Properties

Label 177.6.a.d
Level $177$
Weight $6$
Character orbit 177.a
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $13$
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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + 81977088 x^{5} - 3773728 x^{4} - 1245415104 x^{3} + 453320896 x^{2} + 6872784896 x - 6400833792\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -9 q^{3} + ( 19 + \beta_{2} ) q^{4} + ( -1 + \beta_{7} ) q^{5} -9 \beta_{1} q^{6} + ( 29 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} + ( 10 + 13 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{9} ) q^{8} + 81 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -9 q^{3} + ( 19 + \beta_{2} ) q^{4} + ( -1 + \beta_{7} ) q^{5} -9 \beta_{1} q^{6} + ( 29 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} + ( 10 + 13 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{9} ) q^{8} + 81 q^{9} + ( 10 - 10 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{10} + ( 19 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{11} + ( -171 - 9 \beta_{2} ) q^{12} + ( 83 + 5 \beta_{1} + 15 \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{9} - \beta_{10} + 3 \beta_{12} ) q^{13} + ( -42 + 45 \beta_{1} + 7 \beta_{2} - 2 \beta_{4} + \beta_{5} + 4 \beta_{7} + \beta_{8} + 5 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 5 \beta_{12} ) q^{14} + ( 9 - 9 \beta_{7} ) q^{15} + ( 74 + 24 \beta_{1} + 25 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + 5 \beta_{8} - 2 \beta_{10} - \beta_{11} + 3 \beta_{12} ) q^{16} + ( 19 + 47 \beta_{1} - \beta_{3} + 3 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 4 \beta_{9} + 9 \beta_{10} - 5 \beta_{11} - 3 \beta_{12} ) q^{17} + 81 \beta_{1} q^{18} + ( 49 + 56 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} - 7 \beta_{8} - 5 \beta_{9} + 4 \beta_{10} + 5 \beta_{11} - \beta_{12} ) q^{19} + ( -419 + 59 \beta_{1} - 19 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} + 8 \beta_{5} - 3 \beta_{6} + 15 \beta_{7} - 7 \beta_{8} - 5 \beta_{9} - 5 \beta_{10} + 4 \beta_{12} ) q^{20} + ( -261 + 9 \beta_{1} - 9 \beta_{2} + 9 \beta_{5} ) q^{21} + ( 75 + 25 \beta_{1} - 21 \beta_{2} - 4 \beta_{3} + 14 \beta_{4} - 11 \beta_{5} - 12 \beta_{6} - 5 \beta_{7} + 9 \beta_{8} - 11 \beta_{9} - 12 \beta_{10} + 6 \beta_{11} - 3 \beta_{12} ) q^{22} + ( 302 + 225 \beta_{1} + 7 \beta_{2} + 14 \beta_{3} - 10 \beta_{5} + 2 \beta_{7} + 10 \beta_{8} - 4 \beta_{9} - 8 \beta_{10} + 13 \beta_{11} + 13 \beta_{12} ) q^{23} + ( -90 - 117 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} + 9 \beta_{6} - 9 \beta_{9} ) q^{24} + ( 1206 + 186 \beta_{1} - 27 \beta_{2} - 17 \beta_{3} - \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + 7 \beta_{7} - 11 \beta_{8} + 5 \beta_{9} + 3 \beta_{10} - 12 \beta_{11} - 12 \beta_{12} ) q^{25} + ( 352 + 451 \beta_{1} + 2 \beta_{2} - 13 \beta_{3} - 10 \beta_{4} + \beta_{5} - 19 \beta_{6} - 20 \beta_{7} + \beta_{8} + 9 \beta_{9} + 5 \beta_{10} + 15 \beta_{11} - 2 \beta_{12} ) q^{26} -729 q^{27} + ( 1631 + 168 \beta_{1} + 40 \beta_{2} + 10 \beta_{3} - 16 \beta_{4} + \beta_{5} + 9 \beta_{6} - 49 \beta_{7} - 8 \beta_{8} + 3 \beta_{9} + 32 \beta_{10} + 5 \beta_{11} ) q^{28} + ( -815 + 147 \beta_{1} - 41 \beta_{2} + 12 \beta_{3} - 26 \beta_{5} - 20 \beta_{6} + 19 \beta_{7} - 26 \beta_{9} - 16 \beta_{10} - 7 \beta_{11} + 15 \beta_{12} ) q^{29} + ( -90 + 90 \beta_{1} - 18 \beta_{2} - 9 \beta_{4} + 18 \beta_{5} + 9 \beta_{6} - 27 \beta_{7} - 9 \beta_{8} + 27 \beta_{9} + 9 \beta_{10} - 18 \beta_{11} ) q^{30} + ( 1984 + 232 \beta_{1} + 9 \beta_{2} - 47 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} + 9 \beta_{8} + 29 \beta_{9} + 12 \beta_{10} - 43 \beta_{11} - 20 \beta_{12} ) q^{31} + ( 1231 + 270 \beta_{1} + 9 \beta_{2} - 12 \beta_{3} - 24 \beta_{4} + 14 \beta_{5} + 4 \beta_{6} - 30 \beta_{7} - 2 \beta_{8} + 18 \beta_{9} + 28 \beta_{10} - 10 \beta_{12} ) q^{32} + ( -171 - 18 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} - 9 \beta_{4} - 18 \beta_{5} - 9 \beta_{6} - 18 \beta_{7} + 18 \beta_{8} - 18 \beta_{10} + 9 \beta_{11} ) q^{33} + ( 2423 + 5 \beta_{1} + 24 \beta_{2} + 33 \beta_{3} + 23 \beta_{4} - 33 \beta_{5} - 8 \beta_{6} - 60 \beta_{7} + 26 \beta_{8} + 2 \beta_{9} - 20 \beta_{10} + 24 \beta_{11} - 8 \beta_{12} ) q^{34} + ( 533 + 94 \beta_{1} - 104 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 40 \beta_{6} + 73 \beta_{7} + 3 \beta_{9} - 37 \beta_{10} + 10 \beta_{11} + 27 \beta_{12} ) q^{35} + ( 1539 + 81 \beta_{2} ) q^{36} + ( 1355 + 172 \beta_{1} + 19 \beta_{2} + 77 \beta_{3} - 7 \beta_{4} - 33 \beta_{5} + 10 \beta_{6} - 8 \beta_{7} - 17 \beta_{8} + 5 \beta_{9} - 42 \beta_{10} + 44 \beta_{11} + 19 \beta_{12} ) q^{37} + ( 2687 + 114 \beta_{1} - 32 \beta_{2} - 42 \beta_{3} + 34 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 41 \beta_{7} + 34 \beta_{8} + 28 \beta_{9} - 38 \beta_{10} - 31 \beta_{11} - 24 \beta_{12} ) q^{38} + ( -747 - 45 \beta_{1} - 135 \beta_{2} - 9 \beta_{5} - 18 \beta_{6} - 9 \beta_{9} + 9 \beta_{10} - 27 \beta_{12} ) q^{39} + ( 2446 - 504 \beta_{1} - 31 \beta_{2} - 3 \beta_{3} - 16 \beta_{4} + 50 \beta_{5} - 4 \beta_{6} + 47 \beta_{7} - 67 \beta_{8} - 38 \beta_{9} - 31 \beta_{10} - 32 \beta_{11} + 54 \beta_{12} ) q^{40} + ( 923 + 100 \beta_{1} - 100 \beta_{2} - 10 \beta_{3} + 24 \beta_{4} + 15 \beta_{5} + 21 \beta_{6} + 36 \beta_{7} + 25 \beta_{8} + 6 \beta_{9} + 90 \beta_{10} - \beta_{11} - 41 \beta_{12} ) q^{41} + ( 378 - 405 \beta_{1} - 63 \beta_{2} + 18 \beta_{4} - 9 \beta_{5} - 36 \beta_{7} - 9 \beta_{8} - 45 \beta_{9} + 18 \beta_{10} + 27 \beta_{11} + 45 \beta_{12} ) q^{42} + ( 2058 - 517 \beta_{1} + 44 \beta_{2} - 21 \beta_{3} - 19 \beta_{4} + 69 \beta_{5} - 12 \beta_{6} + 69 \beta_{7} - 11 \beta_{8} + 19 \beta_{9} + 50 \beta_{10} - 65 \beta_{11} - 26 \beta_{12} ) q^{43} + ( 502 - 296 \beta_{1} - 136 \beta_{2} + 28 \beta_{3} + 10 \beta_{4} + 17 \beta_{5} + 14 \beta_{6} + 29 \beta_{7} - 61 \beta_{8} - 65 \beta_{9} + 12 \beta_{10} + 20 \beta_{11} + 39 \beta_{12} ) q^{44} + ( -81 + 81 \beta_{7} ) q^{45} + ( 11447 + 180 \beta_{1} + 244 \beta_{2} - 72 \beta_{3} + 12 \beta_{4} + 62 \beta_{5} + 20 \beta_{6} - 43 \beta_{7} + 44 \beta_{8} + 50 \beta_{9} + 86 \beta_{10} + 7 \beta_{11} - 20 \beta_{12} ) q^{46} + ( 4008 + 36 \beta_{1} - 69 \beta_{2} + 23 \beta_{3} + 23 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} + 154 \beta_{7} - 16 \beta_{8} - 40 \beta_{9} - 15 \beta_{10} - 39 \beta_{11} - 43 \beta_{12} ) q^{47} + ( -666 - 216 \beta_{1} - 225 \beta_{2} - 18 \beta_{3} + 36 \beta_{4} + 18 \beta_{5} - 9 \beta_{6} - 45 \beta_{8} + 18 \beta_{10} + 9 \beta_{11} - 27 \beta_{12} ) q^{48} + ( 7318 - 134 \beta_{1} + 69 \beta_{2} + 133 \beta_{3} + 19 \beta_{4} - 103 \beta_{5} + 53 \beta_{6} - 58 \beta_{7} + 2 \beta_{8} - 44 \beta_{9} - 41 \beta_{10} + 132 \beta_{11} + 90 \beta_{12} ) q^{49} + ( 9423 + 731 \beta_{1} + 137 \beta_{2} + 115 \beta_{3} - 11 \beta_{4} - 76 \beta_{5} + 12 \beta_{6} + 118 \beta_{7} - 57 \beta_{8} - 65 \beta_{9} - 108 \beta_{10} + 7 \beta_{11} - 17 \beta_{12} ) q^{50} + ( -171 - 423 \beta_{1} + 9 \beta_{3} - 27 \beta_{4} - 45 \beta_{5} - 36 \beta_{6} + 18 \beta_{7} + 9 \beta_{8} - 36 \beta_{9} - 81 \beta_{10} + 45 \beta_{11} + 27 \beta_{12} ) q^{51} + ( 20325 + 192 \beta_{1} + 675 \beta_{2} + 29 \beta_{3} - 28 \beta_{4} - 9 \beta_{5} + 23 \beta_{6} + 46 \beta_{7} + 113 \beta_{8} + 13 \beta_{9} - 5 \beta_{10} - 47 \beta_{11} + 8 \beta_{12} ) q^{52} + ( 1411 - 334 \beta_{1} - 52 \beta_{2} + 42 \beta_{3} + 100 \beta_{4} - 12 \beta_{5} - 34 \beta_{6} - 51 \beta_{7} + 20 \beta_{8} - 58 \beta_{9} + 136 \beta_{10} + 76 \beta_{11} - 102 \beta_{12} ) q^{53} -729 \beta_{1} q^{54} + ( 9333 - 749 \beta_{1} - 79 \beta_{2} - 270 \beta_{3} - 72 \beta_{4} + 144 \beta_{5} - 2 \beta_{6} - 69 \beta_{7} + 46 \beta_{8} + 133 \beta_{9} + 85 \beta_{10} - 51 \beta_{11} - 34 \beta_{12} ) q^{55} + ( 9796 + 1176 \beta_{1} + 252 \beta_{2} - 132 \beta_{3} - 56 \beta_{4} + 7 \beta_{5} + 6 \beta_{6} - 211 \beta_{7} + 193 \beta_{8} + 195 \beta_{9} - 82 \beta_{10} - 62 \beta_{11} + 91 \beta_{12} ) q^{56} + ( -441 - 504 \beta_{1} - 36 \beta_{2} - 72 \beta_{3} - 36 \beta_{4} + 9 \beta_{5} - 27 \beta_{6} + 9 \beta_{7} + 63 \beta_{8} + 45 \beta_{9} - 36 \beta_{10} - 45 \beta_{11} + 9 \beta_{12} ) q^{57} + ( 6647 - 2018 \beta_{1} - 26 \beta_{2} - 120 \beta_{3} + 27 \beta_{4} + 132 \beta_{5} - 107 \beta_{6} + 208 \beta_{7} - 109 \beta_{8} - 99 \beta_{9} - 9 \beta_{10} - 133 \beta_{11} - 108 \beta_{12} ) q^{58} + 3481 q^{59} + ( 3771 - 531 \beta_{1} + 171 \beta_{2} + 72 \beta_{3} + 27 \beta_{4} - 72 \beta_{5} + 27 \beta_{6} - 135 \beta_{7} + 63 \beta_{8} + 45 \beta_{9} + 45 \beta_{10} - 36 \beta_{12} ) q^{60} + ( -849 - 1334 \beta_{1} - 34 \beta_{2} + 164 \beta_{3} - 80 \beta_{4} - 108 \beta_{5} - 69 \beta_{6} - 57 \beta_{7} - 105 \beta_{8} - 90 \beta_{9} - 208 \beta_{10} + 138 \beta_{11} + 156 \beta_{12} ) q^{61} + ( 12735 + 2548 \beta_{1} + 652 \beta_{2} + 290 \beta_{3} - 117 \beta_{4} - 120 \beta_{5} - 47 \beta_{6} - 130 \beta_{7} - 119 \beta_{8} - 71 \beta_{9} - 39 \beta_{10} + 75 \beta_{11} + 8 \beta_{12} ) q^{62} + ( 2349 - 81 \beta_{1} + 81 \beta_{2} - 81 \beta_{5} ) q^{63} + ( 11722 + 715 \beta_{1} + 43 \beta_{2} - 51 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} + 33 \beta_{6} - 70 \beta_{7} + 38 \beta_{8} + 201 \beta_{9} - 64 \beta_{10} - 16 \beta_{11} - 50 \beta_{12} ) q^{64} + ( -2170 - 251 \beta_{1} - 279 \beta_{2} - 248 \beta_{3} + 4 \beta_{4} + 61 \beta_{5} - 120 \beta_{6} - 74 \beta_{7} - 88 \beta_{8} - 24 \beta_{9} - 35 \beta_{10} - 64 \beta_{11} + 11 \beta_{12} ) q^{65} + ( -675 - 225 \beta_{1} + 189 \beta_{2} + 36 \beta_{3} - 126 \beta_{4} + 99 \beta_{5} + 108 \beta_{6} + 45 \beta_{7} - 81 \beta_{8} + 99 \beta_{9} + 108 \beta_{10} - 54 \beta_{11} + 27 \beta_{12} ) q^{66} + ( 67 - 2927 \beta_{1} + 3 \beta_{2} - 90 \beta_{3} + 98 \beta_{4} + 82 \beta_{5} - 26 \beta_{6} + 101 \beta_{7} + 2 \beta_{8} + 49 \beta_{9} - 13 \beta_{10} - 43 \beta_{11} - 4 \beta_{12} ) q^{67} + ( -325 + 1617 \beta_{1} - 144 \beta_{2} + 35 \beta_{3} - 51 \beta_{4} + 25 \beta_{5} + 105 \beta_{6} - 353 \beta_{7} - 25 \beta_{8} + 184 \beta_{9} + 112 \beta_{10} + 18 \beta_{11} + 31 \beta_{12} ) q^{68} + ( -2718 - 2025 \beta_{1} - 63 \beta_{2} - 126 \beta_{3} + 90 \beta_{5} - 18 \beta_{7} - 90 \beta_{8} + 36 \beta_{9} + 72 \beta_{10} - 117 \beta_{11} - 117 \beta_{12} ) q^{69} + ( 4162 - 2692 \beta_{1} - 539 \beta_{2} + 123 \beta_{3} + 135 \beta_{4} - 102 \beta_{5} + 78 \beta_{6} + 143 \beta_{7} - 77 \beta_{8} - 309 \beta_{9} + 48 \beta_{10} + 286 \beta_{11} - 57 \beta_{12} ) q^{70} + ( 1656 - 821 \beta_{1} + 274 \beta_{2} + 29 \beta_{3} + 35 \beta_{4} + 115 \beta_{5} + 211 \beta_{6} + 13 \beta_{7} + 22 \beta_{8} + 24 \beta_{9} - 5 \beta_{10} + 91 \beta_{11} + 307 \beta_{12} ) q^{71} + ( 810 + 1053 \beta_{1} + 81 \beta_{2} - 81 \beta_{3} - 81 \beta_{4} - 81 \beta_{6} + 81 \beta_{9} ) q^{72} + ( 6176 - 2739 \beta_{1} - 86 \beta_{2} + 33 \beta_{3} + 63 \beta_{4} - 244 \beta_{5} - 196 \beta_{6} - 172 \beta_{7} + 143 \beta_{8} - 54 \beta_{9} + 5 \beta_{10} - 168 \beta_{11} - 178 \beta_{12} ) q^{73} + ( 7812 + 947 \beta_{1} + 65 \beta_{2} - 328 \beta_{3} + 121 \beta_{4} + 87 \beta_{5} + 191 \beta_{6} - 361 \beta_{7} + 182 \beta_{8} + 292 \beta_{9} + 303 \beta_{10} - 229 \beta_{11} - 223 \beta_{12} ) q^{74} + ( -10854 - 1674 \beta_{1} + 243 \beta_{2} + 153 \beta_{3} + 9 \beta_{4} - 27 \beta_{5} - 54 \beta_{6} - 63 \beta_{7} + 99 \beta_{8} - 45 \beta_{9} - 27 \beta_{10} + 108 \beta_{11} + 108 \beta_{12} ) q^{75} + ( 5573 + 524 \beta_{1} - 404 \beta_{2} + 246 \beta_{3} - 30 \beta_{4} - 110 \beta_{5} - 5 \beta_{6} - 220 \beta_{7} - 209 \beta_{8} - 254 \beta_{9} + 236 \beta_{10} + 189 \beta_{11} + 139 \beta_{12} ) q^{76} + ( -7872 - 2664 \beta_{1} - 77 \beta_{2} + 113 \beta_{3} - 191 \beta_{4} + 185 \beta_{5} + 204 \beta_{6} + 107 \beta_{7} - 303 \beta_{8} - 85 \beta_{9} + 66 \beta_{10} - 206 \beta_{11} + 69 \beta_{12} ) q^{77} + ( -3168 - 4059 \beta_{1} - 18 \beta_{2} + 117 \beta_{3} + 90 \beta_{4} - 9 \beta_{5} + 171 \beta_{6} + 180 \beta_{7} - 9 \beta_{8} - 81 \beta_{9} - 45 \beta_{10} - 135 \beta_{11} + 18 \beta_{12} ) q^{78} + ( -1168 - 3140 \beta_{1} - 297 \beta_{2} + 65 \beta_{3} + 161 \beta_{4} - 233 \beta_{5} - 65 \beta_{6} + 73 \beta_{7} + 300 \beta_{8} + 15 \beta_{9} - 94 \beta_{10} - 8 \beta_{11} - 115 \beta_{12} ) q^{79} + ( -14936 + 361 \beta_{1} - 831 \beta_{2} - 92 \beta_{3} + 195 \beta_{4} - 210 \beta_{5} - 414 \beta_{6} - 153 \beta_{7} - 40 \beta_{8} - 435 \beta_{9} - 143 \beta_{10} - 7 \beta_{11} - 79 \beta_{12} ) q^{80} + 6561 q^{81} + ( 6012 - 2365 \beta_{1} + 307 \beta_{2} + 332 \beta_{3} + 208 \beta_{4} - 273 \beta_{5} + 142 \beta_{6} + 174 \beta_{7} + 385 \beta_{8} - 35 \beta_{9} - 284 \beta_{10} + 255 \beta_{11} + 45 \beta_{12} ) q^{82} + ( -5853 + 448 \beta_{1} - 26 \beta_{2} + 308 \beta_{3} - 130 \beta_{4} + 115 \beta_{5} - 15 \beta_{6} + 6 \beta_{7} - 133 \beta_{8} - 82 \beta_{9} - 236 \beta_{10} + 201 \beta_{11} + 189 \beta_{12} ) q^{83} + ( -14679 - 1512 \beta_{1} - 360 \beta_{2} - 90 \beta_{3} + 144 \beta_{4} - 9 \beta_{5} - 81 \beta_{6} + 441 \beta_{7} + 72 \beta_{8} - 27 \beta_{9} - 288 \beta_{10} - 45 \beta_{11} ) q^{84} + ( 730 - 5138 \beta_{1} - 690 \beta_{2} - 468 \beta_{3} - 12 \beta_{4} + 386 \beta_{5} + 219 \beta_{6} - 30 \beta_{7} + 79 \beta_{8} + 226 \beta_{9} + 356 \beta_{10} - 160 \beta_{11} - 110 \beta_{12} ) q^{85} + ( -24797 + 3189 \beta_{1} - 319 \beta_{2} - 26 \beta_{3} - 26 \beta_{4} - 365 \beta_{5} - 380 \beta_{6} - 175 \beta_{7} + 83 \beta_{8} - 365 \beta_{9} - 198 \beta_{10} + 194 \beta_{11} + 151 \beta_{12} ) q^{86} + ( 7335 - 1323 \beta_{1} + 369 \beta_{2} - 108 \beta_{3} + 234 \beta_{5} + 180 \beta_{6} - 171 \beta_{7} + 234 \beta_{9} + 144 \beta_{10} + 63 \beta_{11} - 135 \beta_{12} ) q^{87} + ( -21075 - 3785 \beta_{1} - 642 \beta_{2} - 171 \beta_{3} - 141 \beta_{4} + 463 \beta_{5} + 183 \beta_{6} + 879 \beta_{7} - 223 \beta_{8} + 24 \beta_{9} - 112 \beta_{10} - 296 \beta_{11} + 49 \beta_{12} ) q^{88} + ( -14799 + 1187 \beta_{1} - 164 \beta_{2} + 5 \beta_{3} + 59 \beta_{4} + 39 \beta_{5} - 176 \beta_{6} - 257 \beta_{7} + 33 \beta_{8} + 37 \beta_{9} + 189 \beta_{10} - 45 \beta_{11} - 279 \beta_{12} ) q^{89} + ( 810 - 810 \beta_{1} + 162 \beta_{2} + 81 \beta_{4} - 162 \beta_{5} - 81 \beta_{6} + 243 \beta_{7} + 81 \beta_{8} - 243 \beta_{9} - 81 \beta_{10} + 162 \beta_{11} ) q^{90} + ( 6328 + 935 \beta_{1} + 334 \beta_{2} + 301 \beta_{3} - 489 \beta_{4} + 11 \beta_{5} - 63 \beta_{6} - 249 \beta_{7} - 144 \beta_{8} + 167 \beta_{9} + 206 \beta_{10} - 13 \beta_{11} + 94 \beta_{12} ) q^{91} + ( 3639 + 10620 \beta_{1} + 1248 \beta_{2} - 134 \beta_{3} - 222 \beta_{4} + 92 \beta_{5} - 47 \beta_{6} - 310 \beta_{7} + 47 \beta_{8} + 384 \beta_{9} + 92 \beta_{10} + 103 \beta_{11} - 9 \beta_{12} ) q^{92} + ( -17856 - 2088 \beta_{1} - 81 \beta_{2} + 423 \beta_{3} + 45 \beta_{4} - 54 \beta_{5} + 72 \beta_{6} + 36 \beta_{7} - 81 \beta_{8} - 261 \beta_{9} - 108 \beta_{10} + 387 \beta_{11} + 180 \beta_{12} ) q^{93} + ( 2758 + 2234 \beta_{1} - 1285 \beta_{2} - 73 \beta_{3} + 193 \beta_{4} - 274 \beta_{5} - 242 \beta_{6} + 513 \beta_{7} - 321 \beta_{8} - 607 \beta_{9} - 46 \beta_{10} + 80 \beta_{11} - 67 \beta_{12} ) q^{94} + ( 729 + 1452 \beta_{1} - 151 \beta_{2} - 199 \beta_{3} + 57 \beta_{4} - 114 \beta_{5} + 76 \beta_{6} + 383 \beta_{7} + 285 \beta_{8} + 15 \beta_{9} - 236 \beta_{10} + 305 \beta_{11} - 180 \beta_{12} ) q^{95} + ( -11079 - 2430 \beta_{1} - 81 \beta_{2} + 108 \beta_{3} + 216 \beta_{4} - 126 \beta_{5} - 36 \beta_{6} + 270 \beta_{7} + 18 \beta_{8} - 162 \beta_{9} - 252 \beta_{10} + 90 \beta_{12} ) q^{96} + ( 12797 + 2074 \beta_{1} - 812 \beta_{2} + 146 \beta_{3} - 472 \beta_{4} + 290 \beta_{5} + 307 \beta_{6} + 177 \beta_{7} - 533 \beta_{8} + 415 \beta_{9} - 355 \beta_{10} - 142 \beta_{11} + 135 \beta_{12} ) q^{97} + ( -9095 + 7076 \beta_{1} - 546 \beta_{2} - 633 \beta_{3} + 179 \beta_{4} + 521 \beta_{5} + 314 \beta_{6} + 504 \beta_{8} + 876 \beta_{9} + 230 \beta_{10} - 298 \beta_{11} - 492 \beta_{12} ) q^{98} + ( 1539 + 162 \beta_{1} + 81 \beta_{2} + 81 \beta_{3} + 81 \beta_{4} + 162 \beta_{5} + 81 \beta_{6} + 162 \beta_{7} - 162 \beta_{8} + 162 \beta_{10} - 81 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + O(q^{10}) \) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + 137q^{10} + 250q^{11} - 2214q^{12} + 1054q^{13} - 575q^{14} + 126q^{15} + 922q^{16} + 271q^{17} + 671q^{19} - 5491q^{20} - 3357q^{21} + 1094q^{22} + 3975q^{23} - 1107q^{24} + 15569q^{25} + 4622q^{26} - 9477q^{27} + 21214q^{28} - 10613q^{29} - 1233q^{30} + 25597q^{31} + 15966q^{32} - 2250q^{33} + 31796q^{34} + 6729q^{35} + 19926q^{36} + 17585q^{37} + 34903q^{38} - 9486q^{39} + 31382q^{40} + 12537q^{41} + 5175q^{42} + 26644q^{43} + 6654q^{44} - 1134q^{45} + 149005q^{46} + 52087q^{47} - 8298q^{48} + 95384q^{49} + 121821q^{50} - 2439q^{51} + 263630q^{52} + 20014q^{53} + 120932q^{55} + 126688q^{56} - 6039q^{57} + 86066q^{58} + 45253q^{59} + 49419q^{60} - 11667q^{61} + 164794q^{62} + 30213q^{63} + 151893q^{64} - 28674q^{65} - 9846q^{66} + 1106q^{67} - 4043q^{68} - 35775q^{69} + 56066q^{70} + 21230q^{71} + 9963q^{72} + 81131q^{73} + 102042q^{74} - 140121q^{75} + 73900q^{76} - 104655q^{77} - 41598q^{78} - 13470q^{79} - 191969q^{80} + 85293q^{81} + 79909q^{82} - 76149q^{83} - 190926q^{84} + 10035q^{85} - 321496q^{86} + 95517q^{87} - 276779q^{88} - 190205q^{89} + 11097q^{90} + 80601q^{91} + 45672q^{92} - 230373q^{93} + 36768q^{94} + 9875q^{95} - 143694q^{96} + 160850q^{97} - 116644q^{98} + 20250q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + 81977088 x^{5} - 3773728 x^{4} - 1245415104 x^{3} + 453320896 x^{2} + 6872784896 x - 6400833792\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 51 \)
\(\beta_{3}\)\(=\)\((\)\(39783793826895 \nu^{12} - 489453600148070 \nu^{11} - 16976260499266057 \nu^{10} + 173400552887267555 \nu^{9} + 2618126653315728364 \nu^{8} - 22576174422036271781 \nu^{7} - 176867473001742472306 \nu^{6} + 1306552202038622730424 \nu^{5} + 4910935561337209934672 \nu^{4} - 31773753569039047097600 \nu^{3} - 38787690119447100113728 \nu^{2} + 237612643533434276329408 \nu - 118486951803141085340032\)\()/ \)\(33\!\cdots\!28\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-826874692028041 \nu^{12} + 8707178998092938 \nu^{11} + 242959735353246943 \nu^{10} - 2502171917891064085 \nu^{9} - 26027976733390868724 \nu^{8} + 260206357988407773859 \nu^{7} + 1233471997339628863070 \nu^{6} - 11989595846763284566024 \nu^{5} - 24149330057839261034800 \nu^{4} + 243059241457176650722048 \nu^{3} + 86337265125117685358272 \nu^{2} - 1680944477628499755966016 \nu + 1386711865436038322772608\)\()/ \)\(33\!\cdots\!28\)\( \)
\(\beta_{5}\)\(=\)\((\)\(54340442593255 \nu^{12} - 52352870381654 \nu^{11} - 15931966674141105 \nu^{10} + 15155402107655195 \nu^{9} + 1681914535894438860 \nu^{8} - 1836338903824460461 \nu^{7} - 77824735047652982082 \nu^{6} + 113874528744436418552 \nu^{5} + 1545634558033536178384 \nu^{4} - 3227490580352160364288 \nu^{3} - 9356145936197742145856 \nu^{2} + 30544485864777114674624 \nu - 18662664520399065197952\)\()/ 20908883724336939008 \)
\(\beta_{6}\)\(=\)\((\)\(-1113563145590487 \nu^{12} + 6427918252947126 \nu^{11} + 326295802354114241 \nu^{10} - 1845485253816644939 \nu^{9} - 34658890543075405836 \nu^{8} + 193969902953198097789 \nu^{7} + 1626022103525751273570 \nu^{6} - 9169927390548757681912 \nu^{5} - 32507250145090709824720 \nu^{4} + 191762095301201226757376 \nu^{3} + 170014501795408920389952 \nu^{2} - 1345711626600217924786624 \nu + 1018785390979270801831296\)\()/ \)\(33\!\cdots\!28\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-1320253842628871 \nu^{12} + 10651286371910934 \nu^{11} + 383178975216834065 \nu^{10} - 3043793163821328443 \nu^{9} - 40236367280509962572 \nu^{8} + 315566609760609461901 \nu^{7} + 1854835450786592746626 \nu^{6} - 14548054411744976306296 \nu^{5} - 35512388633512947978704 \nu^{4} + 295619309481243808580864 \nu^{3} + 138197767088569099963712 \nu^{2} - 2049104176941380359374272 \nu + 1734892138667907983707520\)\()/ \)\(33\!\cdots\!28\)\( \)
\(\beta_{8}\)\(=\)\((\)\(1650337456541775 \nu^{12} - 4955971839841766 \nu^{11} - 479446960098985993 \nu^{10} + 1379487991946146147 \nu^{9} + 50124887187548432748 \nu^{8} - 142514024656920468389 \nu^{7} - 2294287024897826325490 \nu^{6} + 6794387848742615930040 \nu^{5} + 44883965212892001517392 \nu^{4} - 147967947121967418612992 \nu^{3} - 258860825301534993875776 \nu^{2} + 1135044579866026071873472 \nu - 702003306072892036759936\)\()/ \)\(33\!\cdots\!28\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-1900654043791633 \nu^{12} + 14645643650891994 \nu^{11} + 552279277208095127 \nu^{10} - 4174256618820441469 \nu^{9} - 58068740623150546196 \nu^{8} + 431600086519569599867 \nu^{7} + 2682626627863637664334 \nu^{6} - 19852971035273419517512 \nu^{5} - 51745644641592760924848 \nu^{4} + 403382125328928221405952 \nu^{3} + 217229534661490114610368 \nu^{2} - 2814803205443666513281088 \nu + 2300726532335333071253120\)\()/ \)\(33\!\cdots\!28\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-2234669099134249 \nu^{12} + 17389793189846090 \nu^{11} + 651521949057218879 \nu^{10} - 4975805206076548021 \nu^{9} - 68821058401010218740 \nu^{8} + 517212108318267617411 \nu^{7} + 3195955150486150034078 \nu^{6} - 23951284218566533208328 \nu^{5} - 61655502668973420593968 \nu^{4} + 489658650043767858080512 \nu^{3} + 244584360626935218411200 \nu^{2} - 3419981077478770845607488 \nu + 2916983411794306463500928\)\()/ \)\(33\!\cdots\!28\)\( \)
\(\beta_{11}\)\(=\)\((\)\(3238642461084169 \nu^{12} - 11725381212024970 \nu^{11} - 943070311405386079 \nu^{10} + 3298033769987841429 \nu^{9} + 98929005255715337204 \nu^{8} - 342457909205840852771 \nu^{7} - 4542165672376554424414 \nu^{6} + 16260704122077087417352 \nu^{5} + 88250090225962246224176 \nu^{4} - 349864054826960975178496 \nu^{3} - 463226775486699800821440 \nu^{2} + 2636128671139686083411520 \nu - 2034294686657283740883584\)\()/ \)\(33\!\cdots\!28\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-3338997355694837 \nu^{12} + 25179492484794626 \nu^{11} + 975627037896290643 \nu^{10} - 7192015752657721873 \nu^{9} - 103401850677514915428 \nu^{8} + 745927154545806510151 \nu^{7} + 4830796354465410322630 \nu^{6} - 34457126346509573099624 \nu^{5} - 94532611462907602667376 \nu^{4} + 703345219988242057572096 \nu^{3} + 411093247456719188021184 \nu^{2} - 4920984472601971111299392 \nu + 4073958545866859657402496\)\()/ \)\(33\!\cdots\!28\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 51\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 77 \beta_{1} + 10\)
\(\nu^{4}\)\(=\)\(3 \beta_{12} - \beta_{11} - 2 \beta_{10} + 5 \beta_{8} + \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 121 \beta_{2} + 24 \beta_{1} + 3946\)
\(\nu^{5}\)\(=\)\(-10 \beta_{12} + 28 \beta_{10} + 146 \beta_{9} - 2 \beta_{8} - 30 \beta_{7} - 124 \beta_{6} + 14 \beta_{5} - 152 \beta_{4} - 140 \beta_{3} + 137 \beta_{2} + 7054 \beta_{1} + 2511\)
\(\nu^{6}\)\(=\)\(430 \beta_{12} - 176 \beta_{11} - 384 \beta_{10} + 201 \beta_{9} + 838 \beta_{8} - 70 \beta_{7} + 193 \beta_{6} - 318 \beta_{5} - 647 \beta_{4} + 269 \beta_{3} + 13259 \beta_{2} + 4555 \beta_{1} + 362506\)
\(\nu^{7}\)\(=\)\(-1613 \beta_{12} + 413 \beta_{11} + 5482 \beta_{10} + 17480 \beta_{9} - 295 \beta_{8} - 6826 \beta_{7} - 12847 \beta_{6} + 1744 \beta_{5} - 18150 \beta_{4} - 15200 \beta_{3} + 18219 \beta_{2} + 700598 \beta_{1} + 379332\)
\(\nu^{8}\)\(=\)\(44114 \beta_{12} - 26902 \beta_{11} - 49184 \beta_{10} + 49688 \beta_{9} + 111710 \beta_{8} - 21616 \beta_{7} + 27170 \beta_{6} - 36688 \beta_{5} - 81208 \beta_{4} + 26628 \beta_{3} + 1424503 \beta_{2} + 673526 \beta_{1} + 36059531\)
\(\nu^{9}\)\(=\)\(-192940 \beta_{12} + 99996 \beta_{11} + 801488 \beta_{10} + 1993913 \beta_{9} - 23868 \beta_{8} - 1125152 \beta_{7} - 1269021 \beta_{6} + 152808 \beta_{5} - 2033353 \beta_{4} - 1525641 \beta_{3} + 2358625 \beta_{2} + 72247173 \beta_{1} + 50738702\)
\(\nu^{10}\)\(=\)\(4003155 \beta_{12} - 3748361 \beta_{11} - 5521578 \beta_{10} + 8462900 \beta_{9} + 13829037 \beta_{8} - 4305344 \beta_{7} + 3479885 \beta_{6} - 3826458 \beta_{5} - 9499248 \beta_{4} + 2333422 \beta_{3} + 152672313 \beta_{2} + 91041284 \beta_{1} + 3721822926\)
\(\nu^{11}\)\(=\)\(-21290390 \beta_{12} + 16375980 \beta_{11} + 105394716 \beta_{10} + 225028878 \beta_{9} - 387470 \beta_{8} - 162036366 \beta_{7} - 123324220 \beta_{6} + 10925486 \beta_{5} - 223625532 \beta_{4} - 149019168 \beta_{3} + 297761833 \beta_{2} + 7592343586 \beta_{1} + 6455842751\)
\(\nu^{12}\)\(=\)\(338534002 \beta_{12} - 488092428 \beta_{11} - 585026064 \beta_{10} + 1236018841 \beta_{9} + 1655369514 \beta_{8} - 703594110 \beta_{7} + 428704365 \beta_{6} - 386978134 \beta_{5} - 1086964311 \beta_{4} + 190581437 \beta_{3} + 16407966139 \beta_{2} + 11756119187 \beta_{1} + 391375714806\)

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
−10.3652
−9.47242
−6.40804
−6.29094
−6.06926
−3.96558
1.05732
2.69937
5.29201
6.46075
6.57049
9.82137
10.6702
−10.3652 −9.00000 75.4381 −24.0297 93.2871 −107.618 −450.246 81.0000 249.073
1.2 −9.47242 −9.00000 57.7268 −42.5216 85.2518 243.042 −243.695 81.0000 402.782
1.3 −6.40804 −9.00000 9.06300 −77.9071 57.6724 112.353 146.981 81.0000 499.232
1.4 −6.29094 −9.00000 7.57589 82.0909 56.6184 51.5315 153.651 81.0000 −516.429
1.5 −6.06926 −9.00000 4.83596 26.0687 54.6234 −152.673 164.866 81.0000 −158.218
1.6 −3.96558 −9.00000 −16.2741 45.8225 35.6903 214.983 191.435 81.0000 −181.713
1.7 1.05732 −9.00000 −30.8821 33.7931 −9.51585 −85.6226 −66.4863 81.0000 35.7300
1.8 2.69937 −9.00000 −24.7134 −80.3454 −24.2943 −162.093 −153.090 81.0000 −216.882
1.9 5.29201 −9.00000 −3.99466 108.071 −47.6281 212.139 −190.484 81.0000 571.910
1.10 6.46075 −9.00000 9.74133 −95.9365 −58.1468 79.3965 −143.808 81.0000 −619.822
1.11 6.57049 −9.00000 11.1714 −5.02964 −59.1344 −195.140 −136.854 81.0000 −33.0472
1.12 9.82137 −9.00000 64.4594 77.2018 −88.3924 −23.9833 318.795 81.0000 758.228
1.13 10.6702 −9.00000 81.8525 −61.2778 −96.0315 186.685 531.935 81.0000 −653.844
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.d 13
3.b odd 2 1 531.6.a.e 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.a.d 13 1.a even 1 1 trivial
531.6.a.e 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{13} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(177))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6400833792 + 6872784896 T + 453320896 T^{2} - 1245415104 T^{3} - 3773728 T^{4} + 81977088 T^{5} - 312100 T^{6} - 2592364 T^{7} + 7229 T^{8} + 41990 T^{9} - 41 T^{10} - 331 T^{11} + T^{13} \)
$3$ \( ( 9 + T )^{13} \)
$5$ \( \)\(52\!\cdots\!36\)\( + \)\(10\!\cdots\!32\)\( T - 23375173885239524240 T^{2} - 3091280158657842328 T^{3} + 37619818584681916 T^{4} + 3125931734458834 T^{5} - 22001737218208 T^{6} - 1386405245449 T^{7} + 5045389402 T^{8} + 289773709 T^{9} - 470420 T^{10} - 27999 T^{11} + 14 T^{12} + T^{13} \)
$7$ \( -\)\(10\!\cdots\!96\)\( - \)\(21\!\cdots\!76\)\( T + \)\(11\!\cdots\!36\)\( T^{2} + \)\(56\!\cdots\!68\)\( T^{3} - \)\(31\!\cdots\!28\)\( T^{4} - 405924796987275303 T^{5} + 34436175950426693 T^{6} - 11255659844621 T^{7} - 1744279803188 T^{8} + 2359402408 T^{9} + 41466900 T^{10} - 87373 T^{11} - 373 T^{12} + T^{13} \)
$11$ \( \)\(10\!\cdots\!04\)\( + \)\(37\!\cdots\!32\)\( T + \)\(20\!\cdots\!56\)\( T^{2} - \)\(65\!\cdots\!84\)\( T^{3} - \)\(15\!\cdots\!80\)\( T^{4} + \)\(32\!\cdots\!15\)\( T^{5} + 7379070231939210362 T^{6} - 54558228039546686 T^{7} - 77156139404902 T^{8} + 377760095872 T^{9} + 272512974 T^{10} - 1089246 T^{11} - 250 T^{12} + T^{13} \)
$13$ \( -\)\(63\!\cdots\!20\)\( - \)\(15\!\cdots\!40\)\( T + \)\(50\!\cdots\!24\)\( T^{2} - \)\(12\!\cdots\!42\)\( T^{3} - \)\(89\!\cdots\!00\)\( T^{4} + \)\(65\!\cdots\!39\)\( T^{5} + \)\(60\!\cdots\!46\)\( T^{6} - 540229701408146714 T^{7} - 1823511255501462 T^{8} + 1723342170240 T^{9} + 2372853290 T^{10} - 2267532 T^{11} - 1054 T^{12} + T^{13} \)
$17$ \( -\)\(22\!\cdots\!44\)\( + \)\(43\!\cdots\!40\)\( T - \)\(13\!\cdots\!00\)\( T^{2} - \)\(44\!\cdots\!48\)\( T^{3} + \)\(70\!\cdots\!26\)\( T^{4} + \)\(17\!\cdots\!87\)\( T^{5} - \)\(98\!\cdots\!11\)\( T^{6} - 27920442940333238619 T^{7} + 4555237991780262 T^{8} + 20875755610462 T^{9} - 52697264 T^{10} - 7344461 T^{11} - 271 T^{12} + T^{13} \)
$19$ \( \)\(31\!\cdots\!80\)\( - \)\(21\!\cdots\!00\)\( T - \)\(15\!\cdots\!24\)\( T^{2} + \)\(27\!\cdots\!04\)\( T^{3} + \)\(98\!\cdots\!32\)\( T^{4} + \)\(98\!\cdots\!40\)\( T^{5} - \)\(84\!\cdots\!56\)\( T^{6} - \)\(13\!\cdots\!81\)\( T^{7} + 18862425317957193 T^{8} + 64723384907196 T^{9} + 1876487765 T^{10} - 13417016 T^{11} - 671 T^{12} + T^{13} \)
$23$ \( -\)\(43\!\cdots\!40\)\( - \)\(17\!\cdots\!00\)\( T + \)\(24\!\cdots\!68\)\( T^{2} - \)\(76\!\cdots\!32\)\( T^{3} - \)\(26\!\cdots\!12\)\( T^{4} + \)\(92\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!60\)\( T^{6} - \)\(35\!\cdots\!85\)\( T^{7} - 1963632820749122763 T^{8} + 570653053949692 T^{9} + 148880618161 T^{10} - 39871972 T^{11} - 3975 T^{12} + T^{13} \)
$29$ \( \)\(17\!\cdots\!68\)\( + \)\(65\!\cdots\!24\)\( T + \)\(64\!\cdots\!12\)\( T^{2} + \)\(24\!\cdots\!20\)\( T^{3} + \)\(88\!\cdots\!76\)\( T^{4} - \)\(17\!\cdots\!62\)\( T^{5} - \)\(34\!\cdots\!14\)\( T^{6} + \)\(26\!\cdots\!29\)\( T^{7} + \)\(13\!\cdots\!11\)\( T^{8} + 4316817762489306 T^{9} - 2006011641749 T^{10} - 145462188 T^{11} + 10613 T^{12} + T^{13} \)
$31$ \( \)\(27\!\cdots\!04\)\( + \)\(70\!\cdots\!76\)\( T - \)\(11\!\cdots\!76\)\( T^{2} + \)\(11\!\cdots\!72\)\( T^{3} + \)\(34\!\cdots\!52\)\( T^{4} - \)\(11\!\cdots\!26\)\( T^{5} - \)\(13\!\cdots\!84\)\( T^{6} + \)\(94\!\cdots\!95\)\( T^{7} - 52774112289554867761 T^{8} - 21325831432298616 T^{9} + 2614898150843 T^{10} + 87184894 T^{11} - 25597 T^{12} + T^{13} \)
$37$ \( \)\(50\!\cdots\!88\)\( - \)\(10\!\cdots\!56\)\( T + \)\(23\!\cdots\!04\)\( T^{2} + \)\(25\!\cdots\!62\)\( T^{3} - \)\(94\!\cdots\!98\)\( T^{4} - \)\(52\!\cdots\!89\)\( T^{5} + \)\(46\!\cdots\!81\)\( T^{6} - \)\(94\!\cdots\!41\)\( T^{7} - \)\(83\!\cdots\!16\)\( T^{8} + 35335075421890110 T^{9} + 6365734685442 T^{10} - 330373639 T^{11} - 17585 T^{12} + T^{13} \)
$41$ \( -\)\(34\!\cdots\!56\)\( + \)\(23\!\cdots\!40\)\( T + \)\(47\!\cdots\!96\)\( T^{2} - \)\(36\!\cdots\!12\)\( T^{3} - \)\(27\!\cdots\!78\)\( T^{4} + \)\(17\!\cdots\!39\)\( T^{5} + \)\(99\!\cdots\!47\)\( T^{6} - \)\(29\!\cdots\!63\)\( T^{7} - \)\(13\!\cdots\!74\)\( T^{8} + 219802815190504094 T^{9} + 7180500861880 T^{10} - 759508625 T^{11} - 12537 T^{12} + T^{13} \)
$43$ \( -\)\(77\!\cdots\!52\)\( + \)\(69\!\cdots\!20\)\( T - \)\(26\!\cdots\!68\)\( T^{2} - \)\(32\!\cdots\!64\)\( T^{3} - \)\(47\!\cdots\!80\)\( T^{4} + \)\(87\!\cdots\!49\)\( T^{5} + \)\(18\!\cdots\!24\)\( T^{6} - \)\(10\!\cdots\!60\)\( T^{7} - \)\(24\!\cdots\!24\)\( T^{8} + 81119668953999138 T^{9} + 14081912881636 T^{10} - 473321576 T^{11} - 26644 T^{12} + T^{13} \)
$47$ \( -\)\(29\!\cdots\!00\)\( - \)\(17\!\cdots\!40\)\( T + \)\(17\!\cdots\!56\)\( T^{2} + \)\(20\!\cdots\!56\)\( T^{3} + \)\(32\!\cdots\!28\)\( T^{4} - \)\(72\!\cdots\!40\)\( T^{5} - \)\(79\!\cdots\!64\)\( T^{6} + \)\(10\!\cdots\!57\)\( T^{7} - \)\(36\!\cdots\!27\)\( T^{8} - 556549006415000706 T^{9} + 36397922814697 T^{10} + 98497234 T^{11} - 52087 T^{12} + T^{13} \)
$53$ \( \)\(71\!\cdots\!32\)\( + \)\(26\!\cdots\!28\)\( T + \)\(99\!\cdots\!12\)\( T^{2} - \)\(51\!\cdots\!76\)\( T^{3} - \)\(30\!\cdots\!44\)\( T^{4} + \)\(41\!\cdots\!30\)\( T^{5} + \)\(18\!\cdots\!72\)\( T^{6} - \)\(16\!\cdots\!09\)\( T^{7} - \)\(46\!\cdots\!02\)\( T^{8} + 3153035955687710365 T^{9} + 50768372948316 T^{10} - 2888098031 T^{11} - 20014 T^{12} + T^{13} \)
$59$ \( ( -3481 + T )^{13} \)
$61$ \( \)\(58\!\cdots\!52\)\( - \)\(59\!\cdots\!16\)\( T - \)\(30\!\cdots\!76\)\( T^{2} + \)\(11\!\cdots\!76\)\( T^{3} + \)\(47\!\cdots\!32\)\( T^{4} + \)\(53\!\cdots\!70\)\( T^{5} - \)\(20\!\cdots\!82\)\( T^{6} - \)\(59\!\cdots\!11\)\( T^{7} + \)\(24\!\cdots\!53\)\( T^{8} + 9595262504441923414 T^{9} - 103085062965559 T^{10} - 5344371572 T^{11} + 11667 T^{12} + T^{13} \)
$67$ \( \)\(18\!\cdots\!48\)\( + \)\(27\!\cdots\!84\)\( T - \)\(11\!\cdots\!60\)\( T^{2} - \)\(28\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!16\)\( T^{4} + \)\(10\!\cdots\!72\)\( T^{5} + \)\(58\!\cdots\!16\)\( T^{6} - \)\(19\!\cdots\!47\)\( T^{7} - \)\(34\!\cdots\!06\)\( T^{8} + 17630620220645668183 T^{9} + 20183839502912 T^{10} - 7104474057 T^{11} - 1106 T^{12} + T^{13} \)
$71$ \( \)\(18\!\cdots\!88\)\( + \)\(56\!\cdots\!24\)\( T + \)\(26\!\cdots\!12\)\( T^{2} - \)\(33\!\cdots\!22\)\( T^{3} - \)\(11\!\cdots\!10\)\( T^{4} + \)\(81\!\cdots\!33\)\( T^{5} + \)\(15\!\cdots\!36\)\( T^{6} - \)\(91\!\cdots\!36\)\( T^{7} - \)\(88\!\cdots\!20\)\( T^{8} + 49338040167541938330 T^{9} + 217386882871932 T^{10} - 11993175470 T^{11} - 21230 T^{12} + T^{13} \)
$73$ \( \)\(26\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T + \)\(19\!\cdots\!08\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} - \)\(73\!\cdots\!84\)\( T^{4} + \)\(20\!\cdots\!48\)\( T^{5} + \)\(86\!\cdots\!42\)\( T^{6} - \)\(16\!\cdots\!89\)\( T^{7} - \)\(43\!\cdots\!17\)\( T^{8} + 65990190235573998760 T^{9} + 972005202651711 T^{10} - 12883475056 T^{11} - 81131 T^{12} + T^{13} \)
$79$ \( -\)\(85\!\cdots\!24\)\( + \)\(45\!\cdots\!60\)\( T - \)\(27\!\cdots\!40\)\( T^{2} - \)\(62\!\cdots\!52\)\( T^{3} + \)\(72\!\cdots\!56\)\( T^{4} + \)\(17\!\cdots\!27\)\( T^{5} - \)\(47\!\cdots\!10\)\( T^{6} - \)\(18\!\cdots\!34\)\( T^{7} + \)\(20\!\cdots\!30\)\( T^{8} + 81177198977036525552 T^{9} - 76986108102410 T^{10} - 15631067218 T^{11} + 13470 T^{12} + T^{13} \)
$83$ \( -\)\(87\!\cdots\!96\)\( - \)\(22\!\cdots\!40\)\( T - \)\(22\!\cdots\!76\)\( T^{2} + \)\(22\!\cdots\!92\)\( T^{3} + \)\(99\!\cdots\!04\)\( T^{4} - \)\(58\!\cdots\!41\)\( T^{5} - \)\(35\!\cdots\!95\)\( T^{6} + \)\(31\!\cdots\!25\)\( T^{7} + \)\(39\!\cdots\!02\)\( T^{8} + 26513909351613906582 T^{9} - 1123199410604722 T^{10} - 12442340881 T^{11} + 76149 T^{12} + T^{13} \)
$89$ \( \)\(16\!\cdots\!08\)\( - \)\(33\!\cdots\!32\)\( T - \)\(26\!\cdots\!68\)\( T^{2} + \)\(12\!\cdots\!72\)\( T^{3} + \)\(52\!\cdots\!52\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{5} - \)\(14\!\cdots\!10\)\( T^{6} + \)\(14\!\cdots\!49\)\( T^{7} + \)\(17\!\cdots\!71\)\( T^{8} - 54186797840931281646 T^{9} - 963132271123413 T^{10} + 5343884482 T^{11} + 190205 T^{12} + T^{13} \)
$97$ \( -\)\(79\!\cdots\!20\)\( + \)\(29\!\cdots\!40\)\( T + \)\(13\!\cdots\!44\)\( T^{2} - \)\(39\!\cdots\!68\)\( T^{3} - \)\(68\!\cdots\!80\)\( T^{4} + \)\(30\!\cdots\!76\)\( T^{5} + \)\(15\!\cdots\!90\)\( T^{6} - \)\(83\!\cdots\!77\)\( T^{7} - \)\(16\!\cdots\!82\)\( T^{8} + \)\(10\!\cdots\!39\)\( T^{9} + 8537953816499402 T^{10} - 52711916979 T^{11} - 160850 T^{12} + T^{13} \)
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