Properties

Label 16.6.e.a
Level 16
Weight 6
Character orbit 16.e
Analytic conductor 2.566
Analytic rank 0
Dimension 18
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.56614111701\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{36} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} -\beta_{4} q^{3} + ( -1 + \beta_{1} + \beta_{9} ) q^{4} + ( \beta_{5} - \beta_{6} + \beta_{12} ) q^{5} + ( 6 + 14 \beta_{1} + \beta_{10} ) q^{6} + ( -1 - 11 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{15} + \beta_{16} ) q^{7} + ( 15 + 33 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{11} + \beta_{15} - \beta_{17} ) q^{8} + ( -63 \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{5} - 5 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} -\beta_{4} q^{3} + ( -1 + \beta_{1} + \beta_{9} ) q^{4} + ( \beta_{5} - \beta_{6} + \beta_{12} ) q^{5} + ( 6 + 14 \beta_{1} + \beta_{10} ) q^{6} + ( -1 - 11 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{15} + \beta_{16} ) q^{7} + ( 15 + 33 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{11} + \beta_{15} - \beta_{17} ) q^{8} + ( -63 \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{5} - 5 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{9} + ( -24 + 26 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} + 5 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} ) q^{10} + ( -34 - 32 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 10 \beta_{6} + \beta_{7} + \beta_{8} - 7 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{11} + ( 2 - 30 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 10 \beta_{4} + \beta_{5} + 15 \beta_{6} - 6 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} - 6 \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} ) q^{12} + ( -8 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} - 3 \beta_{4} - 11 \beta_{5} - 13 \beta_{6} + \beta_{7} + 3 \beta_{8} - 8 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + 4 \beta_{16} - 2 \beta_{17} ) q^{13} + ( -10 + 22 \beta_{1} - 8 \beta_{3} + 24 \beta_{4} - 12 \beta_{5} - 14 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{14} + ( 93 + 13 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 54 \beta_{5} - 14 \beta_{6} - 3 \beta_{8} - 9 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{15} + ( -52 + 98 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + 16 \beta_{4} + 17 \beta_{5} + 15 \beta_{6} - 2 \beta_{7} + 30 \beta_{8} - \beta_{9} - 5 \beta_{11} + 6 \beta_{12} + \beta_{13} + 2 \beta_{14} + 4 \beta_{16} - 2 \beta_{17} ) q^{16} + ( 6 + 4 \beta_{1} + 20 \beta_{2} + \beta_{3} + 5 \beta_{4} - 7 \beta_{5} + 61 \beta_{6} - 8 \beta_{7} - 8 \beta_{8} + 14 \beta_{9} + 5 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} + 5 \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + 3 \beta_{17} ) q^{17} + ( -170 - 158 \beta_{1} + 12 \beta_{2} - 30 \beta_{4} - 4 \beta_{5} - 37 \beta_{6} + 8 \beta_{7} - 48 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} + 12 \beta_{12} - 4 \beta_{13} + 6 \beta_{14} + 2 \beta_{15} + 2 \beta_{17} ) q^{18} + ( -132 + 122 \beta_{1} - 9 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} + 18 \beta_{5} - 88 \beta_{6} + 7 \beta_{7} + 10 \beta_{8} + 19 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 8 \beta_{13} - 9 \beta_{14} + 3 \beta_{15} + \beta_{16} + \beta_{17} ) q^{19} + ( 176 + 118 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} - 64 \beta_{4} + 14 \beta_{5} + 28 \beta_{6} + 2 \beta_{7} + 32 \beta_{8} - 3 \beta_{9} - 6 \beta_{11} + 8 \beta_{12} + 4 \beta_{13} - 14 \beta_{14} - 2 \beta_{15} - 4 \beta_{16} + 2 \beta_{17} ) q^{20} + ( 64 + 4 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} - 9 \beta_{4} + 174 \beta_{5} + 56 \beta_{6} + 9 \beta_{7} + 7 \beta_{8} + 40 \beta_{9} + 2 \beta_{11} - 6 \beta_{12} + 8 \beta_{13} + 7 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} - 4 \beta_{17} ) q^{21} + ( 238 - 298 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} - 66 \beta_{4} + 10 \beta_{5} - 38 \beta_{6} - 14 \beta_{7} + 38 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} + 10 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 4 \beta_{15} + \beta_{16} ) q^{22} + ( 41 + 335 \beta_{1} + 27 \beta_{2} - 8 \beta_{3} + \beta_{4} + 150 \beta_{5} + 98 \beta_{6} - 2 \beta_{7} - 16 \beta_{8} + 8 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} - 4 \beta_{12} - 10 \beta_{13} + 16 \beta_{14} + 7 \beta_{15} - 3 \beta_{16} - 2 \beta_{17} ) q^{23} + ( 484 - 434 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 30 \beta_{4} - 11 \beta_{5} + 7 \beta_{6} - 12 \beta_{7} - 120 \beta_{8} + \beta_{9} - 4 \beta_{10} - 5 \beta_{11} - 2 \beta_{12} + 5 \beta_{13} + 16 \beta_{14} - 8 \beta_{15} + 4 \beta_{16} + 6 \beta_{17} ) q^{24} + ( -12 + 317 \beta_{1} - 36 \beta_{2} - 8 \beta_{3} + 40 \beta_{4} + 146 \beta_{5} - 262 \beta_{6} - 26 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} + 6 \beta_{13} - 8 \beta_{14} + 10 \beta_{15} + 2 \beta_{16} - 6 \beta_{17} ) q^{25} + ( 366 + 352 \beta_{1} + 15 \beta_{2} - 2 \beta_{3} + 31 \beta_{4} - 14 \beta_{5} - 42 \beta_{6} + 27 \beta_{7} + 99 \beta_{8} - 23 \beta_{9} + 5 \beta_{10} + 9 \beta_{11} - 2 \beta_{12} + \beta_{13} + 13 \beta_{14} - 16 \beta_{15} + 7 \beta_{16} + 2 \beta_{17} ) q^{26} + ( 230 + 292 \beta_{1} + \beta_{2} - 14 \beta_{3} + 2 \beta_{4} - 184 \beta_{5} + 370 \beta_{6} + 17 \beta_{7} + 34 \beta_{8} - 29 \beta_{9} + 19 \beta_{10} - 8 \beta_{11} - 22 \beta_{12} - 2 \beta_{13} - 33 \beta_{14} - 5 \beta_{15} + 5 \beta_{16} - 9 \beta_{17} ) q^{27} + ( -432 - 952 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 116 \beta_{4} - 30 \beta_{5} + 8 \beta_{6} + 14 \beta_{7} + 8 \beta_{8} - 6 \beta_{9} - 20 \beta_{10} - 10 \beta_{11} + 24 \beta_{12} - 38 \beta_{14} - 2 \beta_{15} + 16 \beta_{16} - 14 \beta_{17} ) q^{28} + ( 172 - 188 \beta_{1} + 12 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} - 405 \beta_{5} - 161 \beta_{6} + 34 \beta_{7} - 46 \beta_{8} - 4 \beta_{9} + 16 \beta_{10} + 8 \beta_{11} - 12 \beta_{12} + 4 \beta_{13} + 18 \beta_{14} + 4 \beta_{15} - 28 \beta_{16} + 16 \beta_{17} ) q^{29} + ( -1706 + 514 \beta_{1} + 26 \beta_{2} + 36 \beta_{3} + 146 \beta_{4} + 34 \beta_{5} - 12 \beta_{6} - 40 \beta_{7} - 56 \beta_{8} - 40 \beta_{9} - 5 \beta_{10} + 8 \beta_{11} + 24 \beta_{12} - 10 \beta_{13} - 14 \beta_{14} + 14 \beta_{15} - 7 \beta_{16} + 18 \beta_{17} ) q^{30} + ( -710 + 22 \beta_{1} + 8 \beta_{2} + 36 \beta_{3} - 8 \beta_{4} - 494 \beta_{5} - 198 \beta_{6} - 16 \beta_{7} - 40 \beta_{8} + 6 \beta_{9} - 20 \beta_{10} + 6 \beta_{11} - 8 \beta_{12} - 10 \beta_{13} + 50 \beta_{14} + 14 \beta_{15} + 14 \beta_{16} + 8 \beta_{17} ) q^{31} + ( -1318 + 246 \beta_{1} - 6 \beta_{2} - 20 \beta_{3} + 54 \beta_{4} - 102 \beta_{5} + 32 \beta_{6} - 24 \beta_{7} + 58 \beta_{8} + 16 \beta_{9} - 4 \beta_{10} + 10 \beta_{11} - 32 \beta_{12} + 4 \beta_{13} + 52 \beta_{14} + 2 \beta_{15} - 36 \beta_{16} + 14 \beta_{17} ) q^{32} + ( 22 + 80 \beta_{1} - 16 \beta_{2} - 5 \beta_{3} + 89 \beta_{4} - 243 \beta_{5} + 581 \beta_{6} - 48 \beta_{7} + 112 \beta_{8} - 18 \beta_{9} - 39 \beta_{10} - 9 \beta_{11} + 19 \beta_{12} - 7 \beta_{13} - 25 \beta_{14} + 7 \beta_{15} + 7 \beta_{16} - 25 \beta_{17} ) q^{33} + ( -78 - 2134 \beta_{1} - 74 \beta_{2} - 12 \beta_{3} - 80 \beta_{4} - 8 \beta_{5} + 92 \beta_{6} + 54 \beta_{7} - 138 \beta_{8} - 4 \beta_{9} + 12 \beta_{10} - 12 \beta_{11} - 64 \beta_{12} + 18 \beta_{13} + 16 \beta_{14} - 14 \beta_{15} - 2 \beta_{16} - 18 \beta_{17} ) q^{34} + ( 412 - 600 \beta_{1} - 10 \beta_{2} - 56 \beta_{3} - 8 \beta_{4} + 280 \beta_{5} - 720 \beta_{6} + 16 \beta_{7} + 54 \beta_{8} - 34 \beta_{9} - 6 \beta_{10} - 16 \beta_{12} + 24 \beta_{13} - 64 \beta_{14} - 22 \beta_{15} - 10 \beta_{16} - 6 \beta_{17} ) q^{35} + ( 323 + 3207 \beta_{1} + 13 \beta_{2} + 20 \beta_{3} - 128 \beta_{4} - 78 \beta_{5} - 122 \beta_{6} + 36 \beta_{7} + 152 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 6 \beta_{11} - 28 \beta_{12} - 14 \beta_{13} - 56 \beta_{14} + 16 \beta_{15} + 36 \beta_{16} - 12 \beta_{17} ) q^{36} + ( -492 - 608 \beta_{1} + 40 \beta_{2} + 14 \beta_{3} - 69 \beta_{4} + 681 \beta_{5} + 357 \beta_{6} + 73 \beta_{7} - 165 \beta_{8} - 84 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} + 5 \beta_{12} - 16 \beta_{13} + 23 \beta_{14} - 18 \beta_{15} + 18 \beta_{16} + 32 \beta_{17} ) q^{37} + ( 3010 - 250 \beta_{1} + 54 \beta_{2} - 60 \beta_{3} - 30 \beta_{4} - 66 \beta_{5} + 98 \beta_{6} - 70 \beta_{7} + 54 \beta_{8} + 34 \beta_{9} + 3 \beta_{10} - 30 \beta_{11} - 12 \beta_{12} - 6 \beta_{13} - 38 \beta_{14} + 32 \beta_{15} - 6 \beta_{16} - 4 \beta_{17} ) q^{38} + ( 147 - 1327 \beta_{1} - 59 \beta_{2} + 72 \beta_{3} + 17 \beta_{4} + 1064 \beta_{5} + 280 \beta_{6} - 48 \beta_{7} - 84 \beta_{8} - 24 \beta_{9} - 16 \beta_{10} + 24 \beta_{11} + 40 \beta_{12} + 24 \beta_{13} + 80 \beta_{14} - 13 \beta_{15} - 19 \beta_{16} + 16 \beta_{17} ) q^{39} + ( 4230 - 700 \beta_{1} + 25 \beta_{2} + 58 \beta_{3} + 16 \beta_{4} + 127 \beta_{5} + 159 \beta_{6} - 20 \beta_{7} - 142 \beta_{8} + 13 \beta_{9} + 40 \beta_{10} + 21 \beta_{11} - 2 \beta_{12} - 15 \beta_{13} + 92 \beta_{14} + 18 \beta_{15} - 32 \beta_{16} - 4 \beta_{17} ) q^{40} + ( -132 - 168 \beta_{1} + 136 \beta_{2} + 18 \beta_{3} - 24 \beta_{4} + 252 \beta_{5} - 960 \beta_{6} - 68 \beta_{7} + 160 \beta_{8} - 12 \beta_{9} - 4 \beta_{10} + 10 \beta_{12} - 24 \beta_{13} - 40 \beta_{14} - 40 \beta_{15} + 32 \beta_{16} + 4 \beta_{17} ) q^{41} + ( 1976 + 5504 \beta_{1} - 100 \beta_{2} - 56 \beta_{3} - 64 \beta_{4} + 26 \beta_{5} + 82 \beta_{6} + 80 \beta_{7} - 32 \beta_{8} + 132 \beta_{9} + 4 \beta_{10} - 44 \beta_{11} + 56 \beta_{12} + 4 \beta_{13} + 24 \beta_{14} + 40 \beta_{15} - 8 \beta_{16} - 16 \beta_{17} ) q^{42} + ( -760 - 704 \beta_{1} - 44 \beta_{2} + 24 \beta_{3} - 32 \beta_{4} - 240 \beta_{5} + 792 \beta_{6} + 12 \beta_{7} + 69 \beta_{8} + 140 \beta_{9} - 20 \beta_{10} + 32 \beta_{11} + 104 \beta_{12} + 24 \beta_{13} - 76 \beta_{14} - 4 \beta_{15} + 4 \beta_{16} + 28 \beta_{17} ) q^{43} + ( -2226 - 4678 \beta_{1} + 18 \beta_{2} - 34 \beta_{3} - 94 \beta_{4} + 299 \beta_{5} - 297 \beta_{6} + 42 \beta_{7} + 78 \beta_{8} - 2 \beta_{9} + 78 \beta_{10} + 37 \beta_{11} + 18 \beta_{12} - 7 \beta_{13} - 68 \beta_{14} + 14 \beta_{15} - 42 \beta_{16} + 20 \beta_{17} ) q^{44} + ( 196 - 304 \beta_{1} - 72 \beta_{2} - 11 \beta_{3} - 325 \beta_{4} - 1185 \beta_{5} - 399 \beta_{6} + 107 \beta_{7} + 21 \beta_{8} + 108 \beta_{9} - 42 \beta_{10} - 24 \beta_{11} + 6 \beta_{12} - 26 \beta_{13} + 27 \beta_{14} - 36 \beta_{15} + 48 \beta_{16} - 42 \beta_{17} ) q^{45} + ( -5062 + 3106 \beta_{1} - 120 \beta_{2} + 8 \beta_{3} + 24 \beta_{4} + 116 \beta_{5} + 230 \beta_{6} - 102 \beta_{7} + 150 \beta_{8} + 222 \beta_{9} - 3 \beta_{10} - 50 \beta_{11} - 44 \beta_{12} + 40 \beta_{13} - 56 \beta_{14} - 22 \beta_{15} + 9 \beta_{16} - 54 \beta_{17} ) q^{46} + ( 2338 + 6 \beta_{1} + 160 \beta_{2} - 124 \beta_{3} + 76 \beta_{4} - 898 \beta_{5} - 554 \beta_{6} - 64 \beta_{7} - 120 \beta_{8} + 54 \beta_{9} + 24 \beta_{10} - 6 \beta_{11} + 80 \beta_{12} + 58 \beta_{13} + 86 \beta_{14} - 22 \beta_{15} - 22 \beta_{16} - 20 \beta_{17} ) q^{47} + ( -8262 + 4298 \beta_{1} + 56 \beta_{2} - 40 \beta_{3} + 50 \beta_{4} + 332 \beta_{5} - 394 \beta_{6} - 24 \beta_{7} - 270 \beta_{8} - 30 \beta_{9} + 36 \beta_{10} + 24 \beta_{11} + 12 \beta_{12} - 26 \beta_{13} + 108 \beta_{14} - 14 \beta_{15} + 116 \beta_{16} - 22 \beta_{17} ) q^{48} + ( -709 + 384 \beta_{1} - 192 \beta_{2} - 6 \beta_{3} - 166 \beta_{4} - 718 \beta_{5} + 1098 \beta_{6} - 80 \beta_{7} - 304 \beta_{8} - 148 \beta_{9} + 90 \beta_{10} - 10 \beta_{11} - 14 \beta_{12} - 38 \beta_{13} - 42 \beta_{14} - 10 \beta_{15} - 10 \beta_{16} + 70 \beta_{17} ) q^{49} + ( -4712 - 8104 \beta_{1} + 232 \beta_{2} + 64 \beta_{3} + 32 \beta_{4} - 8 \beta_{5} + 197 \beta_{6} + 88 \beta_{7} + 208 \beta_{8} + 80 \beta_{9} - 4 \beta_{10} - 32 \beta_{11} + 48 \beta_{12} - 24 \beta_{13} + 24 \beta_{14} + 24 \beta_{15} + 12 \beta_{16} + 56 \beta_{17} ) q^{50} + ( -328 - 74 \beta_{1} + 167 \beta_{2} + 202 \beta_{3} - 74 \beta_{4} + 750 \beta_{5} - 936 \beta_{6} + 29 \beta_{7} + 104 \beta_{8} - 125 \beta_{9} + \beta_{10} + 26 \beta_{11} + 38 \beta_{12} + 16 \beta_{13} - 67 \beta_{14} + 39 \beta_{15} + 37 \beta_{16} + \beta_{17} ) q^{51} + ( 5170 + 7452 \beta_{1} + 45 \beta_{2} - 16 \beta_{3} + 312 \beta_{4} + 200 \beta_{5} + 458 \beta_{6} + 30 \beta_{7} - 184 \beta_{8} + 19 \beta_{9} - 36 \beta_{10} + 56 \beta_{11} - 52 \beta_{12} - 2 \beta_{13} - 70 \beta_{14} - 34 \beta_{15} - 120 \beta_{16} + 6 \beta_{17} ) q^{52} + ( 1424 + 1220 \beta_{1} - 4 \beta_{2} - 18 \beta_{3} - 153 \beta_{4} + 773 \beta_{5} + 289 \beta_{6} + 113 \beta_{7} + 567 \beta_{8} - 240 \beta_{9} - 40 \beta_{10} - 30 \beta_{11} + 33 \beta_{12} - 56 \beta_{13} + 31 \beta_{14} + 58 \beta_{15} - 58 \beta_{16} - 76 \beta_{17} ) q^{53} + ( 11512 - 6084 \beta_{1} - 378 \beta_{2} + 36 \beta_{3} + 658 \beta_{4} + 86 \beta_{5} + 150 \beta_{6} - 98 \beta_{7} - 86 \beta_{8} - 98 \beta_{9} + 46 \beta_{10} - 18 \beta_{11} - 52 \beta_{12} + 58 \beta_{13} - 62 \beta_{14} - 76 \beta_{15} + 2 \beta_{16} + 32 \beta_{17} ) q^{54} + ( 103 + 1957 \beta_{1} - 199 \beta_{2} - 232 \beta_{3} + 121 \beta_{4} + 548 \beta_{5} + 428 \beta_{6} - 36 \beta_{7} - 44 \beta_{8} - 40 \beta_{9} + 36 \beta_{10} - 4 \beta_{11} - 160 \beta_{12} + 52 \beta_{13} + 80 \beta_{14} - 13 \beta_{15} + 85 \beta_{16} - 36 \beta_{17} ) q^{55} + ( 10358 - 6186 \beta_{1} + 8 \beta_{2} - 72 \beta_{3} + 246 \beta_{4} - 384 \beta_{5} - 1230 \beta_{6} - 28 \beta_{7} + 570 \beta_{8} - 50 \beta_{9} - 152 \beta_{10} + 36 \beta_{11} + 60 \beta_{12} - 14 \beta_{13} + 84 \beta_{14} + 18 \beta_{15} + 80 \beta_{16} - 38 \beta_{17} ) q^{56} + ( 112 + 70 \beta_{1} + 82 \beta_{2} + 23 \beta_{3} + 416 \beta_{4} + 51 \beta_{5} - 467 \beta_{6} - 39 \beta_{7} - 535 \beta_{8} + 112 \beta_{9} - 39 \beta_{10} - 9 \beta_{11} - 55 \beta_{12} - 41 \beta_{13} - 32 \beta_{14} + 73 \beta_{15} - 151 \beta_{16} + 39 \beta_{17} ) q^{57} + ( 5922 + 11592 \beta_{1} + 205 \beta_{2} + 186 \beta_{3} - 471 \beta_{4} + 200 \beta_{5} + 264 \beta_{6} + 37 \beta_{7} - 635 \beta_{8} - 381 \beta_{9} - 53 \beta_{10} + 3 \beta_{11} - 70 \beta_{12} + 19 \beta_{13} - 5 \beta_{14} - 51 \beta_{16} + 38 \beta_{17} ) q^{58} + ( -1488 - 1756 \beta_{1} + 50 \beta_{2} + 40 \beta_{3} - 106 \beta_{4} - 64 \beta_{5} + 832 \beta_{6} + 12 \beta_{7} + 35 \beta_{8} + 214 \beta_{9} - 94 \beta_{10} - 8 \beta_{11} - 272 \beta_{12} + 24 \beta_{13} - 28 \beta_{14} + 54 \beta_{15} - 54 \beta_{16} - 14 \beta_{17} ) q^{59} + ( -8608 - 18016 \beta_{1} + 50 \beta_{2} + 96 \beta_{3} - 840 \beta_{4} - 1332 \beta_{5} + 574 \beta_{6} + 14 \beta_{7} + 132 \beta_{8} + 10 \beta_{9} - 136 \beta_{10} + 60 \beta_{11} - 148 \beta_{12} - 22 \beta_{13} - 2 \beta_{14} - 18 \beta_{15} + 20 \beta_{16} + 70 \beta_{17} ) q^{60} + ( -2756 + 2784 \beta_{1} - 128 \beta_{2} - 29 \beta_{3} + 797 \beta_{4} - 167 \beta_{5} - 201 \beta_{6} + 37 \beta_{7} + 83 \beta_{8} - 36 \beta_{9} + 18 \beta_{10} - 40 \beta_{11} + 90 \beta_{12} - 54 \beta_{13} - 11 \beta_{14} + 108 \beta_{15} + 72 \beta_{16} + 18 \beta_{17} ) q^{61} + ( -15412 + 6620 \beta_{1} + 156 \beta_{2} - 216 \beta_{3} - 676 \beta_{4} - 460 \beta_{5} - 64 \beta_{6} + 8 \beta_{7} + 384 \beta_{8} - 584 \beta_{9} + 48 \beta_{10} - 24 \beta_{11} + 16 \beta_{12} - 12 \beta_{13} - 28 \beta_{14} - 52 \beta_{15} + 44 \beta_{16} + 20 \beta_{17} ) q^{62} + ( -719 + 57 \beta_{1} - 32 \beta_{2} + 196 \beta_{3} + 172 \beta_{4} + 94 \beta_{5} + 502 \beta_{6} + 21 \beta_{8} + 131 \beta_{9} + 107 \beta_{10} + 26 \beta_{11} - 304 \beta_{12} + 26 \beta_{13} - 10 \beta_{14} - 54 \beta_{15} - 54 \beta_{16} - \beta_{17} ) q^{63} + ( -15708 + 6136 \beta_{1} + 38 \beta_{2} + 164 \beta_{3} - 508 \beta_{4} - 1282 \beta_{5} + 614 \beta_{6} + 36 \beta_{7} - 688 \beta_{8} - 114 \beta_{9} - 112 \beta_{10} - 30 \beta_{11} + 204 \beta_{12} - 14 \beta_{13} - 36 \beta_{14} + 20 \beta_{15} - 120 \beta_{16} - 56 \beta_{17} ) q^{64} + ( 1370 - 164 \beta_{1} + 76 \beta_{2} + 66 \beta_{3} + 800 \beta_{4} + 416 \beta_{5} - 436 \beta_{6} + 40 \beta_{7} + 744 \beta_{8} + 296 \beta_{9} + 32 \beta_{10} + 56 \beta_{11} - 138 \beta_{12} + 32 \beta_{13} + 28 \beta_{14} - 36 \beta_{15} - 36 \beta_{16} - 40 \beta_{17} ) q^{65} + ( -8634 - 20546 \beta_{1} - 566 \beta_{2} - 36 \beta_{3} + 96 \beta_{4} + 252 \beta_{5} - 484 \beta_{6} - 62 \beta_{7} + 986 \beta_{8} - 276 \beta_{9} - 104 \beta_{10} + 36 \beta_{11} + 256 \beta_{12} - 2 \beta_{13} + 8 \beta_{14} + 38 \beta_{15} - 2 \beta_{16} - 38 \beta_{17} ) q^{66} + ( -3912 + 4354 \beta_{1} - 51 \beta_{2} - 338 \beta_{3} - \beta_{4} - 662 \beta_{5} + 632 \beta_{6} - 25 \beta_{7} - 112 \beta_{8} + 145 \beta_{9} + 59 \beta_{10} - 2 \beta_{11} + 18 \beta_{12} - 80 \beta_{13} + 71 \beta_{14} + 77 \beta_{15} - 41 \beta_{16} + 59 \beta_{17} ) q^{67} + ( 7058 + 20738 \beta_{1} - 50 \beta_{2} - 52 \beta_{3} + 840 \beta_{4} - 594 \beta_{5} - 2130 \beta_{6} - 64 \beta_{7} - 576 \beta_{8} - 24 \beta_{9} + 116 \beta_{10} - 22 \beta_{11} + 244 \beta_{12} + 58 \beta_{13} + 68 \beta_{14} - 52 \beta_{15} + 156 \beta_{16} + 88 \beta_{17} ) q^{68} + ( 1288 + 1460 \beta_{1} - 36 \beta_{2} - 86 \beta_{3} + 285 \beta_{4} - 818 \beta_{5} - 524 \beta_{6} - 141 \beta_{7} - 1363 \beta_{8} + 376 \beta_{9} + 144 \beta_{10} - 26 \beta_{11} - 62 \beta_{12} + 104 \beta_{13} - 35 \beta_{14} - 58 \beta_{15} + 58 \beta_{16} - 28 \beta_{17} ) q^{69} + ( 22756 - 8692 \beta_{1} + 860 \beta_{2} + 200 \beta_{3} + 572 \beta_{4} + 196 \beta_{5} - 436 \beta_{6} + 124 \beta_{7} - 492 \beta_{8} + 260 \beta_{9} - 74 \beta_{10} + 68 \beta_{11} + 232 \beta_{12} - 28 \beta_{13} + 108 \beta_{14} - 32 \beta_{15} + 52 \beta_{16} - 72 \beta_{17} ) q^{70} + ( -191 - 617 \beta_{1} + 259 \beta_{2} + 248 \beta_{3} + 41 \beta_{4} - 1290 \beta_{5} - 574 \beta_{6} + 94 \beta_{7} + 80 \beta_{8} + 200 \beta_{9} + 34 \beta_{10} - 102 \beta_{11} + 316 \beta_{12} - 106 \beta_{13} - 176 \beta_{14} + 63 \beta_{15} + 5 \beta_{16} - 34 \beta_{17} ) q^{71} + ( 26391 - 9281 \beta_{1} - 76 \beta_{2} - 296 \beta_{3} + 129 \beta_{4} + 650 \beta_{5} + 2883 \beta_{6} + 24 \beta_{7} + 831 \beta_{8} - 109 \beta_{9} + 232 \beta_{10} - 108 \beta_{11} - 70 \beta_{12} + 59 \beta_{13} - 224 \beta_{14} - 127 \beta_{15} - 8 \beta_{16} + 57 \beta_{17} ) q^{72} + ( 192 + 580 \beta_{1} - 510 \beta_{2} - 165 \beta_{3} - 1168 \beta_{4} - 29 \beta_{5} + 2053 \beta_{6} + 97 \beta_{7} + 1169 \beta_{8} + 65 \beta_{10} - 17 \beta_{11} - 35 \beta_{12} + 127 \beta_{13} + 112 \beta_{14} - 15 \beta_{15} + 145 \beta_{16} - 65 \beta_{17} ) q^{73} + ( 8272 + 23566 \beta_{1} - 225 \beta_{2} - 18 \beta_{3} + 13 \beta_{4} - 578 \beta_{5} - 330 \beta_{6} - 157 \beta_{7} - 497 \beta_{8} + 743 \beta_{9} + 17 \beta_{10} + 119 \beta_{11} - 238 \beta_{12} - 63 \beta_{13} - 5 \beta_{14} - 146 \beta_{15} + 133 \beta_{16} + 16 \beta_{17} ) q^{74} + ( 8236 + 8452 \beta_{1} + 40 \beta_{2} - 148 \beta_{3} + 158 \beta_{4} + 816 \beta_{5} - 2572 \beta_{6} + 14 \beta_{7} - 103 \beta_{8} - 712 \beta_{9} + 172 \beta_{10} - 104 \beta_{11} + 404 \beta_{12} - 108 \beta_{13} + 162 \beta_{14} - 24 \beta_{15} + 24 \beta_{16} - 124 \beta_{17} ) q^{75} + ( -4690 - 23418 \beta_{1} - 94 \beta_{2} + 122 \beta_{3} - 70 \beta_{4} + 3325 \beta_{5} - 617 \beta_{6} - 124 \beta_{7} + 394 \beta_{8} + 2 \beta_{9} + 50 \beta_{10} - 205 \beta_{11} - 22 \beta_{12} + 57 \beta_{13} + 98 \beta_{14} - 84 \beta_{15} + 70 \beta_{16} - 202 \beta_{17} ) q^{76} + ( 2804 - 2696 \beta_{1} + 616 \beta_{2} + 102 \beta_{3} - 1691 \beta_{4} + 3148 \beta_{5} + 1178 \beta_{6} - 315 \beta_{7} - 213 \beta_{8} - 236 \beta_{9} + 90 \beta_{10} + 168 \beta_{11} - 134 \beta_{12} + 138 \beta_{13} - 11 \beta_{14} - 44 \beta_{15} - 224 \beta_{16} + 90 \beta_{17} ) q^{77} + ( -34626 + 9518 \beta_{1} - 160 \beta_{2} + 88 \beta_{3} - 952 \beta_{4} - 60 \beta_{5} - 902 \beta_{6} + 262 \beta_{7} - 526 \beta_{8} + 874 \beta_{9} - 23 \beta_{10} + 234 \beta_{11} + 44 \beta_{12} - 96 \beta_{13} + 208 \beta_{14} + 150 \beta_{15} - 127 \beta_{16} + 214 \beta_{17} ) q^{78} + ( -2484 - 84 \beta_{1} - 584 \beta_{2} - 136 \beta_{3} - 444 \beta_{4} + 2608 \beta_{5} + 144 \beta_{6} + 240 \beta_{7} + 448 \beta_{8} - 356 \beta_{9} - 252 \beta_{10} + 472 \beta_{12} - 272 \beta_{13} - 232 \beta_{14} + 168 \beta_{15} + 168 \beta_{16} + 84 \beta_{17} ) q^{79} + ( -30242 + 11002 \beta_{1} - 350 \beta_{2} + 132 \beta_{3} - 434 \beta_{4} + 4294 \beta_{5} - 592 \beta_{6} - 4 \beta_{7} + 18 \beta_{8} + 268 \beta_{9} + 84 \beta_{10} - 34 \beta_{11} - 288 \beta_{12} + 148 \beta_{13} - 328 \beta_{14} + 70 \beta_{15} - 132 \beta_{16} + 186 \beta_{17} ) q^{80} + ( -507 - 1280 \beta_{1} + 736 \beta_{2} - 25 \beta_{3} - 2031 \beta_{4} + 2485 \beta_{5} - 3547 \beta_{6} + 256 \beta_{7} - 1408 \beta_{8} + 62 \beta_{9} - 367 \beta_{10} - 49 \beta_{11} + 247 \beta_{12} + 113 \beta_{13} + 111 \beta_{14} + 111 \beta_{15} + 111 \beta_{16} - 145 \beta_{17} ) q^{81} + ( -5340 - 27908 \beta_{1} + 960 \beta_{2} - 320 \beta_{3} + 788 \beta_{4} - 240 \beta_{5} - 500 \beta_{6} - 296 \beta_{7} - 64 \beta_{8} + 524 \beta_{9} + 152 \beta_{10} + 108 \beta_{11} - 376 \beta_{12} + 96 \beta_{13} - 140 \beta_{14} - 132 \beta_{15} - 116 \beta_{16} - 164 \beta_{17} ) q^{82} + ( 13044 - 11684 \beta_{1} - 780 \beta_{2} + 244 \beta_{3} + 317 \beta_{4} - 2148 \beta_{5} + 4480 \beta_{6} - 98 \beta_{7} - 358 \beta_{8} + 372 \beta_{9} - 84 \beta_{10} - 132 \beta_{11} - 204 \beta_{12} + 40 \beta_{13} + 238 \beta_{14} - 288 \beta_{15} - 120 \beta_{16} - 84 \beta_{17} ) q^{83} + ( 11556 + 37968 \beta_{1} - 150 \beta_{2} + 104 \beta_{3} - 24 \beta_{4} + 1940 \beta_{5} + 5388 \beta_{6} - 64 \beta_{7} + 648 \beta_{8} - 66 \beta_{9} - 120 \beta_{10} - 244 \beta_{11} + 136 \beta_{12} - 108 \beta_{13} + 312 \beta_{14} + 288 \beta_{15} + 24 \beta_{16} - 120 \beta_{17} ) q^{84} + ( -8388 - 7528 \beta_{1} - 16 \beta_{2} + 238 \beta_{3} + 323 \beta_{4} - 3998 \beta_{5} - 1028 \beta_{6} - 471 \beta_{7} + 2435 \beta_{8} + 788 \beta_{9} - 148 \beta_{10} + 194 \beta_{11} - 34 \beta_{12} + 128 \beta_{13} - 169 \beta_{14} - 90 \beta_{15} + 90 \beta_{16} + 328 \beta_{17} ) q^{85} + ( 26406 - 9678 \beta_{1} - 888 \beta_{2} - 144 \beta_{3} - 1288 \beta_{4} - 472 \beta_{5} + 40 \beta_{6} + 520 \beta_{7} - 72 \beta_{8} - 536 \beta_{9} - 120 \beta_{10} + 40 \beta_{11} - 48 \beta_{12} - 264 \beta_{13} + 216 \beta_{14} + 368 \beta_{15} - 77 \beta_{16} - 64 \beta_{17} ) q^{86} + ( -799 + 1815 \beta_{1} + 835 \beta_{2} + 184 \beta_{3} - 767 \beta_{4} - 5210 \beta_{5} - 2510 \beta_{6} + 174 \beta_{7} + 328 \beta_{8} - 120 \beta_{9} - 238 \beta_{10} + 170 \beta_{11} - 292 \beta_{12} - 154 \beta_{13} - 400 \beta_{14} - 49 \beta_{15} - 427 \beta_{16} + 238 \beta_{17} ) q^{87} + ( 32246 - 11740 \beta_{1} + 97 \beta_{2} + 530 \beta_{3} - 152 \beta_{4} - 1969 \beta_{5} - 3973 \beta_{6} + 264 \beta_{7} - 578 \beta_{8} + 429 \beta_{9} + 52 \beta_{10} - 99 \beta_{11} - 354 \beta_{12} - 11 \beta_{13} - 444 \beta_{14} + 110 \beta_{15} - 220 \beta_{16} + 96 \beta_{17} ) q^{88} + ( -432 - 2692 \beta_{1} - 34 \beta_{2} + 173 \beta_{3} + 1776 \beta_{4} - 1971 \beta_{5} + 3707 \beta_{6} + 431 \beta_{7} - 1937 \beta_{8} - 720 \beta_{9} + 79 \beta_{10} + 145 \beta_{11} + 331 \beta_{12} + 97 \beta_{13} + 176 \beta_{14} - 209 \beta_{15} + 367 \beta_{16} - 79 \beta_{17} ) q^{89} + ( 14986 + 42880 \beta_{1} + 269 \beta_{2} - 678 \beta_{3} + 1629 \beta_{4} - 262 \beta_{5} - 930 \beta_{6} - 431 \beta_{7} + 793 \beta_{8} - 1093 \beta_{9} + 255 \beta_{10} - 37 \beta_{11} + 346 \beta_{12} - 29 \beta_{13} - 105 \beta_{14} + 176 \beta_{15} + 101 \beta_{16} - 186 \beta_{17} ) q^{90} + ( -13844 - 13052 \beta_{1} - 120 \beta_{2} + 44 \beta_{3} + 478 \beta_{4} - 16 \beta_{5} - 5964 \beta_{6} - 274 \beta_{7} - 600 \beta_{8} - 744 \beta_{9} + 204 \beta_{10} + 152 \beta_{11} - 236 \beta_{12} - 108 \beta_{13} + 450 \beta_{14} - 248 \beta_{15} + 248 \beta_{16} + 292 \beta_{17} ) q^{91} + ( -12400 - 24848 \beta_{1} - 302 \beta_{2} - 572 \beta_{3} + 1332 \beta_{4} - 5914 \beta_{5} + 2516 \beta_{6} - 202 \beta_{7} - 440 \beta_{8} + 86 \beta_{9} + 196 \beta_{10} - 158 \beta_{11} + 320 \beta_{12} + 132 \beta_{13} + 474 \beta_{14} + 222 \beta_{15} - 56 \beta_{16} - 70 \beta_{17} ) q^{92} + ( 10336 - 9912 \beta_{1} - 128 \beta_{2} - 40 \beta_{3} + 3934 \beta_{4} + 5868 \beta_{5} + 2912 \beta_{6} - 610 \beta_{7} + 98 \beta_{8} - 248 \beta_{9} - 132 \beta_{10} - 260 \beta_{12} + 300 \beta_{13} - 226 \beta_{14} - 392 \beta_{15} - 128 \beta_{16} - 132 \beta_{17} ) q^{93} + ( -26100 + 13788 \beta_{1} + 620 \beta_{2} + 552 \beta_{3} + 828 \beta_{4} + 1620 \beta_{5} + 208 \beta_{6} + 488 \beta_{7} - 176 \beta_{8} - 760 \beta_{9} - 192 \beta_{10} + 24 \beta_{11} - 304 \beta_{12} + 68 \beta_{13} + 340 \beta_{14} + 76 \beta_{15} - 36 \beta_{16} - 364 \beta_{17} ) q^{94} + ( -13121 - 269 \beta_{1} - 176 \beta_{2} + 32 \beta_{3} - 808 \beta_{4} + 5400 \beta_{5} + 4024 \beta_{6} + 160 \beta_{7} + 221 \beta_{8} - 979 \beta_{9} - 181 \beta_{10} - 280 \beta_{11} + 16 \beta_{12} + 8 \beta_{13} - 664 \beta_{14} + 24 \beta_{15} + 24 \beta_{16} - 133 \beta_{17} ) q^{95} + ( -34208 + 15428 \beta_{1} - 66 \beta_{2} - 540 \beta_{3} + 1872 \beta_{4} - 8642 \beta_{5} + 3426 \beta_{6} + 212 \beta_{7} + 1660 \beta_{8} + 370 \beta_{9} + 248 \beta_{10} - 134 \beta_{11} - 412 \beta_{12} - 34 \beta_{13} - 588 \beta_{14} - 208 \beta_{15} + 336 \beta_{16} - 28 \beta_{17} ) q^{96} + ( -1026 - 644 \beta_{1} + 108 \beta_{2} - 371 \beta_{3} + 2025 \beta_{4} + 957 \beta_{5} - 6463 \beta_{6} + 408 \beta_{7} + 1912 \beta_{8} - 458 \beta_{9} + 233 \beta_{10} - 65 \beta_{11} + 377 \beta_{12} + 41 \beta_{13} + 291 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} + 239 \beta_{17} ) q^{97} + ( -24764 - 28812 \beta_{1} - 916 \beta_{2} + 424 \beta_{3} + 560 \beta_{4} - 1097 \beta_{5} + 728 \beta_{6} - 452 \beta_{7} - 2052 \beta_{8} - 648 \beta_{9} + 344 \beta_{10} + 40 \beta_{11} - 256 \beta_{12} - 284 \beta_{13} - 144 \beta_{14} + 4 \beta_{15} + 156 \beta_{16} + 380 \beta_{17} ) q^{98} + ( -16460 + 17008 \beta_{1} + 574 \beta_{2} - 48 \beta_{3} + 141 \beta_{4} - 3520 \beta_{5} + 4672 \beta_{6} - 60 \beta_{7} - 142 \beta_{8} + 118 \beta_{9} - 222 \beta_{10} + 24 \beta_{11} + 232 \beta_{12} - 120 \beta_{13} + 596 \beta_{14} + 10 \beta_{15} + 454 \beta_{16} - 222 \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 2q^{2} - 2q^{3} - 24q^{4} - 2q^{5} + 112q^{6} + 244q^{8} + O(q^{10}) \) \( 18q - 2q^{2} - 2q^{3} - 24q^{4} - 2q^{5} + 112q^{6} + 244q^{8} - 436q^{10} - 606q^{11} + 4q^{12} - 2q^{13} - 100q^{14} + 1796q^{15} - 872q^{16} - 4q^{17} - 3138q^{18} - 2362q^{19} + 2972q^{20} + 484q^{21} + 4420q^{22} + 8368q^{24} + 7368q^{26} + 4216q^{27} - 7336q^{28} + 4070q^{29} - 30444q^{30} - 11536q^{31} - 23992q^{32} - 4q^{33} - 1740q^{34} + 8636q^{35} + 6892q^{36} - 10650q^{37} + 53248q^{38} + 75272q^{40} + 33400q^{42} - 15382q^{43} - 40124q^{44} + 5762q^{45} - 92532q^{46} + 44176q^{47} - 147992q^{48} - 14410q^{49} - 85050q^{50} - 2748q^{51} + 91572q^{52} + 24726q^{53} + 208672q^{54} + 191128q^{56} + 106776q^{58} - 29734q^{59} - 154368q^{60} - 48082q^{61} - 273872q^{62} - 12156q^{63} - 283776q^{64} + 27684q^{65} - 153356q^{66} - 75210q^{67} + 133712q^{68} + 22804q^{69} + 412160q^{70} + 470244q^{72} + 147148q^{74} + 154726q^{75} - 87468q^{76} + 41060q^{77} - 631780q^{78} - 52864q^{79} - 554456q^{80} - 13126q^{81} - 93216q^{82} + 227838q^{83} + 190888q^{84} - 138652q^{85} + 470468q^{86} + 590328q^{88} + 280152q^{90} - 231164q^{91} - 221896q^{92} + 180688q^{93} - 460912q^{94} - 250380q^{95} - 597328q^{96} - 4q^{97} - 444646q^{98} - 296770q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 5 x^{16} - 30 x^{15} - 42 x^{14} - 344 x^{13} + 2904 x^{12} + 5344 x^{11} + 16576 x^{10} - 297472 x^{9} + 265216 x^{8} + 1368064 x^{7} + 11894784 x^{6} - 22544384 x^{5} - 44040192 x^{4} - 503316480 x^{3} - 1342177280 x^{2} + 68719476736\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-13201861 \nu^{17} + 91238416 \nu^{16} + 259875289 \nu^{15} - 1089436474 \nu^{14} - 8523891854 \nu^{13} + 7748030488 \nu^{12} + 83679829448 \nu^{11} + 198515420448 \nu^{10} - 805030604224 \nu^{9} + 850916220416 \nu^{8} - 18783609756672 \nu^{7} - 23474141765632 \nu^{6} + 233969063723008 \nu^{5} + 1913442062565376 \nu^{4} - 3867677456072704 \nu^{3} - 34017203921616896 \nu^{2} - 2999266662416384 \nu + 372969715829571584\)\()/ 72115765074984960 \)
\(\beta_{2}\)\(=\)\((\)\(13201861 \nu^{17} - 91238416 \nu^{16} - 259875289 \nu^{15} + 1089436474 \nu^{14} + 8523891854 \nu^{13} - 7748030488 \nu^{12} - 83679829448 \nu^{11} - 198515420448 \nu^{10} + 805030604224 \nu^{9} - 850916220416 \nu^{8} + 18783609756672 \nu^{7} + 23474141765632 \nu^{6} - 233969063723008 \nu^{5} - 1913442062565376 \nu^{4} + 3867677456072704 \nu^{3} + 178248734071586816 \nu^{2} + 2999266662416384 \nu - 445085480904556544\)\()/ 72115765074984960 \)
\(\beta_{3}\)\(=\)\((\)\(30864949 \nu^{17} + 4287808496 \nu^{16} + 405905399 \nu^{15} - 80922119654 \nu^{14} - 265530836434 \nu^{13} + 1016058503528 \nu^{12} + 3686732212408 \nu^{11} + 4374252934368 \nu^{10} - 38841459719744 \nu^{9} - 45927172242944 \nu^{8} - 615259217175552 \nu^{7} + 2339001225945088 \nu^{6} + 16559834536706048 \nu^{5} + 34978214750191616 \nu^{4} - 334527409940332544 \nu^{3} - 1185336518357549056 \nu^{2} + 1770383364142923776 \nu + 16937359649542242304\)\()/ 72115765074984960 \)
\(\beta_{4}\)\(=\)\((\)\(181944863 \nu^{17} + 3770692432 \nu^{16} + 1392584293 \nu^{15} - 76307038258 \nu^{14} - 210630119798 \nu^{13} + 664682282296 \nu^{12} + 3768513696296 \nu^{11} - 136072789344 \nu^{10} - 32588294826688 \nu^{9} - 79890776939008 \nu^{8} - 523798932286464 \nu^{7} + 1761597748207616 \nu^{6} + 15697308294086656 \nu^{5} + 26201952319700992 \nu^{4} - 284310039615766528 \nu^{3} - 1142944001574305792 \nu^{2} + 1853141050470694912 \nu + 13338946344266498048\)\()/ 216347295224954880 \)
\(\beta_{5}\)\(=\)\((\)\(5702401 \nu^{17} + 12116624 \nu^{16} - 92843269 \nu^{15} - 567398126 \nu^{14} + 200411894 \nu^{13} + 7626127112 \nu^{12} + 16816635352 \nu^{11} - 36637284768 \nu^{10} - 192266735936 \nu^{9} - 955141561856 \nu^{8} - 338321937408 \nu^{7} + 24437646794752 \nu^{6} + 100988389916672 \nu^{5} - 278068121829376 \nu^{4} - 2541369883099136 \nu^{3} - 1294906534395904 \nu^{2} + 27817842556534784 \nu + 56701561241337856\)\()/ 4507235317186560 \)
\(\beta_{6}\)\(=\)\((\)\(5702401 \nu^{17} + 12116624 \nu^{16} - 92843269 \nu^{15} - 567398126 \nu^{14} + 200411894 \nu^{13} + 7626127112 \nu^{12} + 16816635352 \nu^{11} - 36637284768 \nu^{10} - 192266735936 \nu^{9} - 955141561856 \nu^{8} - 338321937408 \nu^{7} + 24437646794752 \nu^{6} + 100988389916672 \nu^{5} - 278068121829376 \nu^{4} - 2541369883099136 \nu^{3} - 1294906534395904 \nu^{2} + 18803371922161664 \nu + 56701561241337856\)\()/ 4507235317186560 \)
\(\beta_{7}\)\(=\)\((\)\(-703222783 \nu^{17} + 4318122928 \nu^{16} + 6981618427 \nu^{15} - 58664962702 \nu^{14} - 227923298762 \nu^{13} + 988419494344 \nu^{12} + 1930095055064 \nu^{11} - 3252011695776 \nu^{10} - 50778013064512 \nu^{9} + 164964206402048 \nu^{8} - 850253144626176 \nu^{7} + 518152851759104 \nu^{6} + 7514444433424384 \nu^{5} + 55852397067501568 \nu^{4} - 272021626229358592 \nu^{3} - 941396046557216768 \nu^{2} + 2916881926620971008 \nu + 15576764639243927552\)\()/ 216347295224954880 \)
\(\beta_{8}\)\(=\)\((\)\(784516657 \nu^{17} - 1545631312 \nu^{16} - 15933947893 \nu^{15} - 6288779822 \nu^{14} + 352966098518 \nu^{13} + 170768905544 \nu^{12} - 1305597828776 \nu^{11} - 9730883388576 \nu^{10} + 17010383505088 \nu^{9} - 86644942833152 \nu^{8} + 633468139121664 \nu^{7} + 1955863923073024 \nu^{6} - 244763139014656 \nu^{5} - 96660297232678912 \nu^{4} - 21309971687800832 \nu^{3} + 1009372263499169792 \nu^{2} + 2417451640404574208 \nu - 10899933372914597888\)\()/ 216347295224954880 \)
\(\beta_{9}\)\(=\)\((\)\(400933829 \nu^{17} - 2149838864 \nu^{16} - 12942310361 \nu^{15} + 15166644026 \nu^{14} + 315331989646 \nu^{13} + 471580648 \nu^{12} - 2231229132232 \nu^{11} - 9375786937632 \nu^{10} + 24522248793536 \nu^{9} - 60072993693184 \nu^{8} + 551148322495488 \nu^{7} + 1084580117454848 \nu^{6} - 5018321190551552 \nu^{5} - 75200963604905984 \nu^{4} + 54274265119195136 \nu^{3} + 1024872893617537024 \nu^{2} + 1817449226385227776 \nu - 12840566362237566976\)\()/ 72115765074984960 \)
\(\beta_{10}\)\(=\)\((\)\(-278430493 \nu^{17} + 979507088 \nu^{16} + 7114535057 \nu^{15} - 22861838442 \nu^{14} - 134985082142 \nu^{13} + 3045184344 \nu^{12} + 1919528950664 \nu^{11} + 3876721636384 \nu^{10} - 14524706021312 \nu^{9} + 18485561732608 \nu^{8} - 267728141433856 \nu^{7} - 495130029367296 \nu^{6} + 6473894101417984 \nu^{5} + 30448715629068288 \nu^{4} - 31736969845276672 \nu^{3} - 689838803384270848 \nu^{2} - 385507295255068672 \nu + 8241986350614577152\)\()/ 36057882537492480 \)
\(\beta_{11}\)\(=\)\((\)\(133304867 \nu^{17} + 2322371728 \nu^{16} + 1572507217 \nu^{15} - 54604769002 \nu^{14} - 132197478302 \nu^{13} + 509948203864 \nu^{12} + 2618106519944 \nu^{11} + 1394565317664 \nu^{10} - 23568479768512 \nu^{9} - 56679327764992 \nu^{8} - 289187212901376 \nu^{7} + 911647272034304 \nu^{6} + 11688586753245184 \nu^{5} + 22471425532100608 \nu^{4} - 191088117994749952 \nu^{3} - 771391626294591488 \nu^{2} + 1101423482270384128 \nu + 10111461747706560512\)\()/ 12019294179164160 \)
\(\beta_{12}\)\(=\)\((\)\(-339823113 \nu^{17} + 513251408 \nu^{16} + 8617678637 \nu^{15} + 3874526078 \nu^{14} - 120444071142 \nu^{13} - 168692659976 \nu^{12} + 775323397224 \nu^{11} + 3221155665824 \nu^{10} - 1276115978432 \nu^{9} + 34811506882048 \nu^{8} - 227344372714496 \nu^{7} - 1414073750265856 \nu^{6} + 878999404314624 \nu^{5} + 33913771993858048 \nu^{4} + 46391114720083968 \nu^{3} - 455553755757150208 \nu^{2} - 965168413448077312 \nu + 3500354911235735552\)\()/ 24038588358328320 \)
\(\beta_{13}\)\(=\)\((\)\(-135483127 \nu^{17} - 41254928 \nu^{16} + 2857566163 \nu^{15} + 4671877442 \nu^{14} - 40114246298 \nu^{13} - 52481279864 \nu^{12} + 227613114776 \nu^{11} + 842151513696 \nu^{10} - 3561016299328 \nu^{9} + 18346694277632 \nu^{8} - 6482662339584 \nu^{7} - 494710542229504 \nu^{6} - 745126631604224 \nu^{5} + 11038041400410112 \nu^{4} + 22041263668723712 \nu^{3} - 124642540862308352 \nu^{2} - 332201759822839808 \nu + 1679910156420251648\)\()/ 9014470634373120 \)
\(\beta_{14}\)\(=\)\((\)\(-676748947 \nu^{17} - 2336932112 \nu^{16} + 14995692511 \nu^{15} + 63668656778 \nu^{14} - 61730827010 \nu^{13} - 988339623896 \nu^{12} - 1491948299272 \nu^{11} + 6452007349728 \nu^{10} + 20075425115072 \nu^{9} + 107006521184768 \nu^{8} + 36296436612096 \nu^{7} - 3634479344607232 \nu^{6} - 13570694585679872 \nu^{5} + 35399765828042752 \nu^{4} + 326356912874455040 \nu^{3} + 27390125755334656 \nu^{2} - 3622643046790725632 \nu - 2727212952020058112\)\()/ 43269459044990976 \)
\(\beta_{15}\)\(=\)\((\)\(1070983715 \nu^{17} + 3551126800 \nu^{16} - 17360551343 \nu^{15} - 128743565674 \nu^{14} + 15669155170 \nu^{13} + 1698843612760 \nu^{12} + 4076178324872 \nu^{11} - 6670954606560 \nu^{10} - 51230629216192 \nu^{9} - 205236453552640 \nu^{8} + 68878265140224 \nu^{7} + 5698611387392000 \nu^{6} + 23580548355751936 \nu^{5} - 45807396237344768 \nu^{4} - 617340957211230208 \nu^{3} - 538624601303810048 \nu^{2} + 5367946270725898240 \nu + 15491506454631809024\)\()/ 43269459044990976 \)
\(\beta_{16}\)\(=\)\((\)\(2776978621 \nu^{17} + 6335690864 \nu^{16} - 46386583729 \nu^{15} - 284647188566 \nu^{14} + 220029366494 \nu^{13} + 3342257733032 \nu^{12} + 5654358668152 \nu^{11} - 21591862921248 \nu^{10} - 79001759967296 \nu^{9} - 506385941229056 \nu^{8} + 484097789770752 \nu^{7} + 11766033659502592 \nu^{6} + 52533098246930432 \nu^{5} - 161989701277843456 \nu^{4} - 1192991192583766016 \nu^{3} - 380379545410207744 \nu^{2} + 12146433258552295424 \nu + 18861912813884932096\)\()/ 108173647612477440 \)
\(\beta_{17}\)\(=\)\((\)\(1381341305 \nu^{17} - 1975945424 \nu^{16} - 31615730525 \nu^{15} - 40067920414 \nu^{14} + 524826800518 \nu^{13} + 672054041992 \nu^{12} - 1647572062696 \nu^{11} - 21643209245088 \nu^{10} + 31402450980032 \nu^{9} - 162300985718272 \nu^{8} + 1121497624022016 \nu^{7} + 4259920401539072 \nu^{6} + 3009588912160768 \nu^{5} - 158711379550797824 \nu^{4} - 195230238540562432 \nu^{3} + 1493669443438379008 \nu^{2} + 5980859018745217024 \nu - 16890318702405222400\)\()/ 43269459044990976 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{15} + \beta_{13} - 2 \beta_{12} + \beta_{11} + \beta_{9} + 2 \beta_{8} - \beta_{6} + 3 \beta_{5} + 2 \beta_{3} + \beta_{2} - 48 \beta_{1} + 18\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{17} - 4 \beta_{16} - 2 \beta_{14} - \beta_{13} - 6 \beta_{12} + 5 \beta_{11} + \beta_{9} - 30 \beta_{8} + 2 \beta_{7} - 15 \beta_{6} - 17 \beta_{5} - 16 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} - 98 \beta_{1} + 52\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-6 \beta_{17} + 16 \beta_{16} - 8 \beta_{15} - 38 \beta_{14} + 3 \beta_{13} + 26 \beta_{12} - 7 \beta_{11} + 20 \beta_{10} - 11 \beta_{9} - 2 \beta_{8} - 14 \beta_{7} - 67 \beta_{6} + 35 \beta_{5} - 56 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} - 782 \beta_{1} + 536\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(10 \beta_{17} - 56 \beta_{16} + 28 \beta_{15} + 18 \beta_{14} - 15 \beta_{13} - 82 \beta_{12} + 7 \beta_{11} + 60 \beta_{10} + 19 \beta_{9} - 254 \beta_{8} + 18 \beta_{7} - 641 \beta_{6} - 307 \beta_{5} + 344 \beta_{4} + 102 \beta_{3} + 57 \beta_{2} - 7854 \beta_{1} - 3068\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(182 \beta_{17} + 96 \beta_{16} + 76 \beta_{15} + 6 \beta_{14} + 211 \beta_{13} - 166 \beta_{12} + 109 \beta_{11} - 84 \beta_{10} - 303 \beta_{9} - 34 \beta_{8} - 242 \beta_{7} - 2307 \beta_{6} - 5153 \beta_{5} - 920 \beta_{4} + 242 \beta_{3} + 227 \beta_{2} + 38 \beta_{1} - 3476\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(1010 \beta_{17} - 512 \beta_{16} - 404 \beta_{15} - 430 \beta_{14} - 275 \beta_{13} - 410 \beta_{12} + 75 \beta_{11} + 900 \beta_{10} - 4953 \beta_{9} + 3026 \beta_{8} + 2218 \beta_{7} + 2243 \beta_{6} + 105 \beta_{5} - 4984 \beta_{4} + 30 \beta_{3} - 1987 \beta_{2} - 31182 \beta_{1} - 26228\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(5102 \beta_{17} - 4208 \beta_{16} + 2084 \beta_{15} + 1870 \beta_{14} - 4181 \beta_{13} + 5514 \beta_{12} + 941 \beta_{11} - 1236 \beta_{10} - 1919 \beta_{9} - 19138 \beta_{8} - 2810 \beta_{7} + 7173 \beta_{6} - 24769 \beta_{5} - 14104 \beta_{4} + 7026 \beta_{3} + 3003 \beta_{2} - 106210 \beta_{1} + 469236\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-4110 \beta_{17} - 784 \beta_{16} + 8348 \beta_{15} + 10978 \beta_{14} - 9115 \beta_{13} - 2090 \beta_{12} + 355 \beta_{11} + 1652 \beta_{10} - 11041 \beta_{9} + 74066 \beta_{8} - 16534 \beta_{7} - 300469 \beta_{6} + 238097 \beta_{5} - 108920 \beta_{4} + 21486 \beta_{3} - 6395 \beta_{2} + 763458 \beta_{1} - 1015764\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(34718 \beta_{17} - 127920 \beta_{16} + 32692 \beta_{15} - 7138 \beta_{14} + 18091 \beta_{13} - 22006 \beta_{12} + 13501 \beta_{11} + 26956 \beta_{10} - 2799 \beta_{9} + 172382 \beta_{8} - 68970 \beta_{7} + 765189 \beta_{6} - 155089 \beta_{5} + 68872 \beta_{4} + 3410 \beta_{3} + 258299 \beta_{2} + 2394222 \beta_{1} - 4342012\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-21966 \beta_{17} - 106288 \beta_{16} - 143236 \beta_{15} + 418 \beta_{14} + 157877 \beta_{13} + 225014 \beta_{12} + 163059 \beta_{11} - 317420 \beta_{10} + 342799 \beta_{9} + 375858 \beta_{8} + 102666 \beta_{7} + 3534651 \beta_{6} - 122047 \beta_{5} - 1805144 \beta_{4} + 652750 \beta_{3} - 196715 \beta_{2} - 12553310 \beta_{1} - 14192084\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-324130 \beta_{17} - 1139888 \beta_{16} + 1048372 \beta_{15} + 14526 \beta_{14} - 811765 \beta_{13} - 191926 \beta_{12} + 734493 \beta_{11} + 164876 \beta_{10} + 960465 \beta_{9} + 2695742 \beta_{8} + 1368758 \beta_{7} + 2071525 \beta_{6} - 12055281 \beta_{5} - 3902680 \beta_{4} - 1316974 \beta_{3} - 2185477 \beta_{2} + 9215918 \beta_{1} + 44096964\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-2823918 \beta_{17} - 1272752 \beta_{16} + 3039964 \beta_{15} - 8962910 \beta_{14} + 1841685 \beta_{13} + 9383990 \beta_{12} - 3609549 \beta_{11} - 1489772 \beta_{10} - 1818705 \beta_{9} + 15342290 \beta_{8} - 6542326 \beta_{7} - 24814821 \beta_{6} + 36732737 \beta_{5} + 4285576 \beta_{4} - 1412530 \beta_{3} - 10531307 \beta_{2} - 72797118 \beta_{1} + 283053772\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-6858690 \beta_{17} + 182032 \beta_{16} + 17500116 \beta_{15} + 18498622 \beta_{14} - 7369013 \beta_{13} + 27341642 \beta_{12} - 2094531 \beta_{11} - 11058868 \beta_{10} - 6790383 \beta_{9} + 56444702 \beta_{8} + 24766582 \beta_{7} - 211531547 \beta_{6} + 84564399 \beta_{5} + 46561480 \beta_{4} + 12309074 \beta_{3} + 28938075 \beta_{2} - 1685424754 \beta_{1} - 1544346524\)\()/4\)
\(\nu^{16}\)\(=\)\((\)\(-8987918 \beta_{17} + 13321744 \beta_{16} + 4235900 \beta_{15} - 8775454 \beta_{14} + 13943317 \beta_{13} - 10480970 \beta_{12} + 11391891 \beta_{11} - 87933868 \beta_{10} - 46118289 \beta_{9} + 531615218 \beta_{8} + 913290 \beta_{7} - 262775973 \beta_{6} - 1557007583 \beta_{5} + 161424360 \beta_{4} + 51179150 \beta_{3} + 132021109 \beta_{2} + 3217323938 \beta_{1} + 5144753644\)\()/4\)
\(\nu^{17}\)\(=\)\((\)\(51345502 \beta_{17} - 57938480 \beta_{16} - 172198732 \beta_{15} + 8208510 \beta_{14} - 27333365 \beta_{13} - 27121334 \beta_{12} + 135330845 \beta_{11} - 298852340 \beta_{10} - 1052894319 \beta_{9} + 1098028286 \beta_{8} + 192817142 \beta_{7} - 1348078491 \beta_{6} + 4399848591 \beta_{5} - 137731480 \beta_{4} - 223402094 \beta_{3} - 647892293 \beta_{2} - 14223027282 \beta_{1} + 1912341572\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−1.58716 + 3.67164i
−3.75808 + 1.36997i
0.592893 + 3.95582i
−3.62794 1.68466i
2.81847 + 2.83835i
−2.63220 3.01190i
0.562915 3.96019i
3.95408 0.604357i
3.67701 1.57467i
−1.58716 3.67164i
−3.75808 1.36997i
0.592893 3.95582i
−3.62794 + 1.68466i
2.81847 2.83835i
−2.63220 + 3.01190i
0.562915 + 3.96019i
3.95408 + 0.604357i
3.67701 + 1.57467i
−5.25880 + 2.08448i −3.18000 + 3.18000i 23.3098 21.9238i −67.3647 67.3647i 10.0943 23.3517i 148.379i −76.8820 + 163.882i 222.775i 494.678 + 213.836i
5.2 −5.12805 2.38811i −4.57839 + 4.57839i 20.5939 + 24.4927i 48.5981 + 48.5981i 34.4119 12.5445i 106.338i −47.1150 174.780i 201.077i −133.156 365.272i
5.3 −3.36292 + 4.54871i 16.8936 16.8936i −9.38150 30.5939i 66.0049 + 66.0049i 20.0322 + 133.656i 75.3048i 170.712 + 60.2112i 327.790i −522.206 + 78.2676i
5.4 −1.94328 5.31260i 13.2148 13.2148i −24.4473 + 20.6477i −26.0955 26.0955i −95.8850 44.5248i 106.802i 157.201 + 89.7545i 106.262i −87.9240 + 189.346i
5.5 −0.0198738 + 5.65682i −8.39316 + 8.39316i −31.9992 0.224845i −8.40645 8.40645i −47.3118 47.6454i 149.265i 1.90786 181.009i 102.110i 47.7208 47.3867i
5.6 0.379698 5.64410i −20.6438 + 20.6438i −31.7117 4.28611i −28.3274 28.3274i 108.677 + 124.354i 55.5494i −36.2321 + 177.356i 609.330i −170.638 + 149.127i
5.7 4.52311 3.39728i 3.84023 3.84023i 8.91700 30.7325i 8.73514 + 8.73514i 4.32344 30.4161i 28.0117i −64.0743 169.300i 213.505i 69.1857 + 9.83427i
5.8 4.55844 + 3.34972i 14.5768 14.5768i 9.55871 + 30.5390i −33.9573 33.9573i 115.276 17.6192i 141.886i −58.7245 + 171.229i 181.969i −41.0447 268.540i
5.9 5.25168 + 2.10235i −12.7302 + 12.7302i 23.1603 + 22.0817i 39.8132 + 39.8132i −93.6182 + 40.0917i 248.565i 75.2071 + 164.657i 81.1158i 125.385 + 292.787i
13.1 −5.25880 2.08448i −3.18000 3.18000i 23.3098 + 21.9238i −67.3647 + 67.3647i 10.0943 + 23.3517i 148.379i −76.8820 163.882i 222.775i 494.678 213.836i
13.2 −5.12805 + 2.38811i −4.57839 4.57839i 20.5939 24.4927i 48.5981 48.5981i 34.4119 + 12.5445i 106.338i −47.1150 + 174.780i 201.077i −133.156 + 365.272i
13.3 −3.36292 4.54871i 16.8936 + 16.8936i −9.38150 + 30.5939i 66.0049 66.0049i 20.0322 133.656i 75.3048i 170.712 60.2112i 327.790i −522.206 78.2676i
13.4 −1.94328 + 5.31260i 13.2148 + 13.2148i −24.4473 20.6477i −26.0955 + 26.0955i −95.8850 + 44.5248i 106.802i 157.201 89.7545i 106.262i −87.9240 189.346i
13.5 −0.0198738 5.65682i −8.39316 8.39316i −31.9992 + 0.224845i −8.40645 + 8.40645i −47.3118 + 47.6454i 149.265i 1.90786 + 181.009i 102.110i 47.7208 + 47.3867i
13.6 0.379698 + 5.64410i −20.6438 20.6438i −31.7117 + 4.28611i −28.3274 + 28.3274i 108.677 124.354i 55.5494i −36.2321 177.356i 609.330i −170.638 149.127i
13.7 4.52311 + 3.39728i 3.84023 + 3.84023i 8.91700 + 30.7325i 8.73514 8.73514i 4.32344 + 30.4161i 28.0117i −64.0743 + 169.300i 213.505i 69.1857 9.83427i
13.8 4.55844 3.34972i 14.5768 + 14.5768i 9.55871 30.5390i −33.9573 + 33.9573i 115.276 + 17.6192i 141.886i −58.7245 171.229i 181.969i −41.0447 + 268.540i
13.9 5.25168 2.10235i −12.7302 12.7302i 23.1603 22.0817i 39.8132 39.8132i −93.6182 40.0917i 248.565i 75.2071 164.657i 81.1158i 125.385 292.787i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.6.e.a 18
3.b odd 2 1 144.6.k.a 18
4.b odd 2 1 64.6.e.a 18
8.b even 2 1 128.6.e.b 18
8.d odd 2 1 128.6.e.a 18
12.b even 2 1 576.6.k.a 18
16.e even 4 1 inner 16.6.e.a 18
16.e even 4 1 128.6.e.b 18
16.f odd 4 1 64.6.e.a 18
16.f odd 4 1 128.6.e.a 18
32.g even 8 2 1024.6.a.k 18
32.h odd 8 2 1024.6.a.l 18
48.i odd 4 1 144.6.k.a 18
48.k even 4 1 576.6.k.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.6.e.a 18 1.a even 1 1 trivial
16.6.e.a 18 16.e even 4 1 inner
64.6.e.a 18 4.b odd 2 1
64.6.e.a 18 16.f odd 4 1
128.6.e.a 18 8.d odd 2 1
128.6.e.a 18 16.f odd 4 1
128.6.e.b 18 8.b even 2 1
128.6.e.b 18 16.e even 4 1
144.6.k.a 18 3.b odd 2 1
144.6.k.a 18 48.i odd 4 1
576.6.k.a 18 12.b even 2 1
576.6.k.a 18 48.k even 4 1
1024.6.a.k 18 32.g even 8 2
1024.6.a.l 18 32.h odd 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 14 T^{2} - 56 T^{3} + 152 T^{4} + 4256 T^{5} + 61632 T^{6} + 162304 T^{7} + 691200 T^{8} - 6119424 T^{9} + 22118400 T^{10} + 166199296 T^{11} + 2019557376 T^{12} + 4462739456 T^{13} + 5100273664 T^{14} - 60129542144 T^{15} + 481036337152 T^{16} + 2199023255552 T^{17} + 35184372088832 T^{18} \)
$3$ \( 1 + 2 T + 2 T^{2} - 1242 T^{3} - 68095 T^{4} - 1694480 T^{5} - 2481488 T^{6} - 216052656 T^{7} + 2316653620 T^{8} + 75354919544 T^{9} + 1681068240440 T^{10} + 18830919106920 T^{11} + 275841168192612 T^{12} - 818526941437680 T^{13} - 9703676745696816 T^{14} - 1168069411502412624 T^{15} - 5089895017200009810 T^{16} - \)\(12\!\cdots\!96\)\( T^{17} - \)\(25\!\cdots\!80\)\( T^{18} - \)\(31\!\cdots\!28\)\( T^{19} - \)\(30\!\cdots\!90\)\( T^{20} - \)\(16\!\cdots\!68\)\( T^{21} - \)\(33\!\cdots\!16\)\( T^{22} - \)\(69\!\cdots\!40\)\( T^{23} + \)\(56\!\cdots\!88\)\( T^{24} + \)\(94\!\cdots\!40\)\( T^{25} + \)\(20\!\cdots\!40\)\( T^{26} + \)\(22\!\cdots\!92\)\( T^{27} + \)\(16\!\cdots\!80\)\( T^{28} - \)\(37\!\cdots\!92\)\( T^{29} - \)\(10\!\cdots\!88\)\( T^{30} - \)\(17\!\cdots\!40\)\( T^{31} - \)\(17\!\cdots\!55\)\( T^{32} - \)\(75\!\cdots\!94\)\( T^{33} + \)\(29\!\cdots\!02\)\( T^{34} + \)\(71\!\cdots\!86\)\( T^{35} + \)\(87\!\cdots\!49\)\( T^{36} \)
$5$ \( 1 + 2 T + 2 T^{2} + 113130 T^{3} - 6970831 T^{4} - 777846032 T^{5} + 4857448048 T^{6} - 2837940430160 T^{7} + 18448919077076 T^{8} + 6174607723271992 T^{9} + 27638790927349432 T^{10} + 10853707976731014040 T^{11} + \)\(24\!\cdots\!24\)\( T^{12} - \)\(26\!\cdots\!12\)\( T^{13} + \)\(49\!\cdots\!28\)\( T^{14} + \)\(86\!\cdots\!80\)\( T^{15} - \)\(56\!\cdots\!34\)\( T^{16} - \)\(90\!\cdots\!88\)\( T^{17} + \)\(77\!\cdots\!92\)\( T^{18} - \)\(28\!\cdots\!00\)\( T^{19} - \)\(55\!\cdots\!50\)\( T^{20} + \)\(26\!\cdots\!00\)\( T^{21} + \)\(47\!\cdots\!00\)\( T^{22} - \)\(78\!\cdots\!00\)\( T^{23} + \)\(22\!\cdots\!00\)\( T^{24} + \)\(31\!\cdots\!00\)\( T^{25} + \)\(25\!\cdots\!00\)\( T^{26} + \)\(17\!\cdots\!00\)\( T^{27} + \)\(16\!\cdots\!00\)\( T^{28} - \)\(78\!\cdots\!00\)\( T^{29} + \)\(42\!\cdots\!00\)\( T^{30} - \)\(21\!\cdots\!00\)\( T^{31} - \)\(59\!\cdots\!75\)\( T^{32} + \)\(29\!\cdots\!50\)\( T^{33} + \)\(16\!\cdots\!50\)\( T^{34} + \)\(51\!\cdots\!50\)\( T^{35} + \)\(80\!\cdots\!25\)\( T^{36} \)
$7$ \( 1 - 144058 T^{2} + 10408827913 T^{4} - 492522532520944 T^{6} + 16882861815154133876 T^{8} - \)\(44\!\cdots\!52\)\( T^{10} + \)\(90\!\cdots\!56\)\( T^{12} - \)\(15\!\cdots\!64\)\( T^{14} + \)\(22\!\cdots\!98\)\( T^{16} - \)\(34\!\cdots\!24\)\( T^{18} + \)\(62\!\cdots\!02\)\( T^{20} - \)\(11\!\cdots\!64\)\( T^{22} + \)\(20\!\cdots\!44\)\( T^{24} - \)\(28\!\cdots\!52\)\( T^{26} + \)\(30\!\cdots\!24\)\( T^{28} - \)\(25\!\cdots\!44\)\( T^{30} + \)\(14\!\cdots\!37\)\( T^{32} - \)\(58\!\cdots\!58\)\( T^{34} + \)\(11\!\cdots\!49\)\( T^{36} \)
$11$ \( 1 + 606 T + 183618 T^{2} + 79034538 T^{3} + 37127673265 T^{4} - 7247852140784 T^{5} - 8086278408451152 T^{6} - 5838566410456886736 T^{7} - \)\(27\!\cdots\!64\)\( T^{8} - \)\(87\!\cdots\!28\)\( T^{9} - \)\(15\!\cdots\!76\)\( T^{10} - \)\(25\!\cdots\!04\)\( T^{11} + \)\(50\!\cdots\!64\)\( T^{12} + \)\(23\!\cdots\!16\)\( T^{13} + \)\(97\!\cdots\!04\)\( T^{14} + \)\(24\!\cdots\!64\)\( T^{15} - \)\(29\!\cdots\!50\)\( T^{16} - \)\(39\!\cdots\!24\)\( T^{17} - \)\(65\!\cdots\!16\)\( T^{18} - \)\(64\!\cdots\!24\)\( T^{19} - \)\(77\!\cdots\!50\)\( T^{20} + \)\(10\!\cdots\!64\)\( T^{21} + \)\(65\!\cdots\!04\)\( T^{22} + \)\(25\!\cdots\!16\)\( T^{23} + \)\(87\!\cdots\!64\)\( T^{24} - \)\(72\!\cdots\!04\)\( T^{25} - \)\(70\!\cdots\!76\)\( T^{26} - \)\(63\!\cdots\!28\)\( T^{27} - \)\(32\!\cdots\!64\)\( T^{28} - \)\(11\!\cdots\!36\)\( T^{29} - \)\(24\!\cdots\!52\)\( T^{30} - \)\(35\!\cdots\!84\)\( T^{31} + \)\(29\!\cdots\!65\)\( T^{32} + \)\(10\!\cdots\!38\)\( T^{33} + \)\(37\!\cdots\!18\)\( T^{34} + \)\(19\!\cdots\!06\)\( T^{35} + \)\(53\!\cdots\!01\)\( T^{36} \)
$13$ \( 1 + 2 T + 2 T^{2} + 153133114 T^{3} - 141881992575 T^{4} - 61264536987664 T^{5} + 11602629991678320 T^{6} - 36328073410169002704 T^{7} - \)\(23\!\cdots\!16\)\( T^{8} + \)\(19\!\cdots\!52\)\( T^{9} - \)\(20\!\cdots\!72\)\( T^{10} + \)\(28\!\cdots\!00\)\( T^{11} + \)\(71\!\cdots\!32\)\( T^{12} + \)\(62\!\cdots\!96\)\( T^{13} - \)\(72\!\cdots\!72\)\( T^{14} + \)\(12\!\cdots\!24\)\( T^{15} - \)\(47\!\cdots\!78\)\( T^{16} - \)\(61\!\cdots\!20\)\( T^{17} + \)\(11\!\cdots\!68\)\( T^{18} - \)\(22\!\cdots\!60\)\( T^{19} - \)\(66\!\cdots\!22\)\( T^{20} + \)\(62\!\cdots\!68\)\( T^{21} - \)\(13\!\cdots\!72\)\( T^{22} + \)\(44\!\cdots\!28\)\( T^{23} + \)\(18\!\cdots\!68\)\( T^{24} + \)\(28\!\cdots\!00\)\( T^{25} - \)\(72\!\cdots\!72\)\( T^{26} + \)\(25\!\cdots\!36\)\( T^{27} - \)\(11\!\cdots\!84\)\( T^{28} - \)\(67\!\cdots\!28\)\( T^{29} + \)\(79\!\cdots\!20\)\( T^{30} - \)\(15\!\cdots\!52\)\( T^{31} - \)\(13\!\cdots\!75\)\( T^{32} + \)\(53\!\cdots\!98\)\( T^{33} + \)\(26\!\cdots\!02\)\( T^{34} + \)\(96\!\cdots\!86\)\( T^{35} + \)\(17\!\cdots\!49\)\( T^{36} \)
$17$ \( ( 1 + 2 T + 6455241 T^{2} - 1592311056 T^{3} + 20528805234836 T^{4} - 8775071708574920 T^{5} + 45691426381806375332 T^{6} - \)\(22\!\cdots\!56\)\( T^{7} + \)\(80\!\cdots\!02\)\( T^{8} - \)\(37\!\cdots\!28\)\( T^{9} + \)\(11\!\cdots\!14\)\( T^{10} - \)\(45\!\cdots\!44\)\( T^{11} + \)\(13\!\cdots\!76\)\( T^{12} - \)\(35\!\cdots\!20\)\( T^{13} + \)\(11\!\cdots\!52\)\( T^{14} - \)\(13\!\cdots\!44\)\( T^{15} + \)\(75\!\cdots\!13\)\( T^{16} + \)\(33\!\cdots\!02\)\( T^{17} + \)\(23\!\cdots\!57\)\( T^{18} )^{2} \)
$19$ \( 1 + 2362 T + 2789522 T^{2} + 6676713326 T^{3} + 18957362429153 T^{4} + 27681948489238576 T^{5} + 34792033192278772784 T^{6} + \)\(80\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!32\)\( T^{8} + \)\(57\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!72\)\( T^{10} + \)\(22\!\cdots\!12\)\( T^{11} + \)\(12\!\cdots\!40\)\( T^{12} + \)\(11\!\cdots\!36\)\( T^{13} + \)\(58\!\cdots\!60\)\( T^{14} + \)\(88\!\cdots\!08\)\( T^{15} + \)\(18\!\cdots\!50\)\( T^{16} + \)\(39\!\cdots\!36\)\( T^{17} + \)\(64\!\cdots\!12\)\( T^{18} + \)\(99\!\cdots\!64\)\( T^{19} + \)\(11\!\cdots\!50\)\( T^{20} + \)\(13\!\cdots\!92\)\( T^{21} + \)\(21\!\cdots\!60\)\( T^{22} + \)\(10\!\cdots\!64\)\( T^{23} + \)\(29\!\cdots\!40\)\( T^{24} + \)\(13\!\cdots\!88\)\( T^{25} + \)\(16\!\cdots\!72\)\( T^{26} + \)\(20\!\cdots\!00\)\( T^{27} + \)\(87\!\cdots\!32\)\( T^{28} + \)\(17\!\cdots\!60\)\( T^{29} + \)\(18\!\cdots\!84\)\( T^{30} + \)\(36\!\cdots\!24\)\( T^{31} + \)\(61\!\cdots\!53\)\( T^{32} + \)\(53\!\cdots\!74\)\( T^{33} + \)\(55\!\cdots\!22\)\( T^{34} + \)\(11\!\cdots\!38\)\( T^{35} + \)\(12\!\cdots\!01\)\( T^{36} \)
$23$ \( 1 - 65704890 T^{2} + 2176003725806697 T^{4} - \)\(48\!\cdots\!28\)\( T^{6} + \)\(80\!\cdots\!92\)\( T^{8} - \)\(10\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!28\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(88\!\cdots\!26\)\( T^{16} - \)\(61\!\cdots\!80\)\( T^{18} + \)\(36\!\cdots\!74\)\( T^{20} - \)\(18\!\cdots\!20\)\( T^{22} + \)\(84\!\cdots\!72\)\( T^{24} - \)\(31\!\cdots\!92\)\( T^{26} + \)\(98\!\cdots\!08\)\( T^{28} - \)\(24\!\cdots\!28\)\( T^{30} + \)\(45\!\cdots\!53\)\( T^{32} - \)\(56\!\cdots\!90\)\( T^{34} + \)\(35\!\cdots\!49\)\( T^{36} \)
$29$ \( 1 - 4070 T + 8282450 T^{2} - 236838145262 T^{3} + 574654149127841 T^{4} + 2895398974324775728 T^{5} + \)\(11\!\cdots\!12\)\( T^{6} + \)\(73\!\cdots\!44\)\( T^{7} - \)\(73\!\cdots\!36\)\( T^{8} - \)\(40\!\cdots\!64\)\( T^{9} + \)\(35\!\cdots\!52\)\( T^{10} + \)\(33\!\cdots\!76\)\( T^{11} + \)\(20\!\cdots\!80\)\( T^{12} - \)\(10\!\cdots\!76\)\( T^{13} - \)\(30\!\cdots\!16\)\( T^{14} - \)\(12\!\cdots\!80\)\( T^{15} + \)\(55\!\cdots\!90\)\( T^{16} + \)\(24\!\cdots\!64\)\( T^{17} - \)\(24\!\cdots\!88\)\( T^{18} + \)\(51\!\cdots\!36\)\( T^{19} + \)\(23\!\cdots\!90\)\( T^{20} - \)\(11\!\cdots\!20\)\( T^{21} - \)\(53\!\cdots\!16\)\( T^{22} - \)\(36\!\cdots\!24\)\( T^{23} + \)\(15\!\cdots\!80\)\( T^{24} + \)\(51\!\cdots\!24\)\( T^{25} + \)\(11\!\cdots\!52\)\( T^{26} - \)\(25\!\cdots\!36\)\( T^{27} - \)\(96\!\cdots\!36\)\( T^{28} + \)\(19\!\cdots\!56\)\( T^{29} + \)\(63\!\cdots\!12\)\( T^{30} + \)\(32\!\cdots\!72\)\( T^{31} + \)\(13\!\cdots\!41\)\( T^{32} - \)\(11\!\cdots\!38\)\( T^{33} + \)\(81\!\cdots\!50\)\( T^{34} - \)\(81\!\cdots\!30\)\( T^{35} + \)\(41\!\cdots\!01\)\( T^{36} \)
$31$ \( ( 1 + 5768 T + 125978903 T^{2} + 474315485120 T^{3} + 6741247446930084 T^{4} + 13397153796297127648 T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(47\!\cdots\!68\)\( T^{7} + \)\(41\!\cdots\!66\)\( T^{8} - \)\(94\!\cdots\!20\)\( T^{9} + \)\(11\!\cdots\!66\)\( T^{10} - \)\(38\!\cdots\!68\)\( T^{11} + \)\(46\!\cdots\!68\)\( T^{12} + \)\(90\!\cdots\!48\)\( T^{13} + \)\(12\!\cdots\!84\)\( T^{14} + \)\(26\!\cdots\!20\)\( T^{15} + \)\(19\!\cdots\!53\)\( T^{16} + \)\(26\!\cdots\!68\)\( T^{17} + \)\(12\!\cdots\!51\)\( T^{18} )^{2} \)
$37$ \( 1 + 10650 T + 56711250 T^{2} + 917226642594 T^{3} - 532232178061711 T^{4} - \)\(12\!\cdots\!04\)\( T^{5} - \)\(89\!\cdots\!32\)\( T^{6} - \)\(15\!\cdots\!28\)\( T^{7} - \)\(15\!\cdots\!88\)\( T^{8} - \)\(22\!\cdots\!08\)\( T^{9} + \)\(23\!\cdots\!24\)\( T^{10} + \)\(55\!\cdots\!64\)\( T^{11} + \)\(14\!\cdots\!68\)\( T^{12} + \)\(86\!\cdots\!44\)\( T^{13} + \)\(37\!\cdots\!88\)\( T^{14} + \)\(35\!\cdots\!24\)\( T^{15} - \)\(38\!\cdots\!10\)\( T^{16} - \)\(59\!\cdots\!16\)\( T^{17} - \)\(31\!\cdots\!24\)\( T^{18} - \)\(40\!\cdots\!12\)\( T^{19} - \)\(18\!\cdots\!90\)\( T^{20} + \)\(11\!\cdots\!32\)\( T^{21} + \)\(86\!\cdots\!88\)\( T^{22} + \)\(13\!\cdots\!08\)\( T^{23} + \)\(16\!\cdots\!32\)\( T^{24} + \)\(42\!\cdots\!52\)\( T^{25} + \)\(12\!\cdots\!24\)\( T^{26} - \)\(82\!\cdots\!56\)\( T^{27} - \)\(40\!\cdots\!12\)\( T^{28} - \)\(27\!\cdots\!04\)\( T^{29} - \)\(11\!\cdots\!32\)\( T^{30} - \)\(10\!\cdots\!28\)\( T^{31} - \)\(31\!\cdots\!39\)\( T^{32} + \)\(37\!\cdots\!42\)\( T^{33} + \)\(16\!\cdots\!50\)\( T^{34} + \)\(21\!\cdots\!50\)\( T^{35} + \)\(13\!\cdots\!49\)\( T^{36} \)
$41$ \( 1 - 1023541794 T^{2} + 517343071862859497 T^{4} - \)\(17\!\cdots\!60\)\( T^{6} + \)\(44\!\cdots\!84\)\( T^{8} - \)\(90\!\cdots\!12\)\( T^{10} + \)\(15\!\cdots\!68\)\( T^{12} - \)\(24\!\cdots\!32\)\( T^{14} + \)\(33\!\cdots\!66\)\( T^{16} - \)\(41\!\cdots\!36\)\( T^{18} + \)\(45\!\cdots\!66\)\( T^{20} - \)\(44\!\cdots\!32\)\( T^{22} + \)\(38\!\cdots\!68\)\( T^{24} - \)\(29\!\cdots\!12\)\( T^{26} + \)\(19\!\cdots\!84\)\( T^{28} - \)\(10\!\cdots\!60\)\( T^{30} + \)\(40\!\cdots\!97\)\( T^{32} - \)\(10\!\cdots\!94\)\( T^{34} + \)\(14\!\cdots\!01\)\( T^{36} \)
$43$ \( 1 + 15382 T + 118302962 T^{2} + 3612143928786 T^{3} + 172512293409905 T^{4} - \)\(70\!\cdots\!60\)\( T^{5} - \)\(44\!\cdots\!32\)\( T^{6} - \)\(13\!\cdots\!32\)\( T^{7} - \)\(19\!\cdots\!68\)\( T^{8} + \)\(31\!\cdots\!20\)\( T^{9} + \)\(30\!\cdots\!64\)\( T^{10} + \)\(10\!\cdots\!12\)\( T^{11} + \)\(47\!\cdots\!52\)\( T^{12} + \)\(27\!\cdots\!88\)\( T^{13} + \)\(11\!\cdots\!40\)\( T^{14} + \)\(24\!\cdots\!52\)\( T^{15} - \)\(32\!\cdots\!70\)\( T^{16} - \)\(60\!\cdots\!56\)\( T^{17} - \)\(33\!\cdots\!28\)\( T^{18} - \)\(88\!\cdots\!08\)\( T^{19} - \)\(69\!\cdots\!30\)\( T^{20} + \)\(78\!\cdots\!64\)\( T^{21} + \)\(55\!\cdots\!40\)\( T^{22} + \)\(19\!\cdots\!84\)\( T^{23} + \)\(47\!\cdots\!48\)\( T^{24} + \)\(15\!\cdots\!84\)\( T^{25} + \)\(67\!\cdots\!64\)\( T^{26} + \)\(10\!\cdots\!60\)\( T^{27} - \)\(91\!\cdots\!32\)\( T^{28} - \)\(96\!\cdots\!24\)\( T^{29} - \)\(44\!\cdots\!32\)\( T^{30} - \)\(10\!\cdots\!80\)\( T^{31} + \)\(37\!\cdots\!45\)\( T^{32} + \)\(11\!\cdots\!02\)\( T^{33} + \)\(56\!\cdots\!62\)\( T^{34} + \)\(10\!\cdots\!26\)\( T^{35} + \)\(10\!\cdots\!49\)\( T^{36} \)
$47$ \( ( 1 - 22088 T + 1356331495 T^{2} - 27020989683136 T^{3} + 853008998941628452 T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(33\!\cdots\!56\)\( T^{6} - \)\(49\!\cdots\!76\)\( T^{7} + \)\(97\!\cdots\!78\)\( T^{8} - \)\(12\!\cdots\!16\)\( T^{9} + \)\(22\!\cdots\!46\)\( T^{10} - \)\(26\!\cdots\!24\)\( T^{11} + \)\(40\!\cdots\!08\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{13} + \)\(54\!\cdots\!64\)\( T^{14} - \)\(39\!\cdots\!64\)\( T^{15} + \)\(45\!\cdots\!85\)\( T^{16} - \)\(16\!\cdots\!88\)\( T^{17} + \)\(17\!\cdots\!07\)\( T^{18} )^{2} \)
$53$ \( 1 - 24726 T + 305687538 T^{2} - 10125112960046 T^{3} + 254917191364521361 T^{4} - \)\(21\!\cdots\!64\)\( T^{5} + \)\(27\!\cdots\!04\)\( T^{6} - \)\(63\!\cdots\!44\)\( T^{7} - \)\(55\!\cdots\!48\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} - \)\(14\!\cdots\!48\)\( T^{10} + \)\(52\!\cdots\!88\)\( T^{11} - \)\(14\!\cdots\!40\)\( T^{12} + \)\(86\!\cdots\!12\)\( T^{13} - \)\(39\!\cdots\!52\)\( T^{14} - \)\(92\!\cdots\!88\)\( T^{15} + \)\(25\!\cdots\!74\)\( T^{16} - \)\(44\!\cdots\!92\)\( T^{17} + \)\(49\!\cdots\!16\)\( T^{18} - \)\(18\!\cdots\!56\)\( T^{19} + \)\(44\!\cdots\!26\)\( T^{20} - \)\(67\!\cdots\!16\)\( T^{21} - \)\(11\!\cdots\!52\)\( T^{22} + \)\(11\!\cdots\!16\)\( T^{23} - \)\(79\!\cdots\!60\)\( T^{24} + \)\(11\!\cdots\!16\)\( T^{25} - \)\(13\!\cdots\!48\)\( T^{26} + \)\(52\!\cdots\!00\)\( T^{27} - \)\(91\!\cdots\!52\)\( T^{28} - \)\(43\!\cdots\!08\)\( T^{29} + \)\(78\!\cdots\!04\)\( T^{30} - \)\(26\!\cdots\!52\)\( T^{31} + \)\(12\!\cdots\!89\)\( T^{32} - \)\(21\!\cdots\!22\)\( T^{33} + \)\(26\!\cdots\!38\)\( T^{34} - \)\(90\!\cdots\!18\)\( T^{35} + \)\(15\!\cdots\!49\)\( T^{36} \)
$59$ \( 1 + 29734 T + 442055378 T^{2} - 11048743474590 T^{3} + 1067338799101245777 T^{4} + \)\(43\!\cdots\!64\)\( T^{5} + \)\(88\!\cdots\!20\)\( T^{6} - \)\(20\!\cdots\!96\)\( T^{7} + \)\(35\!\cdots\!12\)\( T^{8} + \)\(35\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!12\)\( T^{10} - \)\(12\!\cdots\!28\)\( T^{11} - \)\(11\!\cdots\!04\)\( T^{12} + \)\(17\!\cdots\!76\)\( T^{13} + \)\(94\!\cdots\!60\)\( T^{14} - \)\(13\!\cdots\!24\)\( T^{15} - \)\(15\!\cdots\!82\)\( T^{16} + \)\(51\!\cdots\!12\)\( T^{17} + \)\(57\!\cdots\!32\)\( T^{18} + \)\(36\!\cdots\!88\)\( T^{19} - \)\(79\!\cdots\!82\)\( T^{20} - \)\(49\!\cdots\!76\)\( T^{21} + \)\(24\!\cdots\!60\)\( T^{22} + \)\(32\!\cdots\!24\)\( T^{23} - \)\(15\!\cdots\!04\)\( T^{24} - \)\(12\!\cdots\!72\)\( T^{25} + \)\(76\!\cdots\!12\)\( T^{26} + \)\(17\!\cdots\!00\)\( T^{27} + \)\(12\!\cdots\!12\)\( T^{28} - \)\(51\!\cdots\!04\)\( T^{29} + \)\(15\!\cdots\!20\)\( T^{30} + \)\(55\!\cdots\!36\)\( T^{31} + \)\(97\!\cdots\!77\)\( T^{32} - \)\(71\!\cdots\!10\)\( T^{33} + \)\(20\!\cdots\!78\)\( T^{34} + \)\(99\!\cdots\!66\)\( T^{35} + \)\(23\!\cdots\!01\)\( T^{36} \)
$61$ \( 1 + 48082 T + 1155939362 T^{2} + 42137800674218 T^{3} + 3237087838918633697 T^{4} + \)\(11\!\cdots\!96\)\( T^{5} + \)\(27\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!24\)\( T^{7} + \)\(41\!\cdots\!04\)\( T^{8} + \)\(10\!\cdots\!44\)\( T^{9} + \)\(22\!\cdots\!60\)\( T^{10} + \)\(77\!\cdots\!28\)\( T^{11} + \)\(17\!\cdots\!60\)\( T^{12} + \)\(16\!\cdots\!16\)\( T^{13} + \)\(25\!\cdots\!08\)\( T^{14} - \)\(46\!\cdots\!56\)\( T^{15} - \)\(79\!\cdots\!18\)\( T^{16} - \)\(25\!\cdots\!40\)\( T^{17} - \)\(51\!\cdots\!12\)\( T^{18} - \)\(21\!\cdots\!40\)\( T^{19} - \)\(56\!\cdots\!18\)\( T^{20} - \)\(27\!\cdots\!56\)\( T^{21} + \)\(12\!\cdots\!08\)\( T^{22} + \)\(70\!\cdots\!16\)\( T^{23} + \)\(64\!\cdots\!60\)\( T^{24} + \)\(23\!\cdots\!28\)\( T^{25} + \)\(58\!\cdots\!60\)\( T^{26} + \)\(22\!\cdots\!44\)\( T^{27} + \)\(76\!\cdots\!04\)\( T^{28} + \)\(15\!\cdots\!24\)\( T^{29} + \)\(36\!\cdots\!20\)\( T^{30} + \)\(13\!\cdots\!96\)\( T^{31} + \)\(30\!\cdots\!97\)\( T^{32} + \)\(33\!\cdots\!18\)\( T^{33} + \)\(77\!\cdots\!62\)\( T^{34} + \)\(27\!\cdots\!82\)\( T^{35} + \)\(47\!\cdots\!01\)\( T^{36} \)
$67$ \( 1 + 75210 T + 2828272050 T^{2} + 170812394979710 T^{3} + 12219123993490668673 T^{4} + \)\(43\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(65\!\cdots\!12\)\( T^{7} + \)\(31\!\cdots\!44\)\( T^{8} + \)\(90\!\cdots\!60\)\( T^{9} + \)\(26\!\cdots\!20\)\( T^{10} + \)\(13\!\cdots\!56\)\( T^{11} + \)\(68\!\cdots\!32\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} + \)\(48\!\cdots\!72\)\( T^{14} + \)\(23\!\cdots\!60\)\( T^{15} + \)\(96\!\cdots\!06\)\( T^{16} + \)\(20\!\cdots\!36\)\( T^{17} + \)\(51\!\cdots\!96\)\( T^{18} + \)\(27\!\cdots\!52\)\( T^{19} + \)\(17\!\cdots\!94\)\( T^{20} + \)\(57\!\cdots\!80\)\( T^{21} + \)\(15\!\cdots\!72\)\( T^{22} + \)\(80\!\cdots\!20\)\( T^{23} + \)\(41\!\cdots\!68\)\( T^{24} + \)\(11\!\cdots\!08\)\( T^{25} + \)\(29\!\cdots\!20\)\( T^{26} + \)\(13\!\cdots\!20\)\( T^{27} + \)\(64\!\cdots\!56\)\( T^{28} + \)\(17\!\cdots\!16\)\( T^{29} + \)\(47\!\cdots\!00\)\( T^{30} + \)\(21\!\cdots\!60\)\( T^{31} + \)\(81\!\cdots\!77\)\( T^{32} + \)\(15\!\cdots\!30\)\( T^{33} + \)\(34\!\cdots\!50\)\( T^{34} + \)\(12\!\cdots\!70\)\( T^{35} + \)\(22\!\cdots\!49\)\( T^{36} \)
$71$ \( 1 - 22544733018 T^{2} + \)\(24\!\cdots\!73\)\( T^{4} - \)\(17\!\cdots\!92\)\( T^{6} + \)\(91\!\cdots\!04\)\( T^{8} - \)\(36\!\cdots\!20\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(32\!\cdots\!12\)\( T^{14} + \)\(73\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!24\)\( T^{18} + \)\(23\!\cdots\!82\)\( T^{20} - \)\(34\!\cdots\!12\)\( T^{22} + \)\(41\!\cdots\!44\)\( T^{24} - \)\(41\!\cdots\!20\)\( T^{26} + \)\(33\!\cdots\!04\)\( T^{28} - \)\(21\!\cdots\!92\)\( T^{30} + \)\(96\!\cdots\!73\)\( T^{32} - \)\(28\!\cdots\!18\)\( T^{34} + \)\(41\!\cdots\!01\)\( T^{36} \)
$73$ \( 1 - 20520707858 T^{2} + \)\(20\!\cdots\!01\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{6} + \)\(71\!\cdots\!32\)\( T^{8} - \)\(28\!\cdots\!92\)\( T^{10} + \)\(97\!\cdots\!12\)\( T^{12} - \)\(27\!\cdots\!16\)\( T^{14} + \)\(69\!\cdots\!66\)\( T^{16} - \)\(15\!\cdots\!56\)\( T^{18} + \)\(30\!\cdots\!34\)\( T^{20} - \)\(51\!\cdots\!16\)\( T^{22} + \)\(77\!\cdots\!88\)\( T^{24} - \)\(98\!\cdots\!92\)\( T^{26} + \)\(10\!\cdots\!68\)\( T^{28} - \)\(88\!\cdots\!56\)\( T^{30} + \)\(56\!\cdots\!49\)\( T^{32} - \)\(23\!\cdots\!58\)\( T^{34} + \)\(50\!\cdots\!49\)\( T^{36} \)
$79$ \( ( 1 + 26432 T + 16735348231 T^{2} + 147423503361536 T^{3} + \)\(12\!\cdots\!52\)\( T^{4} - \)\(13\!\cdots\!88\)\( T^{5} + \)\(53\!\cdots\!44\)\( T^{6} - \)\(16\!\cdots\!52\)\( T^{7} + \)\(17\!\cdots\!94\)\( T^{8} - \)\(75\!\cdots\!96\)\( T^{9} + \)\(53\!\cdots\!06\)\( T^{10} - \)\(15\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!56\)\( T^{12} - \)\(11\!\cdots\!88\)\( T^{13} + \)\(33\!\cdots\!48\)\( T^{14} + \)\(12\!\cdots\!36\)\( T^{15} + \)\(43\!\cdots\!69\)\( T^{16} + \)\(21\!\cdots\!32\)\( T^{17} + \)\(24\!\cdots\!99\)\( T^{18} )^{2} \)
$83$ \( 1 - 227838 T + 25955077122 T^{2} - 1270371885318330 T^{3} - 42899065933744508063 T^{4} + \)\(92\!\cdots\!00\)\( T^{5} - \)\(19\!\cdots\!64\)\( T^{6} - \)\(59\!\cdots\!12\)\( T^{7} + \)\(62\!\cdots\!00\)\( T^{8} - \)\(15\!\cdots\!60\)\( T^{9} - \)\(15\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{11} + \)\(17\!\cdots\!16\)\( T^{12} - \)\(69\!\cdots\!96\)\( T^{13} + \)\(30\!\cdots\!72\)\( T^{14} + \)\(79\!\cdots\!68\)\( T^{15} - \)\(93\!\cdots\!34\)\( T^{16} - \)\(21\!\cdots\!76\)\( T^{17} + \)\(49\!\cdots\!56\)\( T^{18} - \)\(82\!\cdots\!68\)\( T^{19} - \)\(14\!\cdots\!66\)\( T^{20} + \)\(48\!\cdots\!76\)\( T^{21} + \)\(72\!\cdots\!72\)\( T^{22} - \)\(65\!\cdots\!28\)\( T^{23} + \)\(66\!\cdots\!84\)\( T^{24} + \)\(18\!\cdots\!00\)\( T^{25} - \)\(91\!\cdots\!92\)\( T^{26} - \)\(35\!\cdots\!80\)\( T^{27} + \)\(55\!\cdots\!00\)\( T^{28} - \)\(20\!\cdots\!84\)\( T^{29} - \)\(26\!\cdots\!64\)\( T^{30} + \)\(50\!\cdots\!00\)\( T^{31} - \)\(92\!\cdots\!87\)\( T^{32} - \)\(10\!\cdots\!10\)\( T^{33} + \)\(87\!\cdots\!22\)\( T^{34} - \)\(30\!\cdots\!34\)\( T^{35} + \)\(52\!\cdots\!49\)\( T^{36} \)
$89$ \( 1 - 48706783410 T^{2} + \)\(12\!\cdots\!33\)\( T^{4} - \)\(20\!\cdots\!28\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{8} - \)\(26\!\cdots\!24\)\( T^{10} + \)\(22\!\cdots\!84\)\( T^{12} - \)\(16\!\cdots\!64\)\( T^{14} + \)\(10\!\cdots\!38\)\( T^{16} - \)\(60\!\cdots\!40\)\( T^{18} + \)\(32\!\cdots\!38\)\( T^{20} - \)\(15\!\cdots\!64\)\( T^{22} + \)\(67\!\cdots\!84\)\( T^{24} - \)\(24\!\cdots\!24\)\( T^{26} + \)\(76\!\cdots\!60\)\( T^{28} - \)\(18\!\cdots\!28\)\( T^{30} + \)\(34\!\cdots\!33\)\( T^{32} - \)\(43\!\cdots\!10\)\( T^{34} + \)\(27\!\cdots\!01\)\( T^{36} \)
$97$ \( ( 1 + 2 T + 45080913113 T^{2} + 68457506698736 T^{3} + \)\(92\!\cdots\!28\)\( T^{4} + \)\(19\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} + \)\(24\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!78\)\( T^{8} + \)\(23\!\cdots\!24\)\( T^{9} + \)\(98\!\cdots\!46\)\( T^{10} + \)\(18\!\cdots\!48\)\( T^{11} + \)\(75\!\cdots\!32\)\( T^{12} + \)\(10\!\cdots\!44\)\( T^{13} + \)\(43\!\cdots\!96\)\( T^{14} + \)\(27\!\cdots\!64\)\( T^{15} + \)\(15\!\cdots\!09\)\( T^{16} + \)\(59\!\cdots\!02\)\( T^{17} + \)\(25\!\cdots\!57\)\( T^{18} )^{2} \)
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