Properties

Label 483.6.a.b
Level $483$
Weight $6$
Character orbit 483.a
Self dual yes
Analytic conductor $77.465$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4653849697\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} - 268 x^{10} + 83 x^{9} + 25315 x^{8} + 5134 x^{7} - 993368 x^{6} - 511968 x^{5} + 14212480 x^{4} + 10085312 x^{3} - 18833856 x^{2} + 3290240 x + 102912\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + 9 q^{3} + ( 13 + \beta_{1} + \beta_{2} ) q^{4} + ( -13 - \beta_{1} + \beta_{6} ) q^{5} -9 \beta_{1} q^{6} -49 q^{7} + ( -40 - 16 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{8} + 81 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + 9 q^{3} + ( 13 + \beta_{1} + \beta_{2} ) q^{4} + ( -13 - \beta_{1} + \beta_{6} ) q^{5} -9 \beta_{1} q^{6} -49 q^{7} + ( -40 - 16 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{8} + 81 q^{9} + ( 40 + 25 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{10} + ( -119 - 9 \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{11} + ( 117 + 9 \beta_{1} + 9 \beta_{2} ) q^{12} + ( -9 + 9 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{11} ) q^{13} + 49 \beta_{1} q^{14} + ( -117 - 9 \beta_{1} + 9 \beta_{6} ) q^{15} + ( 309 + 83 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + \beta_{10} + \beta_{11} ) q^{16} + ( -36 + 25 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + \beta_{7} + 3 \beta_{8} - 4 \beta_{9} - \beta_{10} - \beta_{11} ) q^{17} -81 \beta_{1} q^{18} + ( -112 + 66 \beta_{1} + 7 \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + 8 \beta_{11} ) q^{19} + ( -711 - 99 \beta_{1} - 43 \beta_{2} - 9 \beta_{3} + 8 \beta_{4} + 4 \beta_{6} - 6 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{20} -441 q^{21} + ( 396 + 242 \beta_{1} + 4 \beta_{2} + 9 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} + 3 \beta_{8} + \beta_{9} + 5 \beta_{10} + 6 \beta_{11} ) q^{22} + 529 q^{23} + ( -360 - 144 \beta_{1} - 18 \beta_{2} - 9 \beta_{3} ) q^{24} + ( 445 + 328 \beta_{1} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - 18 \beta_{6} + 5 \beta_{7} + 6 \beta_{8} - 9 \beta_{9} + 8 \beta_{10} + 3 \beta_{11} ) q^{25} + ( -466 + 347 \beta_{1} + 25 \beta_{2} + 12 \beta_{3} - 4 \beta_{4} + 14 \beta_{5} - 10 \beta_{6} + 7 \beta_{7} - 7 \beta_{8} + 3 \beta_{9} + \beta_{10} - 4 \beta_{11} ) q^{26} + 729 q^{27} + ( -637 - 49 \beta_{1} - 49 \beta_{2} ) q^{28} + ( -465 + 169 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} - 30 \beta_{6} + 19 \beta_{7} - 16 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - 6 \beta_{11} ) q^{29} + ( 360 + 225 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} - 9 \beta_{4} - 27 \beta_{6} + 9 \beta_{7} ) q^{30} + ( 497 + 264 \beta_{1} + 4 \beta_{2} - \beta_{3} - 12 \beta_{4} - 4 \beta_{5} - 25 \beta_{6} + 2 \beta_{7} - 19 \beta_{8} + 10 \beta_{9} - 7 \beta_{10} - 15 \beta_{11} ) q^{31} + ( -2365 - 184 \beta_{1} - 103 \beta_{2} - 18 \beta_{3} + 14 \beta_{4} - 10 \beta_{5} + 34 \beta_{6} - 8 \beta_{7} + \beta_{8} + 9 \beta_{9} - 16 \beta_{10} - 13 \beta_{11} ) q^{32} + ( -1071 - 81 \beta_{1} - 36 \beta_{2} + 9 \beta_{3} + 18 \beta_{4} - 9 \beta_{5} - 18 \beta_{6} - 9 \beta_{7} + 9 \beta_{9} ) q^{33} + ( -1246 - 17 \beta_{1} - 90 \beta_{2} - 7 \beta_{3} - 11 \beta_{4} - 14 \beta_{5} - 3 \beta_{6} + 16 \beta_{7} + 10 \beta_{8} - 4 \beta_{9} - 7 \beta_{10} - \beta_{11} ) q^{34} + ( 637 + 49 \beta_{1} - 49 \beta_{6} ) q^{35} + ( 1053 + 81 \beta_{1} + 81 \beta_{2} ) q^{36} + ( -3006 + 163 \beta_{1} - 36 \beta_{2} - 23 \beta_{3} - 2 \beta_{4} + 45 \beta_{5} - 85 \beta_{6} - \beta_{7} - 15 \beta_{8} + 37 \beta_{9} - 26 \beta_{10} - 18 \beta_{11} ) q^{37} + ( -2698 - 130 \beta_{1} - 142 \beta_{2} - 9 \beta_{3} + 33 \beta_{4} - 26 \beta_{5} + 21 \beta_{6} - 32 \beta_{7} + 33 \beta_{8} + 3 \beta_{9} + 23 \beta_{10} + 30 \beta_{11} ) q^{38} + ( -81 + 81 \beta_{1} - 90 \beta_{2} - 18 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} + 9 \beta_{6} + 9 \beta_{9} - 9 \beta_{11} ) q^{39} + ( 2845 + 1461 \beta_{1} + 195 \beta_{2} + 71 \beta_{3} - 17 \beta_{4} + 34 \beta_{5} - 53 \beta_{6} + 46 \beta_{7} - 30 \beta_{8} - 30 \beta_{9} + 39 \beta_{10} + 25 \beta_{11} ) q^{40} + ( -686 - 274 \beta_{1} - 68 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 50 \beta_{5} - 64 \beta_{6} - 8 \beta_{7} + 16 \beta_{8} - 18 \beta_{9} + 25 \beta_{10} - 3 \beta_{11} ) q^{41} + 441 \beta_{1} q^{42} + ( -5517 - 228 \beta_{1} - 36 \beta_{2} - 18 \beta_{3} - 9 \beta_{4} - 2 \beta_{5} - 45 \beta_{6} - \beta_{7} + 31 \beta_{8} - 18 \beta_{9} + 9 \beta_{10} + \beta_{11} ) q^{43} + ( -6907 - 499 \beta_{1} - 285 \beta_{2} - 21 \beta_{3} - 11 \beta_{4} - 26 \beta_{5} + 77 \beta_{6} - 50 \beta_{7} + 60 \beta_{8} - 4 \beta_{9} + 37 \beta_{10} + 49 \beta_{11} ) q^{44} + ( -1053 - 81 \beta_{1} + 81 \beta_{6} ) q^{45} -529 \beta_{1} q^{46} + ( -1368 + 123 \beta_{1} + 103 \beta_{2} - 24 \beta_{3} - \beta_{4} + 2 \beta_{5} - 76 \beta_{6} - 4 \beta_{7} - 40 \beta_{8} + 15 \beta_{9} - 25 \beta_{10} + 28 \beta_{11} ) q^{47} + ( 2781 + 747 \beta_{1} + 81 \beta_{2} + 36 \beta_{3} - 27 \beta_{4} + 18 \beta_{5} - 63 \beta_{6} + 18 \beta_{7} - 36 \beta_{8} - 36 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} ) q^{48} + 2401 q^{49} + ( -14670 - 1124 \beta_{1} - 351 \beta_{2} - 56 \beta_{3} + 42 \beta_{4} - 30 \beta_{5} + 156 \beta_{6} - 69 \beta_{7} + 31 \beta_{8} + 9 \beta_{9} - 25 \beta_{10} - 8 \beta_{11} ) q^{50} + ( -324 + 225 \beta_{1} + 18 \beta_{3} + 27 \beta_{4} + 9 \beta_{7} + 27 \beta_{8} - 36 \beta_{9} - 9 \beta_{10} - 9 \beta_{11} ) q^{51} + ( -15514 - 844 \beta_{1} - 384 \beta_{2} - 62 \beta_{3} + 65 \beta_{4} + 2 \beta_{5} + 77 \beta_{6} - 36 \beta_{7} + 26 \beta_{8} + 90 \beta_{9} - 61 \beta_{10} - 51 \beta_{11} ) q^{52} + ( -5615 - 334 \beta_{1} - 28 \beta_{2} - 63 \beta_{3} - 9 \beta_{4} + 16 \beta_{5} - 21 \beta_{6} + 29 \beta_{7} - 21 \beta_{8} + 4 \beta_{9} - 64 \beta_{10} - 24 \beta_{11} ) q^{53} -729 \beta_{1} q^{54} + ( -999 + 1015 \beta_{1} + 208 \beta_{2} + 21 \beta_{3} + 25 \beta_{4} + 37 \beta_{5} - 214 \beta_{6} + 60 \beta_{7} - 22 \beta_{8} + 5 \beta_{9} - 25 \beta_{10} - 25 \beta_{11} ) q^{55} + ( 1960 + 784 \beta_{1} + 98 \beta_{2} + 49 \beta_{3} ) q^{56} + ( -1008 + 594 \beta_{1} + 63 \beta_{2} + 9 \beta_{3} - 45 \beta_{4} + 9 \beta_{5} - 45 \beta_{6} - 45 \beta_{7} + 9 \beta_{8} - 18 \beta_{9} + 9 \beta_{10} + 72 \beta_{11} ) q^{57} + ( -7682 - 111 \beta_{1} - 94 \beta_{2} - 113 \beta_{3} - 7 \beta_{4} + 14 \beta_{5} + 177 \beta_{6} - 30 \beta_{7} - 60 \beta_{8} + 12 \beta_{9} - 109 \beta_{10} - 105 \beta_{11} ) q^{58} + ( -2981 - 333 \beta_{1} - 29 \beta_{2} + 62 \beta_{3} - 23 \beta_{4} + 15 \beta_{5} + 239 \beta_{6} + 51 \beta_{7} - 49 \beta_{8} - 34 \beta_{9} + 104 \beta_{10} + 28 \beta_{11} ) q^{59} + ( -6399 - 891 \beta_{1} - 387 \beta_{2} - 81 \beta_{3} + 72 \beta_{4} + 36 \beta_{6} - 54 \beta_{7} + 36 \beta_{8} + 36 \beta_{9} - 54 \beta_{10} - 54 \beta_{11} ) q^{60} + ( -727 + 833 \beta_{1} + 365 \beta_{2} + 22 \beta_{3} - 160 \beta_{4} - 84 \beta_{5} + 185 \beta_{6} + 15 \beta_{7} + 27 \beta_{8} - 133 \beta_{9} + 62 \beta_{10} + 43 \beta_{11} ) q^{61} + ( -11798 - 1091 \beta_{1} - 222 \beta_{2} - 31 \beta_{3} + 39 \beta_{4} + 98 \beta_{5} + 135 \beta_{6} + 88 \beta_{7} - 94 \beta_{8} - 20 \beta_{9} - 5 \beta_{10} - 51 \beta_{11} ) q^{62} -3969 q^{63} + ( -2342 + 3551 \beta_{1} + 346 \beta_{2} + 143 \beta_{3} - 26 \beta_{4} - 6 \beta_{5} - 66 \beta_{6} + 148 \beta_{7} + 27 \beta_{8} - 9 \beta_{9} + 36 \beta_{10} + 17 \beta_{11} ) q^{64} + ( 10211 + 1149 \beta_{1} + 631 \beta_{2} + 127 \beta_{3} - 87 \beta_{4} + 62 \beta_{5} + 160 \beta_{6} + 44 \beta_{7} - 95 \beta_{8} - 73 \beta_{9} + 106 \beta_{10} + 102 \beta_{11} ) q^{65} + ( 3564 + 2178 \beta_{1} + 36 \beta_{2} + 81 \beta_{3} + 27 \beta_{4} - 54 \beta_{5} - 45 \beta_{6} - 72 \beta_{7} + 27 \beta_{8} + 9 \beta_{9} + 45 \beta_{10} + 54 \beta_{11} ) q^{66} + ( -10503 + 2222 \beta_{1} - 218 \beta_{2} + 17 \beta_{3} - 85 \beta_{4} - 28 \beta_{5} + 310 \beta_{6} + 75 \beta_{7} + 67 \beta_{8} - 110 \beta_{9} + 76 \beta_{10} + 49 \beta_{11} ) q^{67} + ( 1377 + 3034 \beta_{1} + 413 \beta_{2} + 51 \beta_{3} - 89 \beta_{4} + 4 \beta_{5} + 383 \beta_{6} + 18 \beta_{7} - 109 \beta_{8} + 7 \beta_{9} - 27 \beta_{10} - 6 \beta_{11} ) q^{68} + 4761 q^{69} + ( -1960 - 1225 \beta_{1} - 49 \beta_{2} - 49 \beta_{3} + 49 \beta_{4} + 147 \beta_{6} - 49 \beta_{7} ) q^{70} + ( -8951 + 516 \beta_{1} + 380 \beta_{2} - 11 \beta_{3} + 18 \beta_{4} - 41 \beta_{5} - 68 \beta_{6} - 3 \beta_{7} + 135 \beta_{8} - 7 \beta_{9} + 12 \beta_{10} - 36 \beta_{11} ) q^{71} + ( -3240 - 1296 \beta_{1} - 162 \beta_{2} - 81 \beta_{3} ) q^{72} + ( -8766 + 1459 \beta_{1} + 367 \beta_{2} - 173 \beta_{3} + 161 \beta_{4} + 118 \beta_{5} - 120 \beta_{6} - 94 \beta_{7} + 26 \beta_{8} + 133 \beta_{9} - 42 \beta_{10} - 5 \beta_{11} ) q^{73} + ( -7670 + 3866 \beta_{1} - 160 \beta_{2} - 45 \beta_{3} - 7 \beta_{4} + 286 \beta_{5} + 69 \beta_{6} + 104 \beta_{7} - 103 \beta_{8} + 149 \beta_{9} - 17 \beta_{10} - 138 \beta_{11} ) q^{74} + ( 4005 + 2952 \beta_{1} + 27 \beta_{3} - 54 \beta_{4} - 27 \beta_{5} - 162 \beta_{6} + 45 \beta_{7} + 54 \beta_{8} - 81 \beta_{9} + 72 \beta_{10} + 27 \beta_{11} ) q^{75} + ( 9633 + 5285 \beta_{1} + 455 \beta_{2} + 347 \beta_{3} + 163 \beta_{4} - 246 \beta_{5} - 261 \beta_{6} - 126 \beta_{7} + 100 \beta_{8} + 4 \beta_{9} + 163 \beta_{10} + 23 \beta_{11} ) q^{76} + ( 5831 + 441 \beta_{1} + 196 \beta_{2} - 49 \beta_{3} - 98 \beta_{4} + 49 \beta_{5} + 98 \beta_{6} + 49 \beta_{7} - 49 \beta_{9} ) q^{77} + ( -4194 + 3123 \beta_{1} + 225 \beta_{2} + 108 \beta_{3} - 36 \beta_{4} + 126 \beta_{5} - 90 \beta_{6} + 63 \beta_{7} - 63 \beta_{8} + 27 \beta_{9} + 9 \beta_{10} - 36 \beta_{11} ) q^{78} + ( -3028 + 23 \beta_{1} + 548 \beta_{2} - 182 \beta_{3} + 11 \beta_{4} + 22 \beta_{5} - 118 \beta_{6} - 109 \beta_{7} - 25 \beta_{8} + 200 \beta_{9} - 123 \beta_{10} - 163 \beta_{11} ) q^{79} + ( -42066 - 7595 \beta_{1} - 1292 \beta_{2} - 456 \beta_{3} - 45 \beta_{4} - 200 \beta_{5} + 507 \beta_{6} - 326 \beta_{7} + 61 \beta_{8} + 253 \beta_{9} - 143 \beta_{10} - 68 \beta_{11} ) q^{80} + 6561 q^{81} + ( 12756 + 1615 \beta_{1} + 610 \beta_{2} + 85 \beta_{3} + 151 \beta_{4} - 118 \beta_{5} + 299 \beta_{6} - 152 \beta_{7} + 16 \beta_{8} - 140 \beta_{9} + 87 \beta_{10} + 135 \beta_{11} ) q^{82} + ( -15356 + 950 \beta_{1} - 210 \beta_{2} + 206 \beta_{3} + 199 \beta_{4} + 11 \beta_{5} - 74 \beta_{6} + 126 \beta_{7} - 36 \beta_{8} + 3 \beta_{9} - 190 \beta_{10} + 21 \beta_{11} ) q^{83} + ( -5733 - 441 \beta_{1} - 441 \beta_{2} ) q^{84} + ( -3803 - 153 \beta_{1} + 608 \beta_{2} - 103 \beta_{3} + 135 \beta_{4} - 235 \beta_{5} + 126 \beta_{6} - 142 \beta_{7} + 98 \beta_{8} + 315 \beta_{9} - 215 \beta_{10} + 31 \beta_{11} ) q^{85} + ( 10202 + 5996 \beta_{1} + 385 \beta_{2} + 4 \beta_{3} + 26 \beta_{4} + 10 \beta_{5} + 156 \beta_{6} - 25 \beta_{7} + 44 \beta_{8} - 30 \beta_{9} + 25 \beta_{10} + 37 \beta_{11} ) q^{86} + ( -4185 + 1521 \beta_{1} + 36 \beta_{2} + 72 \beta_{3} + 45 \beta_{4} + 54 \beta_{5} - 270 \beta_{6} + 171 \beta_{7} - 144 \beta_{8} + 18 \beta_{9} - 27 \beta_{10} - 54 \beta_{11} ) q^{87} + ( 9490 + 9705 \beta_{1} + 844 \beta_{2} + 310 \beta_{3} + 11 \beta_{4} - 181 \beta_{6} - 62 \beta_{7} + 189 \beta_{8} - 27 \beta_{9} + 137 \beta_{10} + 172 \beta_{11} ) q^{88} + ( -22684 - 2539 \beta_{1} - 445 \beta_{2} - 35 \beta_{3} + 13 \beta_{4} - 147 \beta_{5} + 529 \beta_{6} - 37 \beta_{7} + 228 \beta_{8} - 24 \beta_{9} - 29 \beta_{10} + 146 \beta_{11} ) q^{89} + ( 3240 + 2025 \beta_{1} + 81 \beta_{2} + 81 \beta_{3} - 81 \beta_{4} - 243 \beta_{6} + 81 \beta_{7} ) q^{90} + ( 441 - 441 \beta_{1} + 490 \beta_{2} + 98 \beta_{3} + 49 \beta_{4} - 49 \beta_{5} - 49 \beta_{6} - 49 \beta_{9} + 49 \beta_{11} ) q^{91} + ( 6877 + 529 \beta_{1} + 529 \beta_{2} ) q^{92} + ( 4473 + 2376 \beta_{1} + 36 \beta_{2} - 9 \beta_{3} - 108 \beta_{4} - 36 \beta_{5} - 225 \beta_{6} + 18 \beta_{7} - 171 \beta_{8} + 90 \beta_{9} - 63 \beta_{10} - 135 \beta_{11} ) q^{93} + ( -3104 - 2582 \beta_{1} + 714 \beta_{2} - 62 \beta_{3} + 2 \beta_{4} - 28 \beta_{5} + 34 \beta_{6} - 4 \beta_{7} - 178 \beta_{8} - 272 \beta_{9} + 42 \beta_{10} + 76 \beta_{11} ) q^{94} + ( -22823 - 3973 \beta_{1} - 1103 \beta_{2} - 278 \beta_{3} - 287 \beta_{4} - 123 \beta_{5} + 194 \beta_{6} - 57 \beta_{7} - 5 \beta_{8} - 78 \beta_{9} - 42 \beta_{10} - 287 \beta_{11} ) q^{95} + ( -21285 - 1656 \beta_{1} - 927 \beta_{2} - 162 \beta_{3} + 126 \beta_{4} - 90 \beta_{5} + 306 \beta_{6} - 72 \beta_{7} + 9 \beta_{8} + 81 \beta_{9} - 144 \beta_{10} - 117 \beta_{11} ) q^{96} + ( -14242 + 144 \beta_{1} + 150 \beta_{2} + 61 \beta_{3} - 76 \beta_{4} + 114 \beta_{5} - 457 \beta_{6} - 208 \beta_{7} - 8 \beta_{8} + 129 \beta_{10} - 36 \beta_{11} ) q^{97} -2401 \beta_{1} q^{98} + ( -9639 - 729 \beta_{1} - 324 \beta_{2} + 81 \beta_{3} + 162 \beta_{4} - 81 \beta_{5} - 162 \beta_{6} - 81 \beta_{7} + 81 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 108 q^{3} + 153 q^{4} - 162 q^{5} - 9 q^{6} - 588 q^{7} - 492 q^{8} + 972 q^{9} + O(q^{10}) \) \( 12 q - q^{2} + 108 q^{3} + 153 q^{4} - 162 q^{5} - 9 q^{6} - 588 q^{7} - 492 q^{8} + 972 q^{9} + 528 q^{10} - 1425 q^{11} + 1377 q^{12} - 70 q^{13} + 49 q^{14} - 1458 q^{15} + 3865 q^{16} - 398 q^{17} - 81 q^{18} - 1293 q^{19} - 8593 q^{20} - 5292 q^{21} + 4961 q^{22} + 6348 q^{23} - 4428 q^{24} + 5830 q^{25} - 5187 q^{26} + 8748 q^{27} - 7497 q^{28} - 5127 q^{29} + 4752 q^{30} + 6498 q^{31} - 28485 q^{32} - 12825 q^{33} - 14527 q^{34} + 7938 q^{35} + 12393 q^{36} - 35545 q^{37} - 32617 q^{38} - 630 q^{39} + 35789 q^{40} - 7806 q^{41} + 441 q^{42} - 66142 q^{43} - 83253 q^{44} - 13122 q^{45} - 529 q^{46} - 16432 q^{47} + 34785 q^{48} + 28812 q^{49} - 177328 q^{50} - 3582 q^{51} - 187010 q^{52} - 67456 q^{53} - 729 q^{54} - 10453 q^{55} + 24108 q^{56} - 11637 q^{57} - 92677 q^{58} - 36346 q^{59} - 77337 q^{60} - 8768 q^{61} - 141813 q^{62} - 47628 q^{63} - 24604 q^{64} + 121875 q^{65} + 44649 q^{66} - 123617 q^{67} + 17217 q^{68} + 57132 q^{69} - 25872 q^{70} - 108667 q^{71} - 39852 q^{72} - 107406 q^{73} - 87825 q^{74} + 52470 q^{75} + 120191 q^{76} + 69825 q^{77} - 46683 q^{78} - 39470 q^{79} - 513682 q^{80} + 78732 q^{81} + 150219 q^{82} - 181838 q^{83} - 67473 q^{84} - 52633 q^{85} + 125713 q^{86} - 46143 q^{87} + 120642 q^{88} - 277361 q^{89} + 42768 q^{90} + 3430 q^{91} + 80937 q^{92} + 58482 q^{93} - 40880 q^{94} - 272491 q^{95} - 256365 q^{96} - 169005 q^{97} - 2401 q^{98} - 115425 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} - 268 x^{10} + 83 x^{9} + 25315 x^{8} + 5134 x^{7} - 993368 x^{6} - 511968 x^{5} + 14212480 x^{4} + 10085312 x^{3} - 18833856 x^{2} + 3290240 x + 102912\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 45 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 78 \nu + 50 \)
\(\beta_{4}\)\(=\)\((\)\(-2031574625 \nu^{11} + 12856961043 \nu^{10} + 615454775966 \nu^{9} - 2651120115911 \nu^{8} - 69760433587669 \nu^{7} + 151676690716452 \nu^{6} + 3432769237934680 \nu^{5} - 782434345399472 \nu^{4} - 59349206170195264 \nu^{3} - 71487869479169024 \nu^{2} + 22231797633188800 \nu + 4277594394331392\)\()/ 119354720039424 \)
\(\beta_{5}\)\(=\)\((\)\(999513821 \nu^{11} + 1620050505 \nu^{10} - 252615322334 \nu^{9} - 513869754637 \nu^{8} + 21251156457073 \nu^{7} + 44014721477412 \nu^{6} - 670891337665456 \nu^{5} - 959228777624032 \nu^{4} + 6862587935009344 \nu^{3} - 1844596517006656 \nu^{2} - 13513372389366592 \nu + 9509911376711424\)\()/ 39784906679808 \)
\(\beta_{6}\)\(=\)\((\)\(1842383909 \nu^{11} + 602103219 \nu^{10} - 484909014092 \nu^{9} - 446003791225 \nu^{8} + 44154933248983 \nu^{7} + 56330700832374 \nu^{6} - 1622803445294560 \nu^{5} - 2172883505672368 \nu^{4} + 21022179737191984 \nu^{3} + 25615611799669184 \nu^{2} - 18579467749951936 \nu - 1875993171216384\)\()/ 59677360019712 \)
\(\beta_{7}\)\(=\)\((\)\(1377918191 \nu^{11} - 410055561 \nu^{10} - 365385332414 \nu^{9} - 119037349519 \nu^{8} + 33808454073031 \nu^{7} + 24979116156264 \nu^{6} - 1281589210911256 \nu^{5} - 1164801978616336 \nu^{4} + 17531775675222688 \nu^{3} + 16735019927060864 \nu^{2} - 21790828605317440 \nu - 3990040069684992\)\()/ 39784906679808 \)
\(\beta_{8}\)\(=\)\((\)\(-2862015409 \nu^{11} + 829916283 \nu^{10} + 764429966790 \nu^{9} + 280328312169 \nu^{8} - 71541284414973 \nu^{7} - 58995954481076 \nu^{6} + 2757764981495656 \nu^{5} + 2905381260291472 \nu^{4} - 38354280248330496 \nu^{3} - 43745140065019904 \nu^{2} + 41677505055529152 \nu - 206739256755456\)\()/ 39784906679808 \)
\(\beta_{9}\)\(=\)\((\)\(10406218379 \nu^{11} - 12352539861 \nu^{10} - 2824835475374 \nu^{9} + 1221107203445 \nu^{8} + 272209622707243 \nu^{7} + 49195461903792 \nu^{6} - 10985915640311080 \nu^{5} - 7061173584927760 \nu^{4} + 161204245142221984 \nu^{3} + 162942717563844992 \nu^{2} - 172971031210373440 \nu - 17646475645049088\)\()/ 119354720039424 \)
\(\beta_{10}\)\(=\)\((\)\(-15479062333 \nu^{11} + 23245884543 \nu^{10} + 4182351769054 \nu^{9} - 3123057169723 \nu^{8} - 401045593909913 \nu^{7} + 52979304764556 \nu^{6} + 16115815100881208 \nu^{5} + 5294270025729008 \nu^{4} - 237158411749072064 \nu^{3} - 162951151502770432 \nu^{2} + 317764774203150272 \nu - 41866660739087616\)\()/ 119354720039424 \)
\(\beta_{11}\)\(=\)\((\)\(7048452430 \nu^{11} - 5739172515 \nu^{10} - 1885723002493 \nu^{9} + 242863732396 \nu^{8} + 177479693325335 \nu^{7} + 66386986769559 \nu^{6} - 6918088778259296 \nu^{5} - 4644630525018080 \nu^{4} + 97785831651636536 \nu^{3} + 83253837357905920 \nu^{2} - 115676091160141952 \nu + 8013970114475520\)\()/ 29838680009856 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 45\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 80 \beta_{1} + 40\)
\(\nu^{4}\)\(=\)\(\beta_{11} + \beta_{10} - 4 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} - 7 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + 105 \beta_{2} + 179 \beta_{1} + 3605\)
\(\nu^{5}\)\(=\)\(13 \beta_{11} + 16 \beta_{10} - 9 \beta_{9} - \beta_{8} + 8 \beta_{7} - 34 \beta_{6} + 10 \beta_{5} - 14 \beta_{4} + 146 \beta_{3} + 359 \beta_{2} + 7352 \beta_{1} + 7485\)
\(\nu^{6}\)\(=\)\(177 \beta_{11} + 196 \beta_{10} - 649 \beta_{9} - 613 \beta_{8} + 468 \beta_{7} - 1186 \beta_{6} + 314 \beta_{5} - 506 \beta_{4} + 783 \beta_{3} + 11002 \beta_{2} + 26047 \beta_{1} + 330746\)
\(\nu^{7}\)\(=\)\(2786 \beta_{11} + 3437 \beta_{10} - 1997 \beta_{9} - 737 \beta_{8} + 2230 \beta_{7} - 8537 \beta_{6} + 1932 \beta_{5} - 2733 \beta_{4} + 18264 \beta_{3} + 50579 \beta_{2} + 732012 \beta_{1} + 1112549\)
\(\nu^{8}\)\(=\)\(26212 \beta_{11} + 31346 \beta_{10} - 83930 \beta_{9} - 76118 \beta_{8} + 76056 \beta_{7} - 166958 \beta_{6} + 42336 \beta_{5} - 68858 \beta_{4} + 117452 \beta_{3} + 1174175 \beta_{2} + 3487359 \beta_{1} + 32843607\)
\(\nu^{9}\)\(=\)\(438852 \beta_{11} + 540068 \beta_{10} - 343224 \beta_{9} - 168968 \beta_{8} + 423760 \beta_{7} - 1453196 \beta_{6} + 285224 \beta_{5} - 421076 \beta_{4} + 2204473 \beta_{3} + 6605522 \beta_{2} + 76590736 \beta_{1} + 150708552\)
\(\nu^{10}\)\(=\)\(3670109 \beta_{11} + 4608497 \beta_{10} - 10156288 \beta_{9} - 8959040 \beta_{8} + 10721458 \beta_{7} - 22189919 \beta_{6} + 5438202 \beta_{5} - 8744363 \beta_{4} + 16088632 \beta_{3} + 127944709 \beta_{2} + 447729403 \beta_{1} + 3426955569\)
\(\nu^{11}\)\(=\)\(61098725 \beta_{11} + 75134124 \beta_{10} - 52275269 \beta_{9} - 29163405 \beta_{8} + 68091568 \beta_{7} - 212951310 \beta_{6} + 38805146 \beta_{5} - 59536234 \beta_{4} + 263632278 \beta_{3} + 835734739 \beta_{2} + 8286619320 \beta_{1} + 19484246329\)

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
10.9835
9.45158
6.14710
6.11405
0.714833
0.238049
−0.0270357
−1.77899
−5.44023
−7.10224
−8.58736
−9.71327
−10.9835 9.00000 88.6377 −108.950 −98.8517 −49.0000 −622.081 81.0000 1196.65
1.2 −9.45158 9.00000 57.3324 43.5642 −85.0642 −49.0000 −239.431 81.0000 −411.751
1.3 −6.14710 9.00000 5.78687 64.1572 −55.3239 −49.0000 161.135 81.0000 −394.381
1.4 −6.11405 9.00000 5.38156 −80.8988 −55.0264 −49.0000 162.746 81.0000 494.619
1.5 −0.714833 9.00000 −31.4890 65.9172 −6.43350 −49.0000 45.3841 81.0000 −47.1198
1.6 −0.238049 9.00000 −31.9433 −89.8479 −2.14244 −49.0000 15.2217 81.0000 21.3882
1.7 0.0270357 9.00000 −31.9993 −35.6847 0.243322 −49.0000 −1.73027 81.0000 −0.964762
1.8 1.77899 9.00000 −28.8352 −4.92580 16.0109 −49.0000 −108.225 81.0000 −8.76294
1.9 5.44023 9.00000 −2.40387 46.9255 48.9621 −49.0000 −187.165 81.0000 255.285
1.10 7.10224 9.00000 18.4419 9.99558 63.9202 −49.0000 −96.2931 81.0000 70.9911
1.11 8.58736 9.00000 41.7428 −47.8265 77.2863 −49.0000 83.6650 81.0000 −410.704
1.12 9.71327 9.00000 62.3475 −24.4259 87.4194 −49.0000 294.774 81.0000 −237.255
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 483.6.a.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.6.a.b 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(483))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 102912 - 3290240 T - 18833856 T^{2} - 10085312 T^{3} + 14212480 T^{4} + 511968 T^{5} - 993368 T^{6} - 5134 T^{7} + 25315 T^{8} - 83 T^{9} - 268 T^{10} + T^{11} + T^{12} \)
$3$ \( ( -9 + T )^{12} \)
$5$ \( -14052245560235578368 - 2114697072729336480 T + 233899518380155224 T^{2} + 16311109635644788 T^{3} - 256591406313894 T^{4} - 20895411220291 T^{5} + 76550362822 T^{6} + 10247427575 T^{7} + 10919883 T^{8} - 2141129 T^{9} - 8543 T^{10} + 162 T^{11} + T^{12} \)
$7$ \( ( 49 + T )^{12} \)
$11$ \( -\)\(13\!\cdots\!20\)\( - \)\(13\!\cdots\!80\)\( T + \)\(44\!\cdots\!12\)\( T^{2} + \)\(18\!\cdots\!28\)\( T^{3} + \)\(28\!\cdots\!78\)\( T^{4} + 21189081534134720263 T^{5} + 70737645784859321 T^{6} + 28751722081011 T^{7} - 427074516055 T^{8} - 902961803 T^{9} - 17557 T^{10} + 1425 T^{11} + T^{12} \)
$13$ \( \)\(18\!\cdots\!32\)\( + \)\(11\!\cdots\!28\)\( T - \)\(84\!\cdots\!04\)\( T^{2} - \)\(64\!\cdots\!84\)\( T^{3} + \)\(43\!\cdots\!50\)\( T^{4} + 59806541528187345333 T^{5} - 406625172427769858 T^{6} - 149811207661037 T^{7} + 1398644547495 T^{8} + 67414683 T^{9} - 1996195 T^{10} + 70 T^{11} + T^{12} \)
$17$ \( -\)\(58\!\cdots\!00\)\( + \)\(32\!\cdots\!24\)\( T - \)\(35\!\cdots\!08\)\( T^{2} + \)\(37\!\cdots\!32\)\( T^{3} + \)\(17\!\cdots\!20\)\( T^{4} - \)\(16\!\cdots\!96\)\( T^{5} - 36105192948888316147 T^{6} + 15940399258525358 T^{7} + 29431855809575 T^{8} - 4999917876 T^{9} - 9531157 T^{10} + 398 T^{11} + T^{12} \)
$19$ \( -\)\(20\!\cdots\!72\)\( - \)\(73\!\cdots\!08\)\( T - \)\(37\!\cdots\!12\)\( T^{2} + \)\(16\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!74\)\( T^{4} - \)\(10\!\cdots\!73\)\( T^{5} - \)\(63\!\cdots\!67\)\( T^{6} + 268206850581470551 T^{7} + 180965489037285 T^{8} - 31102584751 T^{9} - 22384481 T^{10} + 1293 T^{11} + T^{12} \)
$23$ \( ( -529 + T )^{12} \)
$29$ \( -\)\(13\!\cdots\!00\)\( - \)\(79\!\cdots\!60\)\( T - \)\(12\!\cdots\!60\)\( T^{2} + \)\(36\!\cdots\!64\)\( T^{3} + \)\(86\!\cdots\!66\)\( T^{4} - \)\(48\!\cdots\!91\)\( T^{5} - \)\(11\!\cdots\!95\)\( T^{6} + 26858754721293882269 T^{7} + 6008664212793413 T^{8} - 630597838301 T^{9} - 131011465 T^{10} + 5127 T^{11} + T^{12} \)
$31$ \( \)\(74\!\cdots\!92\)\( + \)\(12\!\cdots\!32\)\( T - \)\(20\!\cdots\!56\)\( T^{2} - \)\(70\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!24\)\( T^{4} + \)\(86\!\cdots\!04\)\( T^{5} - \)\(17\!\cdots\!27\)\( T^{6} - 41240723419361106250 T^{7} + 8050726425219211 T^{8} + 865109094984 T^{9} - 150123073 T^{10} - 6498 T^{11} + T^{12} \)
$37$ \( -\)\(27\!\cdots\!72\)\( - \)\(56\!\cdots\!00\)\( T - \)\(37\!\cdots\!48\)\( T^{2} - \)\(72\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!36\)\( T^{4} + \)\(98\!\cdots\!67\)\( T^{5} + \)\(13\!\cdots\!31\)\( T^{6} + 74007330296769844507 T^{7} - 143603714853346825 T^{8} - 10731767012275 T^{9} + 95220385 T^{10} + 35545 T^{11} + T^{12} \)
$41$ \( \)\(77\!\cdots\!92\)\( - \)\(77\!\cdots\!44\)\( T - \)\(28\!\cdots\!56\)\( T^{2} + \)\(20\!\cdots\!72\)\( T^{3} + \)\(46\!\cdots\!67\)\( T^{4} - \)\(89\!\cdots\!82\)\( T^{5} - \)\(14\!\cdots\!54\)\( T^{6} + \)\(10\!\cdots\!50\)\( T^{7} + 158708489466911868 T^{8} - 4887176120286 T^{9} - 679555382 T^{10} + 7806 T^{11} + T^{12} \)
$43$ \( -\)\(17\!\cdots\!92\)\( - \)\(42\!\cdots\!88\)\( T - \)\(45\!\cdots\!28\)\( T^{2} - \)\(27\!\cdots\!32\)\( T^{3} - \)\(99\!\cdots\!74\)\( T^{4} - \)\(22\!\cdots\!79\)\( T^{5} - \)\(32\!\cdots\!86\)\( T^{6} - \)\(23\!\cdots\!09\)\( T^{7} + 8252011501022519 T^{8} + 18840137061099 T^{9} + 1689326085 T^{10} + 66142 T^{11} + T^{12} \)
$47$ \( \)\(45\!\cdots\!00\)\( + \)\(34\!\cdots\!60\)\( T - \)\(41\!\cdots\!84\)\( T^{2} + \)\(20\!\cdots\!12\)\( T^{3} + \)\(49\!\cdots\!16\)\( T^{4} - \)\(22\!\cdots\!76\)\( T^{5} - \)\(92\!\cdots\!08\)\( T^{6} + \)\(32\!\cdots\!68\)\( T^{7} + 534915538064932224 T^{8} - 13570993384622 T^{9} - 1230785607 T^{10} + 16432 T^{11} + T^{12} \)
$53$ \( \)\(89\!\cdots\!00\)\( - \)\(53\!\cdots\!72\)\( T - \)\(13\!\cdots\!72\)\( T^{2} - \)\(38\!\cdots\!96\)\( T^{3} - \)\(23\!\cdots\!86\)\( T^{4} + \)\(36\!\cdots\!71\)\( T^{5} + \)\(47\!\cdots\!36\)\( T^{6} + \)\(33\!\cdots\!39\)\( T^{7} - 1776969645274209695 T^{8} - 78341136608841 T^{9} + 193220443 T^{10} + 67456 T^{11} + T^{12} \)
$59$ \( \)\(10\!\cdots\!44\)\( - \)\(35\!\cdots\!84\)\( T + \)\(33\!\cdots\!72\)\( T^{2} + \)\(76\!\cdots\!40\)\( T^{3} - \)\(15\!\cdots\!06\)\( T^{4} - \)\(66\!\cdots\!09\)\( T^{5} - \)\(44\!\cdots\!26\)\( T^{6} + \)\(19\!\cdots\!49\)\( T^{7} + 3859408941353981263 T^{8} - 152513724283091 T^{9} - 3783279907 T^{10} + 36346 T^{11} + T^{12} \)
$61$ \( \)\(20\!\cdots\!32\)\( + \)\(88\!\cdots\!72\)\( T - \)\(16\!\cdots\!56\)\( T^{2} - \)\(41\!\cdots\!56\)\( T^{3} + \)\(56\!\cdots\!60\)\( T^{4} - \)\(93\!\cdots\!79\)\( T^{5} - \)\(15\!\cdots\!50\)\( T^{6} + \)\(93\!\cdots\!01\)\( T^{7} + 15431293817518847515 T^{8} - 58670604280583 T^{9} - 6655398457 T^{10} + 8768 T^{11} + T^{12} \)
$67$ \( -\)\(12\!\cdots\!44\)\( + \)\(40\!\cdots\!36\)\( T + \)\(58\!\cdots\!40\)\( T^{2} - \)\(74\!\cdots\!64\)\( T^{3} - \)\(74\!\cdots\!56\)\( T^{4} + \)\(11\!\cdots\!54\)\( T^{5} + \)\(22\!\cdots\!53\)\( T^{6} + \)\(31\!\cdots\!15\)\( T^{7} - 16057148574268452018 T^{8} - 447417655583581 T^{9} + 863160402 T^{10} + 123617 T^{11} + T^{12} \)
$71$ \( -\)\(12\!\cdots\!76\)\( + \)\(10\!\cdots\!56\)\( T - \)\(21\!\cdots\!48\)\( T^{2} + \)\(84\!\cdots\!16\)\( T^{3} + \)\(15\!\cdots\!72\)\( T^{4} - \)\(11\!\cdots\!62\)\( T^{5} - \)\(23\!\cdots\!75\)\( T^{6} + \)\(32\!\cdots\!33\)\( T^{7} - 8405207090465346 T^{8} - 297716154558539 T^{9} - 552515994 T^{10} + 108667 T^{11} + T^{12} \)
$73$ \( \)\(12\!\cdots\!00\)\( - \)\(20\!\cdots\!60\)\( T - \)\(92\!\cdots\!40\)\( T^{2} + \)\(17\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!44\)\( T^{4} - \)\(38\!\cdots\!08\)\( T^{5} - \)\(28\!\cdots\!31\)\( T^{6} + \)\(31\!\cdots\!98\)\( T^{7} + 23366057028265708287 T^{8} - 1017754485714352 T^{9} - 8428102737 T^{10} + 107406 T^{11} + T^{12} \)
$79$ \( \)\(78\!\cdots\!00\)\( + \)\(23\!\cdots\!16\)\( T + \)\(20\!\cdots\!08\)\( T^{2} + \)\(47\!\cdots\!60\)\( T^{3} - \)\(21\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!86\)\( T^{5} - \)\(31\!\cdots\!17\)\( T^{6} + \)\(52\!\cdots\!22\)\( T^{7} + 61604160266882468579 T^{8} - 855161339876970 T^{9} - 14711022731 T^{10} + 39470 T^{11} + T^{12} \)
$83$ \( -\)\(31\!\cdots\!20\)\( + \)\(29\!\cdots\!36\)\( T + \)\(17\!\cdots\!08\)\( T^{2} + \)\(79\!\cdots\!24\)\( T^{3} - \)\(27\!\cdots\!72\)\( T^{4} - \)\(46\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!05\)\( T^{6} + \)\(25\!\cdots\!30\)\( T^{7} - 43381684559265085277 T^{8} - 3912207483902620 T^{9} - 13274931289 T^{10} + 181838 T^{11} + T^{12} \)
$89$ \( \)\(11\!\cdots\!00\)\( - \)\(57\!\cdots\!56\)\( T + \)\(96\!\cdots\!96\)\( T^{2} - \)\(50\!\cdots\!32\)\( T^{3} - \)\(15\!\cdots\!08\)\( T^{4} + \)\(13\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!51\)\( T^{6} - \)\(11\!\cdots\!67\)\( T^{7} - \)\(37\!\cdots\!36\)\( T^{8} - 3085107139780255 T^{9} + 11994557548 T^{10} + 277361 T^{11} + T^{12} \)
$97$ \( \)\(37\!\cdots\!80\)\( - \)\(14\!\cdots\!16\)\( T - \)\(10\!\cdots\!60\)\( T^{2} + \)\(29\!\cdots\!16\)\( T^{3} + \)\(85\!\cdots\!58\)\( T^{4} - \)\(20\!\cdots\!97\)\( T^{5} - \)\(23\!\cdots\!27\)\( T^{6} + \)\(49\!\cdots\!63\)\( T^{7} + \)\(34\!\cdots\!25\)\( T^{8} - 4899630130401427 T^{9} - 29150372197 T^{10} + 169005 T^{11} + T^{12} \)
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