255 \beta_{1} - 12 \beta_{2} + 90 \beta_{3} - 184 \beta_{4} + 110 \beta_{5} - 102 \beta_{6} + 36 \beta_{7} + 120 \beta_{8} - 214 \beta_{9} - 27 \beta_{10} - 56 \beta_{11} ) q^{53} + 729 \beta_{1} q^{54} + ( -10025 + 253 \beta_{1} - 155 \beta_{2} - 76 \beta_{3} - 215 \beta_{4} - 94 \beta_{5} - 100 \beta_{6} - 149 \beta_{7} - 172 \beta_{8} + 68 \beta_{9} + 32 \beta_{10} - 48 \beta_{11} ) q^{55} + ( -16532 + 2189 \beta_{1} - 418 \beta_{2} - 112 \beta_{3} + 58 \beta_{4} - 120 \beta_{5} + 295 \beta_{6} + 66 \beta_{7} - 199 \beta_{8} + 32 \beta_{9} - 17 \beta_{10} - 27 \beta_{11} ) q^{56} + ( 2520 - 387 \beta_{1} + 27 \beta_{2} + 81 \beta_{3} - 63 \beta_{5} - 54 \beta_{6} + 72 \beta_{7} + 18 \beta_{8} - 27 \beta_{9} + 9 \beta_{10} - 18 \beta_{11} ) q^{57} + ( -15716 - 323 \beta_{1} - 384 \beta_{2} + 47 \beta_{3} - 118 \beta_{4} + 95 \beta_{5} - 119 \beta_{6} - 33 \beta_{7} - 58 \beta_{8} - 157 \beta_{9} - 77 \beta_{10} - 46 \beta_{11} ) q^{58} -3481 q^{59} + ( 1215 - 2043 \beta_{1} + 135 \beta_{3} + 153 \beta_{4} + 90 \beta_{5} - 18 \beta_{6} - 27 \beta_{7} - 18 \beta_{8} + 18 \beta_{9} + 27 \beta_{10} + 171 \beta_{11} ) q^{60} + ( -4507 + 408 \beta_{1} + 93 \beta_{2} - 239 \beta_{3} + 279 \beta_{4} - 119 \beta_{5} + 80 \beta_{6} + 20 \beta_{7} - 266 \beta_{8} + 151 \beta_{9} - 146 \beta_{10} - 54 \beta_{11} ) q^{61} + ( 2990 + 217 \beta_{1} + 306 \beta_{2} - 69 \beta_{3} - 138 \beta_{4} - 87 \beta_{5} - 151 \beta_{6} + 111 \beta_{7} + 100 \beta_{8} + 211 \beta_{9} - 13 \beta_{10} + 64 \beta_{11} ) q^{62} + ( -2835 + 162 \beta_{1} - 81 \beta_{2} - 81 \beta_{11} ) q^{63} + ( 3994 - 13 \beta_{1} + 222 \beta_{2} + 80 \beta_{3} - 223 \beta_{4} - 13 \beta_{5} - 534 \beta_{6} - 60 \beta_{7} + 185 \beta_{8} - 176 \beta_{9} + 64 \beta_{10} - 79 \beta_{11} ) q^{64} + ( 4301 - 2231 \beta_{1} + 65 \beta_{2} + 214 \beta_{3} + 392 \beta_{4} + 196 \beta_{5} + 58 \beta_{6} + 41 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 220 \beta_{10} + 210 \beta_{11} ) q^{65} + ( 1827 + 792 \beta_{1} - 63 \beta_{2} - 72 \beta_{3} + 54 \beta_{4} - 36 \beta_{5} - 45 \beta_{6} + 36 \beta_{7} - 153 \beta_{8} - 216 \beta_{9} + 81 \beta_{10} - 63 \beta_{11} ) q^{66} + ( 2589 - 981 \beta_{1} + 313 \beta_{2} + 336 \beta_{3} + 269 \beta_{4} + 306 \beta_{5} - 210 \beta_{6} - 3 \beta_{7} + 36 \beta_{8} - 156 \beta_{9} - 164 \beta_{10} - 24 \beta_{11} ) q^{67} + ( -17899 + 482 \beta_{1} - 573 \beta_{2} - 162 \beta_{3} + 493 \beta_{4} - 397 \beta_{5} + 673 \beta_{6} + 56 \beta_{7} - 126 \beta_{8} + 290 \beta_{9} - 121 \beta_{10} + 166 \beta_{11} ) q^{68} + ( 2520 - 2142 \beta_{1} + 18 \beta_{2} - 117 \beta_{3} - 162 \beta_{4} - 153 \beta_{5} + 63 \beta_{6} - 18 \beta_{7} - 63 \beta_{8} + 234 \beta_{9} + 63 \beta_{10} + 90 \beta_{11} ) q^{69} + ( -1431 - 4795 \beta_{1} + 331 \beta_{2} + 104 \beta_{3} - 405 \beta_{4} + 248 \beta_{5} - 116 \beta_{6} - 44 \beta_{7} + 312 \beta_{8} + 178 \beta_{9} + 38 \beta_{10} + 124 \beta_{11} ) q^{70} + ( 4965 - 143 \beta_{1} + 163 \beta_{2} + 63 \beta_{3} - 283 \beta_{4} - 11 \beta_{5} + 204 \beta_{6} - 123 \beta_{7} - 82 \beta_{8} + 157 \beta_{9} - 32 \beta_{10} + 33 \beta_{11} ) q^{71} + ( -1701 \beta_{1} - 81 \beta_{4} + 81 \beta_{5} + 81 \beta_{8} + 162 \beta_{10} + 81 \beta_{11} ) q^{72} + ( 43 - 2367 \beta_{1} + 117 \beta_{2} - 293 \beta_{3} + 100 \beta_{4} - 147 \beta_{5} + 113 \beta_{6} + \beta_{7} + 161 \beta_{8} - 286 \beta_{9} - 257 \beta_{10} - 176 \beta_{11} ) q^{73} + ( 17261 + 3157 \beta_{1} + 453 \beta_{2} - 159 \beta_{3} + 14 \beta_{4} + 33 \beta_{5} - 382 \beta_{6} - 99 \beta_{7} - 111 \beta_{8} - 205 \beta_{9} - 156 \beta_{10} + 41 \beta_{11} ) q^{74} + ( -5742 - 2025 \beta_{1} - 99 \beta_{2} - 234 \beta_{3} + 81 \beta_{4} - 108 \beta_{5} + 144 \beta_{6} - 81 \beta_{7} + 144 \beta_{9} + 72 \beta_{10} - 144 \beta_{11} ) q^{75} + ( -14181 - 594 \beta_{1} - 300 \beta_{2} + 282 \beta_{3} - 211 \beta_{4} + 62 \beta_{5} + 595 \beta_{6} + 74 \beta_{7} + 33 \beta_{8} - 77 \beta_{9} + 136 \beta_{10} + 46 \beta_{11} ) q^{76} + ( -1543 + 1074 \beta_{1} - 301 \beta_{2} + 46 \beta_{3} + 167 \beta_{4} + 54 \beta_{5} + 124 \beta_{6} + 224 \beta_{7} + 220 \beta_{8} - 266 \beta_{9} - 446 \beta_{10} - 281 \beta_{11} ) q^{77} + ( -576 - 4248 \beta_{1} - 108 \beta_{2} - 27 \beta_{3} - 207 \beta_{4} + 117 \beta_{5} + 126 \beta_{6} - 27 \beta_{7} + 153 \beta_{8} - 9 \beta_{9} + 180 \beta_{10} + 99 \beta_{11} ) q^{78} + ( -2095 - 2359 \beta_{1} + 122 \beta_{2} + 95 \beta_{3} - 716 \beta_{4} - 293 \beta_{5} + 84 \beta_{6} + 72 \beta_{7} - 510 \beta_{8} + 33 \beta_{9} + 401 \beta_{10} + 283 \beta_{11} ) q^{79} + ( 14820 + 10741 \beta_{1} - 270 \beta_{2} + 297 \beta_{3} - 136 \beta_{4} + 124 \beta_{5} - 503 \beta_{6} - 149 \beta_{7} - 241 \beta_{8} - 615 \beta_{9} - 745 \beta_{10} - 263 \beta_{11} ) q^{80} + 6561 q^{81} + ( 4635 - 4013 \beta_{1} + 518 \beta_{2} + 148 \beta_{3} - \beta_{4} + 149 \beta_{5} - 472 \beta_{6} + 90 \beta_{7} + 137 \beta_{8} + 51 \beta_{9} + 67 \beta_{10} + 140 \beta_{11} ) q^{82} + ( -8495 - 2167 \beta_{1} - 110 \beta_{2} - 550 \beta_{3} + 827 \beta_{4} - 148 \beta_{5} + 59 \beta_{6} + 160 \beta_{7} + 121 \beta_{8} - 167 \beta_{9} + 68 \beta_{10} - 299 \beta_{11} ) q^{83} + ( 16641 - 3537 \beta_{1} + 684 \beta_{2} - 36 \beta_{3} + 135 \beta_{4} + 54 \beta_{5} - 72 \beta_{6} + 18 \beta_{7} + 72 \beta_{8} + 297 \beta_{9} + 369 \beta_{10} + 117 \beta_{11} ) q^{84} + ( -7000 - 7107 \beta_{1} + 459 \beta_{2} - 307 \beta_{3} + 671 \beta_{4} + \beta_{5} + 458 \beta_{6} + 114 \beta_{7} + 516 \beta_{8} + 85 \beta_{9} - 407 \beta_{10} - 28 \beta_{11} ) q^{85} + ( 26081 + 454 \beta_{1} + 951 \beta_{2} + 86 \beta_{3} - 226 \beta_{4} + 224 \beta_{5} - 381 \beta_{6} - 206 \beta_{7} + 495 \beta_{8} + 98 \beta_{9} - 203 \beta_{10} - 103 \beta_{11} ) q^{86} + ( 2655 - 3033 \beta_{1} - 126 \beta_{2} + 27 \beta_{3} + 198 \beta_{4} + 207 \beta_{5} - 279 \beta_{6} + 72 \beta_{7} + 117 \beta_{8} - 90 \beta_{9} - 252 \beta_{10} - 126 \beta_{11} ) q^{87} + ( -6343 - 7656 \beta_{1} + 705 \beta_{2} - 346 \beta_{3} - 51 \beta_{4} - 65 \beta_{5} + 417 \beta_{6} + 116 \beta_{7} + 466 \beta_{8} + 652 \beta_{9} + 681 \beta_{10} + 296 \beta_{11} ) q^{88} + ( -3021 - 2491 \beta_{1} - 199 \beta_{2} + 461 \beta_{3} - 615 \beta_{4} + 451 \beta_{5} - 49 \beta_{6} - 277 \beta_{7} - 391 \beta_{8} + 418 \beta_{9} - 71 \beta_{10} + 314 \beta_{11} ) q^{89} + ( -6156 + 162 \beta_{1} - 405 \beta_{2} - 81 \beta_{3} + 243 \beta_{4} - 162 \beta_{5} + 162 \beta_{6} + 81 \beta_{7} - 162 \beta_{8} + 243 \beta_{10} + 81 \beta_{11} ) q^{90} + ( -1571 - 3545 \beta_{1} + 846 \beta_{2} + 246 \beta_{3} + 82 \beta_{4} - 176 \beta_{5} + 53 \beta_{6} + 38 \beta_{7} - 129 \beta_{8} + 359 \beta_{9} + 564 \beta_{10} + 163 \beta_{11} ) q^{91} + ( -8111 + 14922 \beta_{1} - 1082 \beta_{2} + 114 \beta_{3} - 125 \beta_{4} - 364 \beta_{5} - 59 \beta_{6} - 282 \beta_{7} - 1011 \beta_{8} - 301 \beta_{9} - 66 \beta_{10} - 214 \beta_{11} ) q^{92} + ( 8451 + 495 \beta_{1} - 216 \beta_{2} + 369 \beta_{3} - 495 \beta_{4} + 189 \beta_{5} + 99 \beta_{6} + 18 \beta_{7} - 27 \beta_{8} - 180 \beta_{9} - 108 \beta_{11} ) q^{93} + ( 8617 + 6897 \beta_{1} + 481 \beta_{2} + 546 \beta_{3} - 47 \beta_{4} + 542 \beta_{5} - 384 \beta_{6} - 206 \beta_{7} - 222 \beta_{8} - 494 \beta_{9} - 40 \beta_{10} - 44 \beta_{11} ) q^{94} + ( -4762 - 1302 \beta_{1} - 156 \beta_{2} + 507 \beta_{3} - 641 \beta_{4} + 39 \beta_{5} + 185 \beta_{6} + 28 \beta_{7} + 105 \beta_{8} + 430 \beta_{9} + 767 \beta_{10} + 186 \beta_{11} ) q^{95} + ( 9603 + 4104 \beta_{1} - 459 \beta_{2} - 216 \beta_{3} - 342 \beta_{4} - 144 \beta_{6} + 144 \beta_{7} - 36 \beta_{8} - 666 \beta_{9} - 162 \beta_{10} - 414 \beta_{11} ) q^{96} + ( -15969 + 98 \beta_{1} + 534 \beta_{2} - 830 \beta_{3} - 458 \beta_{4} - 188 \beta_{5} + 365 \beta_{6} + 213 \beta_{7} + 279 \beta_{8} - 137 \beta_{9} + 366 \beta_{10} - 298 \beta_{11} ) q^{97} + ( 48528 - 558 \beta_{1} + 1644 \beta_{2} + 334 \beta_{3} - 389 \beta_{4} + 200 \beta_{5} - 727 \beta_{6} - 138 \beta_{7} + 649 \beta_{8} + 552 \beta_{9} + 197 \beta_{10} + 179 \beta_{11} ) q^{98} + ( 2835 + 729 \beta_{1} - 162 \beta_{2} + 81 \beta_{3} - 81 \beta_{4} - 81 \beta_{5} - 324 \beta_{6} - 162 \beta_{7} - 162 \beta_{8} + 81 \beta_{9} + 243 \beta_{10} + 81 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 49 \)
\(\beta_{3}\)\(=\)\((\)\(70995163178963 \nu^{11} + 1260174342721206 \nu^{10} - 15069489363041309 \nu^{9} - 373612354185694543 \nu^{8} + 673646677911366614 \nu^{7} + 38651364669125368297 \nu^{6} + 29283065204018815204 \nu^{5} - 1621232990873406107520 \nu^{4} - 1871642618466862238976 \nu^{3} + 22682859078888124275168 \nu^{2} + 2309661345958546715392 \nu - 44540621931975981150016\)\()/ \)\(37\!\cdots\!24\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-6070866593397 \nu^{11} + 12704329665294 \nu^{10} + 2025127413667451 \nu^{9} - 1017645976025407 \nu^{8} - 237603629271932978 \nu^{7} - 197118219957298639 \nu^{6} + 11187289331975452540 \nu^{5} + 18576113861191679248 \nu^{4} - 169212860659841880416 \nu^{3} - 234904833095610043232 \nu^{2} + 578535839854472803200 \nu + 252251457838879227840\)\()/ 13435985691809412608 \)
\(\beta_{5}\)\(=\)\((\)\(276443349871579 \nu^{11} - 1348141595753418 \nu^{10} - 73035502710430869 \nu^{9} + 374909234235669753 \nu^{8} + 6536419220059167446 \nu^{7} - 37431432449168562815 \nu^{6} - 227080010856454277612 \nu^{5} + 1630276694691093230368 \nu^{4} + 2114165735599360806336 \nu^{3} - 27970911891489684321440 \nu^{2} + 17671121328252462456832 \nu + 75277099833303083684032\)\()/ \)\(37\!\cdots\!24\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-157574427250607 \nu^{11} - 62814114617374 \nu^{10} + 42942157372889537 \nu^{9} + 2124163599740043 \nu^{8} - 4033264827299187998 \nu^{7} + 1830350641272830531 \nu^{6} + 154658714920895602524 \nu^{5} - 172745082388610069312 \nu^{4} - 2151351397113547096704 \nu^{3} + 4029036932992469914912 \nu^{2} + 5255893016983507189504 \nu - 11260185469833302312896\)\()/ \)\(18\!\cdots\!12\)\( \)
\(\beta_{7}\)\(=\)\((\)\(14329992266831 \nu^{11} + 68921999442182 \nu^{10} - 3979652032884737 \nu^{9} - 18209905676036451 \nu^{8} + 370390697832800582 \nu^{7} + 1580859950161255549 \nu^{6} - 12479805914223073028 \nu^{5} - 48722748546299885392 \nu^{4} + 65838396490605035936 \nu^{3} + 258422640881088540704 \nu^{2} + 1080506509259373478272 \nu + 143671022417105045696\)\()/ 13435985691809412608 \)
\(\beta_{8}\)\(=\)\((\)\(-1232144514507415 \nu^{11} + 339054598749730 \nu^{10} + 341906398675822169 \nu^{9} - 83240015220961021 \nu^{8} - 32984685428531840318 \nu^{7} + 9721228830858762571 \nu^{6} + 1309121973508228341708 \nu^{5} - 671477905131811469056 \nu^{4} - 18796017885994915956480 \nu^{3} + 20115068195709835534496 \nu^{2} + 59192065957194059700480 \nu - 55526849730687927914944\)\()/ \)\(37\!\cdots\!24\)\( \)
\(\beta_{9}\)\(=\)\((\)\(1322194336654907 \nu^{11} - 3701116547366298 \nu^{10} - 382838896328099701 \nu^{9} + 887888971518781257 \nu^{8} + 38852118588478225030 \nu^{7} - 70689929145009309215 \nu^{6} - 1616015194601249637852 \nu^{5} + 2326930855637581854912 \nu^{4} + 23223995518001361140352 \nu^{3} - 36921974083861151814688 \nu^{2} - 62943275448972877997824 \nu + 86575115576869461415616\)\()/ \)\(37\!\cdots\!24\)\( \)
\(\beta_{10}\)\(=\)\((\)\(1524851944712805 \nu^{11} - 2687442487401494 \nu^{10} - 429352011744410923 \nu^{9} + 659727714848877735 \nu^{8} + 42190034547873270282 \nu^{7} - 55447889370230709313 \nu^{6} - 1705284178220637720180 \nu^{5} + 1979508163645875132640 \nu^{4} + 24379050756287078102336 \nu^{3} - 30623812468319425334880 \nu^{2} - 59477569094551464042496 \nu + 41000179717318874316608\)\()/ \)\(37\!\cdots\!24\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-1131993494702445 \nu^{11} + 3369846601217454 \nu^{10} + 323268347553059587 \nu^{9} - 819809368020587799 \nu^{8} - 32292352253443995538 \nu^{7} + 66543336099983428489 \nu^{6} + 1320885247542407023692 \nu^{5} - 2198841964368832503824 \nu^{4} - 18595208530012418629600 \nu^{3} + 31263066652870809123104 \nu^{2} + 45134300183044023212160 \nu - 47343784358882143011392\)\()/ \)\(18\!\cdots\!12\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 49\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - 2 \beta_{10} - \beta_{8} - \beta_{5} + \beta_{4} + 85 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} - 4 \beta_{7} - 15 \beta_{6} + 8 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} + 109 \beta_{2} + 28 \beta_{1} + 4204\)
\(\nu^{5}\)\(=\)\(-174 \beta_{11} - 274 \beta_{10} - 74 \beta_{9} - 132 \beta_{8} + 16 \beta_{7} - 16 \beta_{6} - 128 \beta_{5} + 90 \beta_{4} - 24 \beta_{3} - 51 \beta_{2} + 8264 \beta_{1} + 1067\)
\(\nu^{6}\)\(=\)\(241 \beta_{11} - 256 \beta_{10} - 16 \beta_{9} + 345 \beta_{8} - 700 \beta_{7} - 2934 \beta_{6} + 1267 \beta_{5} - 1983 \beta_{4} + 720 \beta_{3} + 11518 \beta_{2} + 4467 \beta_{1} + 408346\)
\(\nu^{7}\)\(=\)\(-23672 \beta_{11} - 33084 \beta_{10} - 15203 \beta_{9} - 16221 \beta_{8} + 2168 \beta_{7} - 4141 \beta_{6} - 14204 \beta_{5} + 5545 \beta_{4} - 4732 \beta_{3} - 10337 \beta_{2} + 856854 \beta_{1} + 152254\)
\(\nu^{8}\)\(=\)\(21552 \beta_{11} - 33848 \beta_{10} - 17550 \beta_{9} + 59270 \beta_{8} - 94636 \beta_{7} - 433282 \beta_{6} + 161360 \beta_{5} - 274062 \beta_{4} + 96776 \beta_{3} + 1242671 \beta_{2} + 559978 \beta_{1} + 42181637\)
\(\nu^{9}\)\(=\)\(-2994209 \beta_{11} - 3901586 \beta_{10} - 2315000 \beta_{9} - 1948793 \beta_{8} + 220864 \beta_{7} - 742552 \beta_{6} - 1516353 \beta_{5} + 155425 \beta_{4} - 659936 \beta_{3} - 1531728 \beta_{2} + 92478861 \beta_{1} + 17198384\)
\(\nu^{10}\)\(=\)\(1651466 \beta_{11} - 4556482 \beta_{10} - 3545407 \beta_{9} + 8261609 \beta_{8} - 11753444 \beta_{7} - 57650911 \beta_{6} + 19302976 \beta_{5} - 34848755 \beta_{4} + 11852676 \beta_{3} + 137093709 \beta_{2} + 67446412 \beta_{1} + 4534829964\)
\(\nu^{11}\)\(=\)\(-366962750 \beta_{11} - 457488818 \beta_{10} - 314484546 \beta_{9} - 231380860 \beta_{8} + 20065392 \beta_{7} - 113760984 \beta_{6} - 161222800 \beta_{5} - 25669038 \beta_{4} - 81723608 \beta_{3} - 200929955 \beta_{2} + 10256141096 \beta_{1} + 1913913907\)

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
10.9029
8.94324
8.80731
4.14510
2.87969
1.62334
0.501674
−1.65902
−5.27208
−7.32600
−8.78000
−10.7661
−10.9029 −9.00000 86.8728 88.1143 98.1259 61.7114 −598.272 81.0000 −960.699
1.2 −8.94324 −9.00000 47.9816 48.3356 80.4892 −32.5863 −142.927 81.0000 −432.277
1.3 −8.80731 −9.00000 45.5687 −88.4760 79.2658 −134.279 −119.504 81.0000 779.236
1.4 −4.14510 −9.00000 −14.8181 −34.9684 37.3059 −110.249 194.066 81.0000 144.948
1.5 −2.87969 −9.00000 −23.7074 −77.5334 25.9172 45.2937 160.420 81.0000 223.272
1.6 −1.62334 −9.00000 −29.3648 103.513 14.6101 −137.577 99.6159 81.0000 −168.037
1.7 −0.501674 −9.00000 −31.7483 −14.0506 4.51506 117.234 31.9809 81.0000 7.04881
1.8 1.65902 −9.00000 −29.2477 59.9319 −14.9312 −87.2478 −101.611 81.0000 99.4282
1.9 5.27208 −9.00000 −4.20512 −17.5332 −47.4488 172.711 −190.876 81.0000 −92.4366
1.10 7.32600 −9.00000 21.6702 46.7845 −65.9340 −85.8026 −75.6759 81.0000 342.743
1.11 8.78000 −9.00000 45.0883 −17.5207 −79.0200 12.9559 114.916 81.0000 −153.831
1.12 10.7661 −9.00000 83.9098 −60.5969 −96.8953 −233.164 558.868 81.0000 −652.394
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 177.6.a.b 12
3.b odd 2 1 531.6.a.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.a.b 12 1.a even 1 1 trivial
531.6.a.d 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(177))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -50564480 - 120303168 T - 16576192 T^{2} + 53008480 T^{3} + 15496192 T^{4} - 3566264 T^{5} - 1130486 T^{6} + 94393 T^{7} + 27968 T^{8} - 1045 T^{9} - 283 T^{10} + 4 T^{11} + T^{12} \)
$3$ \( ( 9 + T )^{12} \)
$5$ \( -77556013394120976384 - 13558574477500130496 T - 628882109042460368 T^{2} + 9113656193745088 T^{3} + 1086082114257678 T^{4} + 2297784312298 T^{5} - 666479026737 T^{6} - 2598739512 T^{7} + 182976233 T^{8} + 560658 T^{9} - 22423 T^{10} - 36 T^{11} + T^{12} \)
$7$ \( -\)\(84\!\cdots\!00\)\( + \)\(40\!\cdots\!16\)\( T + \)\(25\!\cdots\!52\)\( T^{2} - 2676285894510685888 T^{3} - 156332117239711603 T^{4} - 112804534512307 T^{5} + 32342979425819 T^{6} + 200959579356 T^{7} - 1808032128 T^{8} - 18755852 T^{9} - 97 T^{10} + 411 T^{11} + T^{12} \)
$11$ \( \)\(13\!\cdots\!80\)\( - \)\(46\!\cdots\!32\)\( T + \)\(50\!\cdots\!76\)\( T^{2} - \)\(17\!\cdots\!88\)\( T^{3} - \)\(41\!\cdots\!99\)\( T^{4} + 36112064195024788224 T^{5} - 22590287109692432 T^{6} - 223688732070032 T^{7} + 312450092690 T^{8} + 558834096 T^{9} - 995852 T^{10} - 492 T^{11} + T^{12} \)
$13$ \( \)\(62\!\cdots\!40\)\( - \)\(21\!\cdots\!84\)\( T - \)\(80\!\cdots\!74\)\( T^{2} + \)\(17\!\cdots\!14\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} - \)\(23\!\cdots\!68\)\( T^{5} - 180445752650819280 T^{6} + 1010358532701696 T^{7} + 942682963006 T^{8} - 1702867652 T^{9} - 1710774 T^{10} + 974 T^{11} + T^{12} \)
$17$ \( \)\(13\!\cdots\!28\)\( - \)\(22\!\cdots\!16\)\( T - \)\(59\!\cdots\!92\)\( T^{2} + \)\(71\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!81\)\( T^{4} - \)\(86\!\cdots\!69\)\( T^{5} - 90775932888388246567 T^{6} + 49706321890038466 T^{7} + 42847416678522 T^{8} - 13695715862 T^{9} - 10335579 T^{10} + 1463 T^{11} + T^{12} \)
$19$ \( -\)\(24\!\cdots\!00\)\( - \)\(37\!\cdots\!80\)\( T + \)\(25\!\cdots\!40\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!84\)\( T^{4} - \)\(34\!\cdots\!08\)\( T^{5} - 81825081430007783781 T^{6} + 177241756898904313 T^{7} + 49167848150432 T^{8} - 39361101803 T^{9} - 11746340 T^{10} + 3189 T^{11} + T^{12} \)
$23$ \( -\)\(32\!\cdots\!44\)\( + \)\(19\!\cdots\!84\)\( T - \)\(20\!\cdots\!60\)\( T^{2} - \)\(43\!\cdots\!16\)\( T^{3} + \)\(69\!\cdots\!48\)\( T^{4} + \)\(18\!\cdots\!32\)\( T^{5} - \)\(52\!\cdots\!15\)\( T^{6} + 451348397394397325 T^{7} + 792878495593102 T^{8} - 75756515747 T^{9} - 46660802 T^{10} + 2617 T^{11} + T^{12} \)
$29$ \( -\)\(18\!\cdots\!00\)\( + \)\(17\!\cdots\!80\)\( T - \)\(16\!\cdots\!20\)\( T^{2} - \)\(37\!\cdots\!80\)\( T^{3} + \)\(62\!\cdots\!58\)\( T^{4} + \)\(42\!\cdots\!20\)\( T^{5} - \)\(81\!\cdots\!97\)\( T^{6} + 2967444611344690307 T^{7} + 4595802582068772 T^{8} - 156631629145 T^{9} - 113987082 T^{10} + 1963 T^{11} + T^{12} \)
$31$ \( -\)\(10\!\cdots\!00\)\( - \)\(32\!\cdots\!96\)\( T + \)\(90\!\cdots\!88\)\( T^{2} + \)\(44\!\cdots\!84\)\( T^{3} - \)\(11\!\cdots\!06\)\( T^{4} - \)\(80\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} + 50938811924295475645 T^{7} + 2360874291519498 T^{8} - 1329395082931 T^{9} - 93963524 T^{10} + 11929 T^{11} + T^{12} \)
$37$ \( -\)\(16\!\cdots\!04\)\( - \)\(14\!\cdots\!28\)\( T + \)\(84\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!48\)\( T^{3} - \)\(99\!\cdots\!93\)\( T^{4} - \)\(81\!\cdots\!57\)\( T^{5} + \)\(41\!\cdots\!71\)\( T^{6} + \)\(44\!\cdots\!88\)\( T^{7} - 51584382691968206 T^{8} - 8221746998464 T^{9} - 73563915 T^{10} + 28105 T^{11} + T^{12} \)
$41$ \( -\)\(36\!\cdots\!16\)\( - \)\(91\!\cdots\!64\)\( T - \)\(12\!\cdots\!16\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(54\!\cdots\!93\)\( T^{4} - \)\(13\!\cdots\!31\)\( T^{5} - \)\(44\!\cdots\!23\)\( T^{6} + \)\(32\!\cdots\!42\)\( T^{7} + 89607672315652626 T^{8} - 2941921305982 T^{9} - 554741655 T^{10} + 7585 T^{11} + T^{12} \)
$43$ \( \)\(21\!\cdots\!84\)\( + \)\(19\!\cdots\!76\)\( T + \)\(48\!\cdots\!96\)\( T^{2} - \)\(21\!\cdots\!36\)\( T^{3} - \)\(20\!\cdots\!53\)\( T^{4} - \)\(12\!\cdots\!82\)\( T^{5} + \)\(24\!\cdots\!02\)\( T^{6} + \)\(26\!\cdots\!30\)\( T^{7} - 81611162836145932 T^{8} - 17055018223374 T^{9} - 219123078 T^{10} + 33146 T^{11} + T^{12} \)
$47$ \( \)\(12\!\cdots\!96\)\( + \)\(23\!\cdots\!36\)\( T - \)\(28\!\cdots\!20\)\( T^{2} - \)\(72\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!20\)\( T^{4} + \)\(35\!\cdots\!84\)\( T^{5} + \)\(84\!\cdots\!51\)\( T^{6} - \)\(50\!\cdots\!77\)\( T^{7} - 311495546050933804 T^{8} + 14567709122739 T^{9} + 2140527176 T^{10} + 79215 T^{11} + T^{12} \)
$53$ \( -\)\(30\!\cdots\!92\)\( - \)\(20\!\cdots\!48\)\( T - \)\(42\!\cdots\!68\)\( T^{2} + \)\(86\!\cdots\!56\)\( T^{3} + \)\(49\!\cdots\!22\)\( T^{4} - \)\(87\!\cdots\!54\)\( T^{5} - \)\(56\!\cdots\!53\)\( T^{6} + \)\(28\!\cdots\!32\)\( T^{7} + 1994844431748462313 T^{8} - 32939546483046 T^{9} - 2497095083 T^{10} + 12220 T^{11} + T^{12} \)
$59$ \( ( 3481 + T )^{12} \)
$61$ \( \)\(18\!\cdots\!60\)\( - \)\(83\!\cdots\!44\)\( T - \)\(19\!\cdots\!44\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(86\!\cdots\!74\)\( T^{4} - \)\(36\!\cdots\!08\)\( T^{5} - \)\(44\!\cdots\!31\)\( T^{6} + \)\(48\!\cdots\!91\)\( T^{7} + 7397420071469665098 T^{8} - 274114102514393 T^{9} - 4695009764 T^{10} + 54195 T^{11} + T^{12} \)
$67$ \( -\)\(60\!\cdots\!00\)\( + \)\(11\!\cdots\!00\)\( T - \)\(35\!\cdots\!00\)\( T^{2} - \)\(45\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!52\)\( T^{4} + \)\(69\!\cdots\!04\)\( T^{5} - \)\(31\!\cdots\!57\)\( T^{6} - \)\(51\!\cdots\!60\)\( T^{7} + 22861812640933314603 T^{8} + 182632312546636 T^{9} - 7807607315 T^{10} - 24224 T^{11} + T^{12} \)
$71$ \( \)\(75\!\cdots\!56\)\( - \)\(32\!\cdots\!04\)\( T - \)\(64\!\cdots\!58\)\( T^{2} + \)\(32\!\cdots\!62\)\( T^{3} + \)\(16\!\cdots\!49\)\( T^{4} - \)\(62\!\cdots\!00\)\( T^{5} - \)\(62\!\cdots\!56\)\( T^{6} - \)\(88\!\cdots\!36\)\( T^{7} + 7590319158627452730 T^{8} + 165082117040076 T^{9} - 3889081150 T^{10} - 60254 T^{11} + T^{12} \)
$73$ \( -\)\(88\!\cdots\!64\)\( - \)\(24\!\cdots\!40\)\( T - \)\(12\!\cdots\!76\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(83\!\cdots\!84\)\( T^{4} - \)\(19\!\cdots\!24\)\( T^{5} - \)\(12\!\cdots\!45\)\( T^{6} + \)\(88\!\cdots\!45\)\( T^{7} + 67789685252817722120 T^{8} - 182693272957471 T^{9} - 14125934880 T^{10} + 15385 T^{11} + T^{12} \)
$79$ \( -\)\(10\!\cdots\!40\)\( + \)\(30\!\cdots\!00\)\( T - \)\(23\!\cdots\!20\)\( T^{2} + \)\(11\!\cdots\!88\)\( T^{3} + \)\(29\!\cdots\!27\)\( T^{4} - \)\(17\!\cdots\!30\)\( T^{5} - \)\(13\!\cdots\!74\)\( T^{6} + \)\(53\!\cdots\!86\)\( T^{7} + \)\(27\!\cdots\!76\)\( T^{8} - 636015555441178 T^{9} - 26666314370 T^{10} + 27054 T^{11} + T^{12} \)
$83$ \( -\)\(15\!\cdots\!40\)\( - \)\(43\!\cdots\!08\)\( T + \)\(15\!\cdots\!84\)\( T^{2} + \)\(57\!\cdots\!00\)\( T^{3} - \)\(37\!\cdots\!71\)\( T^{4} - \)\(17\!\cdots\!65\)\( T^{5} + \)\(78\!\cdots\!81\)\( T^{6} + \)\(96\!\cdots\!34\)\( T^{7} + 40829717031227045778 T^{8} - 1907329006187386 T^{9} - 13986921103 T^{10} + 117595 T^{11} + T^{12} \)
$89$ \( \)\(33\!\cdots\!00\)\( + \)\(55\!\cdots\!00\)\( T - \)\(28\!\cdots\!00\)\( T^{2} - \)\(41\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!52\)\( T^{4} + \)\(11\!\cdots\!64\)\( T^{5} - \)\(15\!\cdots\!47\)\( T^{6} + \)\(26\!\cdots\!61\)\( T^{7} + \)\(37\!\cdots\!26\)\( T^{8} - 845321619923803 T^{9} - 34708267942 T^{10} + 36033 T^{11} + T^{12} \)
$97$ \( \)\(66\!\cdots\!40\)\( + \)\(16\!\cdots\!08\)\( T - \)\(38\!\cdots\!04\)\( T^{2} - \)\(16\!\cdots\!24\)\( T^{3} + \)\(66\!\cdots\!52\)\( T^{4} - \)\(11\!\cdots\!84\)\( T^{5} - \)\(17\!\cdots\!83\)\( T^{6} + \)\(38\!\cdots\!10\)\( T^{7} + \)\(22\!\cdots\!43\)\( T^{8} - 4635495228758024 T^{9} - 21031363277 T^{10} + 196914 T^{11} + T^{12} \)
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