Properties

Label 177.6.a.a
Level $177$
Weight $6$
Character orbit 177.a
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + 13849341 x^{3} - 23890558 x^{2} - 74443300 x - 14846072\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + 9 q^{3} + ( 14 - \beta_{1} + \beta_{2} ) q^{4} + ( -17 - \beta_{1} - \beta_{2} - \beta_{10} ) q^{5} + ( -9 + 9 \beta_{1} ) q^{6} + ( -31 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{7} + ( -59 + 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{8} + 81 q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + 9 q^{3} + ( 14 - \beta_{1} + \beta_{2} ) q^{4} + ( -17 - \beta_{1} - \beta_{2} - \beta_{10} ) q^{5} + ( -9 + 9 \beta_{1} ) q^{6} + ( -31 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{7} + ( -59 + 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{8} + 81 q^{9} + ( -19 - 41 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - \beta_{8} + 3 \beta_{9} + 5 \beta_{10} ) q^{10} + ( -50 - 24 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{6} + \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{11} + ( 126 - 9 \beta_{1} + 9 \beta_{2} ) q^{12} + ( -140 - 8 \beta_{1} + 3 \beta_{2} - 20 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} - 7 \beta_{9} + 5 \beta_{10} ) q^{13} + ( -139 - 16 \beta_{1} - 13 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 17 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} + 4 \beta_{10} ) q^{14} + ( -153 - 9 \beta_{1} - 9 \beta_{2} - 9 \beta_{10} ) q^{15} + ( -204 - 77 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 5 \beta_{6} - 11 \beta_{7} - 8 \beta_{8} + 7 \beta_{9} + 10 \beta_{10} ) q^{16} + ( -446 - 10 \beta_{1} - 18 \beta_{2} + 8 \beta_{3} + 14 \beta_{4} + 19 \beta_{5} - \beta_{6} + 22 \beta_{7} - 5 \beta_{8} + 18 \beta_{9} ) q^{17} + ( -81 + 81 \beta_{1} ) q^{18} + ( -344 + 9 \beta_{1} + 3 \beta_{2} + 32 \beta_{3} + 23 \beta_{4} + 9 \beta_{5} + 29 \beta_{6} + 7 \beta_{7} - \beta_{8} - 16 \beta_{9} - 6 \beta_{10} ) q^{19} + ( -1010 + 31 \beta_{1} - 48 \beta_{2} + 20 \beta_{3} - \beta_{4} + 11 \beta_{5} + 19 \beta_{6} + 8 \beta_{7} + 5 \beta_{8} - 15 \beta_{9} - 35 \beta_{10} ) q^{20} + ( -279 - 27 \beta_{1} + 18 \beta_{2} + 18 \beta_{3} - 9 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} - 9 \beta_{8} + 18 \beta_{10} ) q^{21} + ( -779 - 155 \beta_{1} + 18 \beta_{2} + 30 \beta_{3} - 22 \beta_{4} - 21 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} - 18 \beta_{8} + 29 \beta_{9} + 10 \beta_{10} ) q^{22} + ( -632 - 89 \beta_{1} + 3 \beta_{2} - 53 \beta_{3} + 8 \beta_{4} - 7 \beta_{5} - \beta_{6} - 45 \beta_{7} + 7 \beta_{8} + 3 \beta_{9} - 8 \beta_{10} ) q^{23} + ( -531 + 18 \beta_{1} - 45 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} + 27 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} - 9 \beta_{9} - 18 \beta_{10} ) q^{24} + ( 751 - 125 \beta_{1} + 53 \beta_{2} - 27 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 127 \beta_{6} + 18 \beta_{7} + 14 \beta_{8} + 2 \beta_{9} + 24 \beta_{10} ) q^{25} + ( -416 - 35 \beta_{1} - 12 \beta_{2} - 70 \beta_{3} - 37 \beta_{4} - 17 \beta_{5} - 39 \beta_{6} - 23 \beta_{7} - 4 \beta_{8} - 68 \beta_{9} - 53 \beta_{10} ) q^{26} + 729 q^{27} + ( 504 - 265 \beta_{1} + 73 \beta_{2} + 14 \beta_{3} - 47 \beta_{4} - 43 \beta_{5} - 43 \beta_{6} - 64 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 64 \beta_{10} ) q^{28} + ( -1395 - 94 \beta_{1} + 72 \beta_{2} - 85 \beta_{3} - 26 \beta_{4} + 17 \beta_{5} + 25 \beta_{6} - 19 \beta_{7} - 13 \beta_{8} + 33 \beta_{9} - 29 \beta_{10} ) q^{29} + ( -171 - 369 \beta_{1} + 9 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} - 27 \beta_{5} + 63 \beta_{6} + 18 \beta_{7} - 9 \beta_{8} + 27 \beta_{9} + 45 \beta_{10} ) q^{30} + ( -414 - 190 \beta_{1} + 17 \beta_{2} + 74 \beta_{3} + 42 \beta_{4} + 19 \beta_{5} + 122 \beta_{6} + 80 \beta_{7} + 29 \beta_{8} + 3 \beta_{9} - 71 \beta_{10} ) q^{31} + ( -1206 - 331 \beta_{1} - 61 \beta_{2} + 124 \beta_{3} + 12 \beta_{4} - 62 \beta_{5} + 44 \beta_{6} + 44 \beta_{7} + 10 \beta_{8} - 12 \beta_{9} - 24 \beta_{10} ) q^{32} + ( -450 - 216 \beta_{1} - 54 \beta_{2} + 27 \beta_{3} - 45 \beta_{4} + 27 \beta_{6} + 9 \beta_{7} + 36 \beta_{8} + 18 \beta_{9} + 18 \beta_{10} ) q^{33} + ( 1079 - 597 \beta_{1} + 80 \beta_{2} + 177 \beta_{3} + 26 \beta_{4} + 67 \beta_{5} + 135 \beta_{6} - 36 \beta_{7} + 2 \beta_{8} + 27 \beta_{9} - 46 \beta_{10} ) q^{34} + ( -2118 - 50 \beta_{1} - 118 \beta_{2} + 40 \beta_{3} + 50 \beta_{4} + 33 \beta_{5} + 240 \beta_{6} + 67 \beta_{7} + 9 \beta_{8} - 25 \beta_{9} - 48 \beta_{10} ) q^{35} + ( 1134 - 81 \beta_{1} + 81 \beta_{2} ) q^{36} + ( 827 - 184 \beta_{1} + 42 \beta_{2} - 99 \beta_{3} - 20 \beta_{4} - 11 \beta_{5} - 60 \beta_{6} + 90 \beta_{7} - 59 \beta_{8} - 74 \beta_{9} + 43 \beta_{10} ) q^{37} + ( 1290 - 77 \beta_{1} + 15 \beta_{2} - 165 \beta_{3} + 90 \beta_{4} + 67 \beta_{5} - 191 \beta_{6} - 19 \beta_{7} + 5 \beta_{8} + 150 \beta_{9} - 2 \beta_{10} ) q^{38} + ( -1260 - 72 \beta_{1} + 27 \beta_{2} - 180 \beta_{3} - 36 \beta_{4} - 36 \beta_{5} - 9 \beta_{6} - 36 \beta_{7} - 63 \beta_{9} + 45 \beta_{10} ) q^{39} + ( 3973 - 550 \beta_{1} + 246 \beta_{2} - 158 \beta_{3} - 52 \beta_{4} - 58 \beta_{5} - 218 \beta_{6} - 39 \beta_{7} - 50 \beta_{8} + 164 \beta_{9} + 99 \beta_{10} ) q^{40} + ( -802 + 311 \beta_{1} + 23 \beta_{2} - 8 \beta_{3} + 59 \beta_{4} + 169 \beta_{5} - 90 \beta_{6} + 56 \beta_{7} - 63 \beta_{8} + 87 \beta_{9} + 66 \beta_{10} ) q^{41} + ( -1251 - 144 \beta_{1} - 117 \beta_{2} + 63 \beta_{3} - 18 \beta_{4} - 18 \beta_{5} - 153 \beta_{6} - 18 \beta_{7} + 45 \beta_{8} - 45 \beta_{9} + 36 \beta_{10} ) q^{42} + ( -2076 + 804 \beta_{1} + 84 \beta_{2} + 72 \beta_{3} - 70 \beta_{4} + 82 \beta_{5} + 11 \beta_{6} - 283 \beta_{7} + 102 \beta_{8} - 177 \beta_{9} - 14 \beta_{10} ) q^{43} + ( -4908 - 289 \beta_{1} - 191 \beta_{2} + 54 \beta_{3} + 60 \beta_{4} - 103 \beta_{5} + 26 \beta_{6} + 101 \beta_{7} - 70 \beta_{8} - 69 \beta_{9} + 22 \beta_{10} ) q^{44} + ( -1377 - 81 \beta_{1} - 81 \beta_{2} - 81 \beta_{10} ) q^{45} + ( -3824 - 863 \beta_{1} - 371 \beta_{2} + 103 \beta_{3} + 254 \beta_{4} + 147 \beta_{5} + 293 \beta_{6} + 179 \beta_{7} + 45 \beta_{8} - 174 \beta_{9} - 248 \beta_{10} ) q^{46} + ( -4382 + 252 \beta_{1} + 2 \beta_{2} + 207 \beta_{3} - 113 \beta_{4} - 273 \beta_{5} - 152 \beta_{6} + 135 \beta_{7} + 49 \beta_{8} + 130 \beta_{9} - 18 \beta_{10} ) q^{47} + ( -1836 - 693 \beta_{1} + 63 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} - 18 \beta_{5} + 45 \beta_{6} - 99 \beta_{7} - 72 \beta_{8} + 63 \beta_{9} + 90 \beta_{10} ) q^{48} + ( -1572 + 1066 \beta_{1} + 85 \beta_{2} - 362 \beta_{3} - 23 \beta_{4} + 129 \beta_{5} - 80 \beta_{6} - 31 \beta_{7} + 71 \beta_{8} - 63 \beta_{9} - 120 \beta_{10} ) q^{49} + ( -9863 + 2268 \beta_{1} + 157 \beta_{2} + 418 \beta_{3} - 146 \beta_{4} + 87 \beta_{5} - 46 \beta_{6} - 506 \beta_{7} + 13 \beta_{8} - 444 \beta_{9} + 170 \beta_{10} ) q^{50} + ( -4014 - 90 \beta_{1} - 162 \beta_{2} + 72 \beta_{3} + 126 \beta_{4} + 171 \beta_{5} - 9 \beta_{6} + 198 \beta_{7} - 45 \beta_{8} + 162 \beta_{9} ) q^{51} + ( 187 - 463 \beta_{1} + 113 \beta_{2} - 218 \beta_{3} - 139 \beta_{4} - 269 \beta_{5} - 155 \beta_{6} + 43 \beta_{7} - 86 \beta_{8} + 158 \beta_{9} + 327 \beta_{10} ) q^{52} + ( -8133 + 2217 \beta_{1} + 441 \beta_{2} - 54 \beta_{3} - 224 \beta_{4} - 420 \beta_{5} - 154 \beta_{6} - 116 \beta_{7} + 16 \beta_{8} + 112 \beta_{9} + 367 \beta_{10} ) q^{53} + ( -729 + 729 \beta_{1} ) q^{54} + ( -954 + 1827 \beta_{1} + 539 \beta_{2} - 121 \beta_{3} - 116 \beta_{4} + 72 \beta_{5} - 293 \beta_{6} - 334 \beta_{7} - 82 \beta_{8} + 68 \beta_{9} + 176 \beta_{10} ) q^{55} + ( -10532 + 1587 \beta_{1} - 155 \beta_{2} + 50 \beta_{3} - 42 \beta_{4} - 9 \beta_{5} + 100 \beta_{6} + 123 \beta_{7} - 76 \beta_{8} - 75 \beta_{9} - 66 \beta_{10} ) q^{56} + ( -3096 + 81 \beta_{1} + 27 \beta_{2} + 288 \beta_{3} + 207 \beta_{4} + 81 \beta_{5} + 261 \beta_{6} + 63 \beta_{7} - 9 \beta_{8} - 144 \beta_{9} - 54 \beta_{10} ) q^{57} + ( -5519 - 352 \beta_{1} - 790 \beta_{2} - 271 \beta_{3} + 191 \beta_{4} + 352 \beta_{5} + 528 \beta_{6} + 537 \beta_{7} + 124 \beta_{8} - 179 \beta_{9} - 499 \beta_{10} ) q^{58} + 3481 q^{59} + ( -9090 + 279 \beta_{1} - 432 \beta_{2} + 180 \beta_{3} - 9 \beta_{4} + 99 \beta_{5} + 171 \beta_{6} + 72 \beta_{7} + 45 \beta_{8} - 135 \beta_{9} - 315 \beta_{10} ) q^{60} + ( -7752 + 882 \beta_{1} - 53 \beta_{2} + 259 \beta_{3} + 23 \beta_{4} - 331 \beta_{5} - 188 \beta_{6} - 28 \beta_{7} + 217 \beta_{8} - 2 \beta_{9} + 107 \beta_{10} ) q^{61} + ( -6961 + 790 \beta_{1} - 298 \beta_{2} - 449 \beta_{3} + 257 \beta_{4} + 340 \beta_{5} + 296 \beta_{6} + 165 \beta_{7} - 76 \beta_{8} + 571 \beta_{9} - 251 \beta_{10} ) q^{62} + ( -2511 - 243 \beta_{1} + 162 \beta_{2} + 162 \beta_{3} - 81 \beta_{5} - 81 \beta_{6} - 81 \beta_{7} - 81 \beta_{8} + 162 \beta_{10} ) q^{63} + ( -4497 - 192 \beta_{1} - 301 \beta_{2} + 46 \beta_{3} - 195 \beta_{4} - 387 \beta_{5} - 153 \beta_{6} + 165 \beta_{7} + 131 \beta_{8} + 219 \beta_{9} + 174 \beta_{10} ) q^{64} + ( -16453 + 3229 \beta_{1} + 76 \beta_{2} + 412 \beta_{3} + 6 \beta_{4} + 130 \beta_{5} + 556 \beta_{6} - 281 \beta_{7} - 46 \beta_{8} - 84 \beta_{9} + 182 \beta_{10} ) q^{65} + ( -7011 - 1395 \beta_{1} + 162 \beta_{2} + 270 \beta_{3} - 198 \beta_{4} - 189 \beta_{5} - 54 \beta_{6} + 27 \beta_{7} - 162 \beta_{8} + 261 \beta_{9} + 90 \beta_{10} ) q^{66} + ( -14956 + 193 \beta_{1} + 299 \beta_{2} + 581 \beta_{3} - 282 \beta_{4} + 70 \beta_{5} + 131 \beta_{6} - 62 \beta_{7} - 364 \beta_{8} + 264 \beta_{9} + 328 \beta_{10} ) q^{67} + ( -13831 + 2328 \beta_{1} - 783 \beta_{2} - 466 \beta_{3} + 285 \beta_{4} + 18 \beta_{5} - 221 \beta_{6} + 182 \beta_{7} + 325 \beta_{8} + 294 \beta_{9} - 70 \beta_{10} ) q^{68} + ( -5688 - 801 \beta_{1} + 27 \beta_{2} - 477 \beta_{3} + 72 \beta_{4} - 63 \beta_{5} - 9 \beta_{6} - 405 \beta_{7} + 63 \beta_{8} + 27 \beta_{9} - 72 \beta_{10} ) q^{69} + ( 7830 - 3424 \beta_{1} + 74 \beta_{2} - 783 \beta_{3} + 176 \beta_{4} + 30 \beta_{5} + 63 \beta_{6} + 293 \beta_{7} - 178 \beta_{8} + 814 \beta_{9} - 538 \beta_{10} ) q^{70} + ( -17230 + 2462 \beta_{1} + 229 \beta_{2} + 797 \beta_{3} - 297 \beta_{4} - 318 \beta_{5} + 227 \beta_{6} - 242 \beta_{7} + 204 \beta_{8} - 330 \beta_{9} + 133 \beta_{10} ) q^{71} + ( -4779 + 162 \beta_{1} - 405 \beta_{2} - 162 \beta_{3} + 81 \beta_{4} + 243 \beta_{5} - 81 \beta_{6} + 81 \beta_{7} + 81 \beta_{8} - 81 \beta_{9} - 162 \beta_{10} ) q^{72} + ( -3417 - 914 \beta_{1} - 709 \beta_{2} - 937 \beta_{3} + 442 \beta_{4} - 53 \beta_{5} - 45 \beta_{6} + 392 \beta_{7} - 365 \beta_{8} - 449 \beta_{9} - 342 \beta_{10} ) q^{73} + ( -11490 + 2807 \beta_{1} + 316 \beta_{2} - 1041 \beta_{3} - 915 \beta_{4} - 437 \beta_{5} - 818 \beta_{6} - 776 \beta_{7} - 24 \beta_{8} - 512 \beta_{9} + 125 \beta_{10} ) q^{74} + ( 6759 - 1125 \beta_{1} + 477 \beta_{2} - 243 \beta_{3} + 36 \beta_{4} - 54 \beta_{5} - 1143 \beta_{6} + 162 \beta_{7} + 126 \beta_{8} + 18 \beta_{9} + 216 \beta_{10} ) q^{75} + ( 4742 + 1428 \beta_{1} - 614 \beta_{2} + 1052 \beta_{3} + 145 \beta_{4} + 776 \beta_{5} + 1071 \beta_{6} - 223 \beta_{7} + 296 \beta_{8} - 567 \beta_{9} - 698 \beta_{10} ) q^{76} + ( -16850 + 2529 \beta_{1} + 393 \beta_{2} - 23 \beta_{3} + 590 \beta_{4} + 421 \beta_{5} + 474 \beta_{6} + 740 \beta_{7} - 283 \beta_{8} + 324 \beta_{9} - 808 \beta_{10} ) q^{77} + ( -3744 - 315 \beta_{1} - 108 \beta_{2} - 630 \beta_{3} - 333 \beta_{4} - 153 \beta_{5} - 351 \beta_{6} - 207 \beta_{7} - 36 \beta_{8} - 612 \beta_{9} - 477 \beta_{10} ) q^{78} + ( -24411 - 1578 \beta_{1} - 603 \beta_{2} + 890 \beta_{3} + 183 \beta_{4} - 760 \beta_{5} + 502 \beta_{6} + 20 \beta_{7} + 172 \beta_{8} - 170 \beta_{9} + 68 \beta_{10} ) q^{79} + ( -5272 + 6119 \beta_{1} - 550 \beta_{2} + 914 \beta_{3} + 92 \beta_{4} + 325 \beta_{5} + 692 \beta_{6} - 3 \beta_{7} + 337 \beta_{8} - 1100 \beta_{9} + 223 \beta_{10} ) q^{80} + 6561 q^{81} + ( 16073 + 686 \beta_{1} + 539 \beta_{2} + 903 \beta_{3} + 176 \beta_{4} + 746 \beta_{5} - 87 \beta_{6} - 144 \beta_{7} + 239 \beta_{8} - 289 \beta_{9} - 374 \beta_{10} ) q^{82} + ( -25698 + 4445 \beta_{1} - 167 \beta_{2} - 496 \beta_{3} + 83 \beta_{4} + 601 \beta_{5} - 658 \beta_{6} + 556 \beta_{7} + 125 \beta_{8} - 189 \beta_{9} - 440 \beta_{10} ) q^{83} + ( 4536 - 2385 \beta_{1} + 657 \beta_{2} + 126 \beta_{3} - 423 \beta_{4} - 387 \beta_{5} - 387 \beta_{6} - 576 \beta_{7} + 36 \beta_{8} + 18 \beta_{9} + 576 \beta_{10} ) q^{84} + ( -381 + 1933 \beta_{1} + 650 \beta_{2} - 387 \beta_{3} - 363 \beta_{4} - 41 \beta_{5} - 1006 \beta_{6} - 244 \beta_{7} - 85 \beta_{8} + 48 \beta_{9} + 410 \beta_{10} ) q^{85} + ( 31159 - 945 \beta_{1} + 1026 \beta_{2} - 1064 \beta_{3} + 660 \beta_{4} + 215 \beta_{5} - 2422 \beta_{6} + 601 \beta_{7} - 84 \beta_{8} + 765 \beta_{9} + 902 \beta_{10} ) q^{86} + ( -12555 - 846 \beta_{1} + 648 \beta_{2} - 765 \beta_{3} - 234 \beta_{4} + 153 \beta_{5} + 225 \beta_{6} - 171 \beta_{7} - 117 \beta_{8} + 297 \beta_{9} - 261 \beta_{10} ) q^{87} + ( 24607 - 3530 \beta_{1} - 41 \beta_{2} - 1344 \beta_{3} - 59 \beta_{4} - 516 \beta_{5} - 79 \beta_{6} - 1038 \beta_{7} + 415 \beta_{8} - 894 \beta_{9} + 284 \beta_{10} ) q^{88} + ( -9660 + 2220 \beta_{1} + 14 \beta_{2} - 4 \beta_{3} + 36 \beta_{4} + 983 \beta_{5} + 1084 \beta_{6} + 677 \beta_{7} + 33 \beta_{8} - 119 \beta_{9} - 830 \beta_{10} ) q^{89} + ( -1539 - 3321 \beta_{1} + 81 \beta_{2} - 162 \beta_{3} + 81 \beta_{4} - 243 \beta_{5} + 567 \beta_{6} + 162 \beta_{7} - 81 \beta_{8} + 243 \beta_{9} + 405 \beta_{10} ) q^{90} + ( 3427 - 1999 \beta_{1} - 64 \beta_{2} + 1203 \beta_{3} + 1137 \beta_{4} + 901 \beta_{5} - 415 \beta_{6} + 627 \beta_{7} + 41 \beta_{8} + 1337 \beta_{9} - 498 \beta_{10} ) q^{91} + ( -752 - 4194 \beta_{1} + 952 \beta_{2} - 160 \beta_{3} + 431 \beta_{4} + 192 \beta_{5} + 61 \beta_{6} + 829 \beta_{7} - 784 \beta_{8} + 1671 \beta_{9} + 572 \beta_{10} ) q^{92} + ( -3726 - 1710 \beta_{1} + 153 \beta_{2} + 666 \beta_{3} + 378 \beta_{4} + 171 \beta_{5} + 1098 \beta_{6} + 720 \beta_{7} + 261 \beta_{8} + 27 \beta_{9} - 639 \beta_{10} ) q^{93} + ( 13190 - 6468 \beta_{1} + 364 \beta_{2} + 1553 \beta_{3} - 1004 \beta_{4} - 1222 \beta_{5} + 1587 \beta_{6} - 565 \beta_{7} - 214 \beta_{8} - 50 \beta_{9} + 1320 \beta_{10} ) q^{94} + ( -13131 - 685 \beta_{1} + 2158 \beta_{2} - 198 \beta_{3} - 578 \beta_{4} - 1049 \beta_{5} + 664 \beta_{6} - 1170 \beta_{7} - 679 \beta_{8} + 557 \beta_{9} + 1542 \beta_{10} ) q^{95} + ( -10854 - 2979 \beta_{1} - 549 \beta_{2} + 1116 \beta_{3} + 108 \beta_{4} - 558 \beta_{5} + 396 \beta_{6} + 396 \beta_{7} + 90 \beta_{8} - 108 \beta_{9} - 216 \beta_{10} ) q^{96} + ( -2523 - 819 \beta_{1} + 688 \beta_{2} + 529 \beta_{3} - 1265 \beta_{4} - 672 \beta_{5} - 264 \beta_{6} - 1281 \beta_{7} - 688 \beta_{8} + 385 \beta_{9} + 1106 \beta_{10} ) q^{97} + ( 40474 + 2386 \beta_{1} + 1272 \beta_{2} - 1143 \beta_{3} + 336 \beta_{4} + 887 \beta_{5} + 577 \beta_{6} + 110 \beta_{7} - 112 \beta_{8} - 653 \beta_{9} - 736 \beta_{10} ) q^{98} + ( -4050 - 1944 \beta_{1} - 486 \beta_{2} + 243 \beta_{3} - 405 \beta_{4} + 243 \beta_{6} + 81 \beta_{7} + 324 \beta_{8} + 162 \beta_{9} + 162 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 6q^{2} + 99q^{3} + 150q^{4} - 192q^{5} - 54q^{6} - 371q^{7} - 621q^{8} + 891q^{9} + O(q^{10}) \) \( 11q - 6q^{2} + 99q^{3} + 150q^{4} - 192q^{5} - 54q^{6} - 371q^{7} - 621q^{8} + 891q^{9} - 399q^{10} - 698q^{11} + 1350q^{12} - 1556q^{13} - 1679q^{14} - 1728q^{15} - 2662q^{16} - 4793q^{17} - 486q^{18} - 3753q^{19} - 11023q^{20} - 3339q^{21} - 9534q^{22} - 7323q^{23} - 5589q^{24} + 7867q^{25} - 4844q^{26} + 8019q^{27} + 3650q^{28} - 15467q^{29} - 3591q^{30} - 5151q^{31} - 15368q^{32} - 6282q^{33} + 8452q^{34} - 23285q^{35} + 12150q^{36} + 8623q^{37} + 15205q^{38} - 14004q^{39} + 41530q^{40} - 6369q^{41} - 15111q^{42} - 20506q^{43} - 55632q^{44} - 15552q^{45} - 45191q^{46} - 47899q^{47} - 23958q^{48} - 10322q^{49} - 102147q^{50} - 43137q^{51} - 292q^{52} - 80048q^{53} - 4374q^{54} - 2114q^{55} - 108126q^{56} - 33777q^{57} - 58294q^{58} + 38291q^{59} - 99207q^{60} - 82527q^{61} - 67438q^{62} - 30051q^{63} - 51411q^{64} - 167646q^{65} - 85806q^{66} - 166976q^{67} - 136533q^{68} - 65907q^{69} + 76140q^{70} - 183560q^{71} - 50301q^{72} - 36809q^{73} - 116686q^{74} + 70803q^{75} + 55580q^{76} - 164885q^{77} - 43596q^{78} - 281518q^{79} - 32683q^{80} + 72171q^{81} + 178815q^{82} - 254691q^{83} + 32850q^{84} + 4763q^{85} + 349324q^{86} - 139203q^{87} + 251285q^{88} - 89687q^{89} - 32319q^{90} + 34897q^{91} - 20240q^{92} - 46359q^{93} + 96548q^{94} - 155113q^{95} - 138312q^{96} - 45828q^{97} + 465864q^{98} - 56538q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + 13849341 x^{3} - 23890558 x^{2} - 74443300 x - 14846072\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 45 \)
\(\beta_{3}\)\(=\)\((\)\(2778460829 \nu^{10} - 187297311739 \nu^{9} - 3169078538 \nu^{8} + 37422999014637 \nu^{7} - 68202867113046 \nu^{6} - 2430423048764513 \nu^{5} + 5261841092179714 \nu^{4} + 54695618337973207 \nu^{3} - 97185495955214979 \nu^{2} - 277273564078787976 \nu - 46744666460014620\)\()/ 4205126743692352 \)
\(\beta_{4}\)\(=\)\((\)\(-14193213478 \nu^{10} + 162977595845 \nu^{9} + 2207753333560 \nu^{8} - 33222926157014 \nu^{7} - 63472531727643 \nu^{6} + 2276300832143201 \nu^{5} - 2949746727927086 \nu^{4} - 58563201370458676 \nu^{3} + 96919307362486299 \nu^{2} + 436511974211010068 \nu + 49137760341652876\)\()/ 4205126743692352 \)
\(\beta_{5}\)\(=\)\((\)\(-47676035497 \nu^{10} + 351855671964 \nu^{9} + 9131611480310 \nu^{8} - 67011465294593 \nu^{7} - 557963364451955 \nu^{6} + 4067068942117866 \nu^{5} + 11268268678038488 \nu^{4} - 79788139597326609 \nu^{3} - 46718511586843450 \nu^{2} + 208390806737984508 \nu + 218108849645718264\)\()/ 4205126743692352 \)
\(\beta_{6}\)\(=\)\((\)\(-28858453840 \nu^{10} + 377512348731 \nu^{9} + 4935210416071 \nu^{8} - 73166083554387 \nu^{7} - 225728352893616 \nu^{6} + 4579736170630205 \nu^{5} + 164851990846319 \nu^{4} - 97755073144838957 \nu^{3} + 91261763313963386 \nu^{2} + 444545767039652472 \nu + 82872747608509216\)\()/ 2102563371846176 \)
\(\beta_{7}\)\(=\)\((\)\(-15568094860 \nu^{10} + 166942662163 \nu^{9} + 2866062504113 \nu^{8} - 32727493696379 \nu^{7} - 159340236044106 \nu^{6} + 2082506296044915 \nu^{5} + 2217875051857707 \nu^{4} - 45698521971969343 \nu^{3} + 14790555254723370 \nu^{2} + 222471726912464284 \nu + 33416666008470872\)\()/ 1051281685923088 \)
\(\beta_{8}\)\(=\)\((\)\(-70699645381 \nu^{10} + 724092912446 \nu^{9} + 12116309216206 \nu^{8} - 138996549095733 \nu^{7} - 545370177084889 \nu^{6} + 8505381357970468 \nu^{5} - 1080392737069084 \nu^{4} - 168561666697995505 \nu^{3} + 290581103124887564 \nu^{2} + 457773222420002388 \nu - 537772628306245264\)\()/ 4205126743692352 \)
\(\beta_{9}\)\(=\)\((\)\(76278029747 \nu^{10} - 810327211110 \nu^{9} - 13430535220258 \nu^{8} + 158556548973571 \nu^{7} + 677824163285755 \nu^{6} - 10099241323288128 \nu^{5} - 5726984781855596 \nu^{4} + 223364021183059031 \nu^{3} - 139765268110871872 \nu^{2} - 1114371303608622340 \nu - 289556126907450288\)\()/ 4205126743692352 \)
\(\beta_{10}\)\(=\)\((\)\(-314311388515 \nu^{10} + 3040305309961 \nu^{9} + 56749599552340 \nu^{8} - 591234225816113 \nu^{7} - 3010055469617252 \nu^{6} + 37113529280203671 \nu^{5} + 33519765392353652 \nu^{4} - 790733613061331263 \nu^{3} + 449709576651727347 \nu^{2} + 3475120040752796400 \nu + 949786364217219244\)\()/ 8410253487384704 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 45\)
\(\nu^{3}\)\(=\)\(-2 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 66 \beta_{1} + 13\)
\(\nu^{4}\)\(=\)\(2 \beta_{10} + 3 \beta_{9} - 4 \beta_{8} - 7 \beta_{7} + \beta_{6} + 10 \beta_{5} + 5 \beta_{4} - 10 \beta_{3} + 89 \beta_{2} + 89 \beta_{1} + 2969\)
\(\nu^{5}\)\(=\)\(-250 \beta_{10} - 115 \beta_{9} + 108 \beta_{8} + 127 \beta_{7} - 69 \beta_{6} + 342 \beta_{5} + 155 \beta_{4} - 162 \beta_{3} - 226 \beta_{2} + 4835 \beta_{1} + 1288\)
\(\nu^{6}\)\(=\)\(204 \beta_{10} + 584 \beta_{9} - 421 \beta_{8} - 708 \beta_{7} + 198 \beta_{6} + 1255 \beta_{5} + 840 \beta_{4} - 1136 \beta_{3} + 7289 \beta_{2} + 7258 \beta_{1} + 218504\)
\(\nu^{7}\)\(=\)\(-25176 \beta_{10} - 10300 \beta_{9} + 9842 \beta_{8} + 12148 \beta_{7} - 1988 \beta_{6} + 32778 \beta_{5} + 17116 \beta_{4} - 9956 \beta_{3} - 21282 \beta_{2} + 372533 \beta_{1} + 116322\)
\(\nu^{8}\)\(=\)\(14096 \beta_{10} + 78956 \beta_{9} - 36456 \beta_{8} - 50500 \beta_{7} + 27296 \beta_{6} + 130176 \beta_{5} + 102748 \beta_{4} - 104188 \beta_{3} + 585597 \beta_{2} + 582289 \beta_{1} + 16953209\)
\(\nu^{9}\)\(=\)\(-2379226 \beta_{10} - 853361 \beta_{9} + 854257 \beta_{8} + 1090113 \beta_{7} + 210331 \beta_{6} + 2997971 \beta_{5} + 1697321 \beta_{4} - 494934 \beta_{3} - 1889958 \beta_{2} + 29516358 \beta_{1} + 10355213\)
\(\nu^{10}\)\(=\)\(632954 \beta_{10} + 9039827 \beta_{9} - 2990368 \beta_{8} - 2910255 \beta_{7} + 3281173 \beta_{6} + 12729134 \beta_{5} + 11048701 \beta_{4} - 9155726 \beta_{3} + 46945409 \beta_{2} + 47232617 \beta_{1} + 1353173089\)

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
−9.21944
−8.44473
−5.70379
−5.62527
−1.75662
−0.216241
4.20625
5.75393
7.91273
8.66878
9.42442
−10.2194 9.00000 72.4370 −99.2561 −91.9750 109.985 −413.244 81.0000 1014.34
1.2 −9.44473 9.00000 57.2030 13.7903 −85.0026 67.4858 −238.036 81.0000 −130.245
1.3 −6.70379 9.00000 12.9408 105.016 −60.3341 −129.262 127.769 81.0000 −704.003
1.4 −6.62527 9.00000 11.8942 −85.2025 −59.6275 −103.010 133.206 81.0000 564.490
1.5 −2.75662 9.00000 −24.4011 5.39522 −24.8096 −153.490 155.476 81.0000 −14.8726
1.6 −1.21624 9.00000 −30.5208 0.914506 −10.9462 61.7584 76.0403 81.0000 −1.11226
1.7 3.20625 9.00000 −21.7200 26.7258 28.8562 39.0273 −172.240 81.0000 85.6897
1.8 4.75393 9.00000 −9.40015 −8.06966 42.7854 −28.8581 −196.813 81.0000 −38.3626
1.9 6.91273 9.00000 15.7858 −11.1343 62.2145 −193.283 −112.084 81.0000 −76.9686
1.10 7.66878 9.00000 26.8102 −109.801 69.0190 156.301 −39.7996 81.0000 −842.036
1.11 8.42442 9.00000 38.9709 −30.3784 75.8198 −197.653 58.7254 81.0000 −255.921
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.a 11
3.b odd 2 1 531.6.a.b 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.a.a 11 1.a even 1 1 trivial
531.6.a.b 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{11} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(177))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -97836992 - 75608576 T + 22980000 T^{2} + 14355856 T^{3} - 2119008 T^{4} - 822892 T^{5} + 75693 T^{6} + 20480 T^{7} - 1135 T^{8} - 233 T^{9} + 6 T^{10} + T^{11} \)
$3$ \( ( -9 + T )^{11} \)
$5$ \( -484006946487552 + 542436315738000 T + 1363030768540 T^{2} - 17016820228014 T^{3} - 348551178600 T^{4} + 99303796341 T^{5} + 1891082980 T^{6} - 100883639 T^{7} - 2285560 T^{8} - 2689 T^{9} + 192 T^{10} + T^{11} \)
$7$ \( -\)\(63\!\cdots\!60\)\( + 20657304973254984912 T + 8508128979608517880 T^{2} - 66186371263557627 T^{3} - 2447122379954507 T^{4} + 13328357459963 T^{5} + 317046263196 T^{6} - 581873792 T^{7} - 18507748 T^{8} - 18457 T^{9} + 371 T^{10} + T^{11} \)
$11$ \( -\)\(75\!\cdots\!92\)\( - \)\(22\!\cdots\!68\)\( T - \)\(25\!\cdots\!96\)\( T^{2} - \)\(13\!\cdots\!65\)\( T^{3} - 24522103760441610218 T^{4} + 62000825694162270 T^{5} + 299982506353750 T^{6} + 141919607268 T^{7} - 841337086 T^{8} - 929402 T^{9} + 698 T^{10} + T^{11} \)
$13$ \( \)\(86\!\cdots\!44\)\( + \)\(14\!\cdots\!74\)\( T + \)\(37\!\cdots\!96\)\( T^{2} - \)\(39\!\cdots\!23\)\( T^{3} - \)\(17\!\cdots\!68\)\( T^{4} + 223957142750548096 T^{5} + 1277921120160796 T^{6} - 32735708322 T^{7} - 2853065200 T^{8} - 1344310 T^{9} + 1556 T^{10} + T^{11} \)
$17$ \( -\)\(45\!\cdots\!24\)\( + \)\(52\!\cdots\!28\)\( T + \)\(74\!\cdots\!26\)\( T^{2} - \)\(64\!\cdots\!99\)\( T^{3} - \)\(33\!\cdots\!11\)\( T^{4} + 19626195559606900301 T^{5} + 12287197761862942 T^{6} - 17528746594058 T^{7} - 14101611276 T^{8} + 3225537 T^{9} + 4793 T^{10} + T^{11} \)
$19$ \( -\)\(93\!\cdots\!28\)\( + \)\(73\!\cdots\!20\)\( T + \)\(16\!\cdots\!40\)\( T^{2} - \)\(43\!\cdots\!60\)\( T^{3} - \)\(28\!\cdots\!64\)\( T^{4} + 5616736068481108803 T^{5} + 167882150916866017 T^{6} + 19866094556832 T^{7} - 42005950547 T^{8} - 8676492 T^{9} + 3753 T^{10} + T^{11} \)
$23$ \( \)\(86\!\cdots\!84\)\( + \)\(22\!\cdots\!20\)\( T + \)\(73\!\cdots\!64\)\( T^{2} - \)\(18\!\cdots\!16\)\( T^{3} - \)\(10\!\cdots\!28\)\( T^{4} + \)\(29\!\cdots\!39\)\( T^{5} + 2674608093378949671 T^{6} - 26171144697072 T^{7} - 240361868317 T^{8} - 20821304 T^{9} + 7323 T^{10} + T^{11} \)
$29$ \( -\)\(27\!\cdots\!08\)\( + \)\(36\!\cdots\!52\)\( T + \)\(46\!\cdots\!88\)\( T^{2} - \)\(29\!\cdots\!62\)\( T^{3} - \)\(17\!\cdots\!66\)\( T^{4} + \)\(49\!\cdots\!55\)\( T^{5} + 23779402378229051797 T^{6} - 1897715249255520 T^{7} - 1094095904967 T^{8} - 17015662 T^{9} + 15467 T^{10} + T^{11} \)
$31$ \( -\)\(75\!\cdots\!48\)\( - \)\(38\!\cdots\!20\)\( T + \)\(14\!\cdots\!32\)\( T^{2} + \)\(16\!\cdots\!14\)\( T^{3} - \)\(63\!\cdots\!08\)\( T^{4} - \)\(16\!\cdots\!59\)\( T^{5} + 38444205878273924771 T^{6} + 9378504825337746 T^{7} - 791188468597 T^{8} - 171167908 T^{9} + 5151 T^{10} + T^{11} \)
$37$ \( -\)\(16\!\cdots\!40\)\( + \)\(47\!\cdots\!26\)\( T - \)\(36\!\cdots\!02\)\( T^{2} + \)\(34\!\cdots\!57\)\( T^{3} + \)\(22\!\cdots\!79\)\( T^{4} - \)\(29\!\cdots\!17\)\( T^{5} - \)\(45\!\cdots\!60\)\( T^{6} + 60123386387949638 T^{7} + 3408962039038 T^{8} - 428737193 T^{9} - 8623 T^{10} + T^{11} \)
$41$ \( \)\(38\!\cdots\!56\)\( + \)\(85\!\cdots\!92\)\( T + \)\(39\!\cdots\!62\)\( T^{2} - \)\(77\!\cdots\!55\)\( T^{3} - \)\(32\!\cdots\!35\)\( T^{4} - \)\(36\!\cdots\!43\)\( T^{5} + \)\(51\!\cdots\!86\)\( T^{6} + 73774416902425874 T^{7} - 3004273848072 T^{8} - 483721503 T^{9} + 6369 T^{10} + T^{11} \)
$43$ \( \)\(12\!\cdots\!40\)\( + \)\(49\!\cdots\!84\)\( T + \)\(30\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!25\)\( T^{3} - \)\(14\!\cdots\!26\)\( T^{4} + \)\(14\!\cdots\!46\)\( T^{5} + \)\(10\!\cdots\!06\)\( T^{6} + 288241914507653804 T^{7} - 25830461314178 T^{8} - 1062521582 T^{9} + 20506 T^{10} + T^{11} \)
$47$ \( \)\(13\!\cdots\!00\)\( - \)\(26\!\cdots\!00\)\( T - \)\(13\!\cdots\!00\)\( T^{2} - \)\(19\!\cdots\!48\)\( T^{3} + \)\(86\!\cdots\!80\)\( T^{4} + \)\(32\!\cdots\!81\)\( T^{5} + \)\(15\!\cdots\!19\)\( T^{6} - 549749944582648698 T^{7} - 56317737705861 T^{8} - 487986614 T^{9} + 47899 T^{10} + T^{11} \)
$53$ \( -\)\(90\!\cdots\!56\)\( + \)\(45\!\cdots\!20\)\( T + \)\(11\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!70\)\( T^{3} - \)\(55\!\cdots\!56\)\( T^{4} + \)\(11\!\cdots\!41\)\( T^{5} + \)\(13\!\cdots\!84\)\( T^{6} - 1033081380561892439 T^{7} - 165079894754680 T^{8} - 590697121 T^{9} + 80048 T^{10} + T^{11} \)
$59$ \( ( -3481 + T )^{11} \)
$61$ \( -\)\(18\!\cdots\!64\)\( + \)\(15\!\cdots\!84\)\( T - \)\(11\!\cdots\!64\)\( T^{2} - \)\(59\!\cdots\!10\)\( T^{3} - \)\(32\!\cdots\!10\)\( T^{4} + \)\(70\!\cdots\!91\)\( T^{5} + \)\(21\!\cdots\!65\)\( T^{6} - 2403040677053750032 T^{7} - 117019095200851 T^{8} + 333956278 T^{9} + 82527 T^{10} + T^{11} \)
$67$ \( -\)\(78\!\cdots\!40\)\( - \)\(70\!\cdots\!32\)\( T - \)\(11\!\cdots\!96\)\( T^{2} - \)\(33\!\cdots\!32\)\( T^{3} + \)\(42\!\cdots\!40\)\( T^{4} + \)\(23\!\cdots\!35\)\( T^{5} - \)\(11\!\cdots\!56\)\( T^{6} - 25557507624458993453 T^{7} - 328765095311920 T^{8} + 6295378377 T^{9} + 166976 T^{10} + T^{11} \)
$71$ \( \)\(44\!\cdots\!24\)\( - \)\(16\!\cdots\!70\)\( T - \)\(40\!\cdots\!14\)\( T^{2} - \)\(15\!\cdots\!25\)\( T^{3} + \)\(19\!\cdots\!10\)\( T^{4} + \)\(12\!\cdots\!34\)\( T^{5} - \)\(18\!\cdots\!58\)\( T^{6} - 21119657326305116276 T^{7} - 179476073794226 T^{8} + 8740285976 T^{9} + 183560 T^{10} + T^{11} \)
$73$ \( -\)\(58\!\cdots\!00\)\( - \)\(23\!\cdots\!72\)\( T + \)\(21\!\cdots\!36\)\( T^{2} + \)\(84\!\cdots\!20\)\( T^{3} - \)\(27\!\cdots\!30\)\( T^{4} - \)\(10\!\cdots\!83\)\( T^{5} + \)\(15\!\cdots\!63\)\( T^{6} + 53577585623492286482 T^{7} - 392140426314301 T^{8} - 12194330754 T^{9} + 36809 T^{10} + T^{11} \)
$79$ \( -\)\(92\!\cdots\!20\)\( + \)\(78\!\cdots\!52\)\( T + \)\(28\!\cdots\!16\)\( T^{2} + \)\(14\!\cdots\!91\)\( T^{3} - \)\(29\!\cdots\!34\)\( T^{4} - \)\(46\!\cdots\!66\)\( T^{5} - \)\(15\!\cdots\!26\)\( T^{6} - \)\(22\!\cdots\!12\)\( T^{7} - 594227927450426 T^{8} + 23543140926 T^{9} + 281518 T^{10} + T^{11} \)
$83$ \( \)\(16\!\cdots\!84\)\( + \)\(59\!\cdots\!56\)\( T - \)\(94\!\cdots\!44\)\( T^{2} - \)\(18\!\cdots\!89\)\( T^{3} + \)\(93\!\cdots\!71\)\( T^{4} + \)\(21\!\cdots\!21\)\( T^{5} + \)\(56\!\cdots\!58\)\( T^{6} - \)\(10\!\cdots\!86\)\( T^{7} - 960228183213870 T^{8} + 14884718739 T^{9} + 254691 T^{10} + T^{11} \)
$89$ \( -\)\(19\!\cdots\!16\)\( + \)\(77\!\cdots\!04\)\( T + \)\(38\!\cdots\!68\)\( T^{2} + \)\(55\!\cdots\!88\)\( T^{3} - \)\(11\!\cdots\!86\)\( T^{4} - \)\(15\!\cdots\!19\)\( T^{5} + \)\(77\!\cdots\!73\)\( T^{6} + \)\(11\!\cdots\!06\)\( T^{7} - 1549938399441535 T^{8} - 19948475170 T^{9} + 89687 T^{10} + T^{11} \)
$97$ \( -\)\(55\!\cdots\!76\)\( + \)\(57\!\cdots\!84\)\( T - \)\(10\!\cdots\!32\)\( T^{2} - \)\(69\!\cdots\!56\)\( T^{3} + \)\(28\!\cdots\!42\)\( T^{4} - \)\(21\!\cdots\!11\)\( T^{5} - \)\(48\!\cdots\!20\)\( T^{6} + \)\(74\!\cdots\!03\)\( T^{7} + 749492215682422 T^{8} - 47557321641 T^{9} + 45828 T^{10} + T^{11} \)
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