Properties

Label 38.5.d.a
Level $38$
Weight $5$
Character orbit 38.d
Analytic conductor $3.928$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 38.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92805859719\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} + 1024 x^{14} - 7028 x^{13} + 404698 x^{12} - 2337188 x^{11} + 77836288 x^{10} - 367923764 x^{9} + 7565874847 x^{8} - 28081380444 x^{7} + 352870494000 x^{6} - 961846520868 x^{5} + 6856327325898 x^{4} - 12141615583188 x^{3} + 32749209391860 x^{2} - 26834137576128 x + 23840536514409\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{5} + \beta_{9} ) q^{2} + ( -1 + \beta_{3} ) q^{3} + 8 \beta_{7} q^{4} + ( -2 + \beta_{2} + 2 \beta_{7} - \beta_{12} ) q^{5} + ( \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{6} + ( 6 + \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} + 8 \beta_{9} q^{8} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 44 \beta_{7} - \beta_{8} - 6 \beta_{9} - \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{5} + \beta_{9} ) q^{2} + ( -1 + \beta_{3} ) q^{3} + 8 \beta_{7} q^{4} + ( -2 + \beta_{2} + 2 \beta_{7} - \beta_{12} ) q^{5} + ( \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{6} + ( 6 + \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} + 8 \beta_{9} q^{8} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 44 \beta_{7} - \beta_{8} - 6 \beta_{9} - \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{9} + ( -2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{12} - \beta_{13} ) q^{10} + ( -9 + \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 17 \beta_{5} + \beta_{6} + 2 \beta_{7} - 8 \beta_{9} ) q^{11} + ( 8 + 8 \beta_{4} - 8 \beta_{7} ) q^{12} + ( 36 + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 13 \beta_{5} - 2 \beta_{6} - 18 \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{13} + ( 11 + \beta_{2} + 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} + 13 \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{11} + \beta_{12} ) q^{14} + ( -23 - \beta_{1} - 2 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} - 19 \beta_{5} + 2 \beta_{6} + 11 \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{15} + ( -64 + 64 \beta_{7} ) q^{16} + ( 70 - 6 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + 8 \beta_{5} - 73 \beta_{7} - \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + \beta_{11} + 6 \beta_{12} + \beta_{15} ) q^{17} + ( 25 - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 52 \beta_{7} + 4 \beta_{8} + 42 \beta_{9} - \beta_{10} + \beta_{11} + 6 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} ) q^{18} + ( -14 - \beta_{1} + 7 \beta_{2} - 9 \beta_{4} + 16 \beta_{5} - \beta_{6} - 20 \beta_{7} + 2 \beta_{8} - 54 \beta_{9} - \beta_{10} - 5 \beta_{12} + \beta_{13} - 4 \beta_{14} ) q^{19} + ( -16 + 8 \beta_{2} ) q^{20} + ( -80 - 6 \beta_{2} - 19 \beta_{3} - 29 \beta_{5} - 4 \beta_{6} - 99 \beta_{7} + 2 \beta_{8} + 27 \beta_{9} + \beta_{11} - 6 \beta_{12} + 7 \beta_{14} ) q^{21} + ( -72 - 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{5} + 4 \beta_{6} - 68 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} + 3 \beta_{14} ) q^{22} + ( 1 + \beta_{1} - 21 \beta_{3} - 19 \beta_{4} + \beta_{5} - 8 \beta_{6} - 15 \beta_{7} + 8 \beta_{8} + \beta_{10} - 2 \beta_{11} + 5 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} + \beta_{15} ) q^{23} + ( 16 - 8 \beta_{5} - 16 \beta_{7} - 8 \beta_{8} ) q^{24} + ( -1 - \beta_{1} + 4 \beta_{3} + 2 \beta_{4} + 65 \beta_{5} + 9 \beta_{6} - 51 \beta_{7} - 9 \beta_{8} - 132 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{25} + ( 97 - 8 \beta_{1} + \beta_{2} - 20 \beta_{3} + 10 \beta_{4} - 35 \beta_{5} + 7 \beta_{6} - 10 \beta_{7} + 21 \beta_{9} - 3 \beta_{13} - 3 \beta_{14} ) q^{26} + ( 77 - \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + 51 \beta_{4} + 13 \beta_{5} - 15 \beta_{6} - 103 \beta_{7} + 30 \beta_{8} - 9 \beta_{9} + \beta_{10} - \beta_{11} + 16 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{27} + ( 8 + 8 \beta_{1} - 16 \beta_{3} + 24 \beta_{7} + 16 \beta_{9} + 8 \beta_{15} ) q^{28} + ( -464 - 2 \beta_{2} + 54 \beta_{3} - 54 \beta_{4} + 232 \beta_{7} + \beta_{12} ) q^{29} + ( 146 - 8 \beta_{1} - 8 \beta_{2} + 32 \beta_{3} - 16 \beta_{4} + 20 \beta_{5} - 4 \beta_{6} + 16 \beta_{7} - 12 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{30} + ( 178 + \beta_{1} - 12 \beta_{2} - 2 \beta_{3} + 17 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} - 339 \beta_{7} - 12 \beta_{8} + 105 \beta_{9} + 24 \beta_{12} + 12 \beta_{13} - 12 \beta_{14} + 2 \beta_{15} ) q^{31} + 64 \beta_{5} q^{32} + ( 331 - 3 \beta_{2} - 7 \beta_{3} - 67 \beta_{5} - 32 \beta_{6} + 324 \beta_{7} + 16 \beta_{8} + 51 \beta_{9} - \beta_{11} - 3 \beta_{12} - \beta_{14} ) q^{33} + ( -108 + 8 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} + 20 \beta_{4} - 73 \beta_{5} + \beta_{6} + 58 \beta_{7} + \beta_{8} + \beta_{10} + 10 \beta_{12} + 9 \beta_{13} - 8 \beta_{15} ) q^{34} + ( 335 + 24 \beta_{2} - 29 \beta_{3} + 59 \beta_{4} - 89 \beta_{5} - 305 \beta_{7} - 35 \beta_{8} - 53 \beta_{9} + 2 \beta_{10} - \beta_{11} - 24 \beta_{12} - 18 \beta_{13} + 9 \beta_{14} - \beta_{15} ) q^{35} + ( -352 + 8 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 40 \beta_{5} + 352 \beta_{7} - 8 \beta_{8} - 24 \beta_{9} - 8 \beta_{12} + 16 \beta_{13} - 8 \beta_{14} - 8 \beta_{15} ) q^{36} + ( -246 + 22 \beta_{2} - 9 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 483 \beta_{7} - 4 \beta_{8} + 141 \beta_{9} + \beta_{10} - \beta_{11} - 44 \beta_{12} + 7 \beta_{13} - 7 \beta_{14} ) q^{37} + ( 257 + 8 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} - 34 \beta_{4} + 20 \beta_{5} - 8 \beta_{6} - 382 \beta_{7} + 17 \beta_{8} - 46 \beta_{9} + \beta_{11} + 16 \beta_{12} - 9 \beta_{13} + 6 \beta_{14} ) q^{38} + ( 450 + 15 \beta_{2} + 26 \beta_{3} - 13 \beta_{4} + 486 \beta_{5} + 60 \beta_{6} + 13 \beta_{7} - 213 \beta_{9} - \beta_{10} - \beta_{11} - 13 \beta_{13} - 13 \beta_{14} ) q^{39} + ( -8 + 8 \beta_{2} + 16 \beta_{3} + 16 \beta_{5} + 8 \beta_{7} - 16 \beta_{9} + 8 \beta_{12} - 8 \beta_{14} ) q^{40} + ( -73 + 2 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} + \beta_{4} - 121 \beta_{5} + 62 \beta_{6} - 69 \beta_{7} - 31 \beta_{8} + 153 \beta_{9} - 5 \beta_{12} + 3 \beta_{14} + \beta_{15} ) q^{41} + ( -8 - 8 \beta_{1} - 4 \beta_{3} - 20 \beta_{4} + 85 \beta_{5} - 3 \beta_{6} + 248 \beta_{7} + 3 \beta_{8} - 186 \beta_{9} - 50 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} - 8 \beta_{15} ) q^{42} + ( -950 + 21 \beta_{2} + 17 \beta_{3} - 34 \beta_{4} - 25 \beta_{5} + 933 \beta_{7} - 42 \beta_{8} + 17 \beta_{9} - 2 \beta_{10} + \beta_{11} - 21 \beta_{12} - 10 \beta_{13} + 5 \beta_{14} ) q^{43} + ( 8 + 8 \beta_{1} + 8 \beta_{3} + 24 \beta_{4} + 72 \beta_{5} + 8 \beta_{6} - 48 \beta_{7} - 8 \beta_{8} - 128 \beta_{9} - 24 \beta_{12} + 8 \beta_{15} ) q^{44} + ( -1046 + 20 \beta_{1} + 8 \beta_{2} + 12 \beta_{3} - 6 \beta_{4} - 458 \beta_{5} + 34 \beta_{6} + 6 \beta_{7} + 246 \beta_{9} + 8 \beta_{13} + 8 \beta_{14} ) q^{45} + ( -70 + 8 \beta_{1} + 12 \beta_{2} - 16 \beta_{3} - 24 \beta_{4} + 32 \beta_{5} - 16 \beta_{6} + 116 \beta_{7} + 32 \beta_{8} - 50 \beta_{9} + \beta_{10} - \beta_{11} - 24 \beta_{12} + 11 \beta_{13} - 11 \beta_{14} + 16 \beta_{15} ) q^{46} + ( -21 - 21 \beta_{1} + 73 \beta_{3} + 31 \beta_{4} - 283 \beta_{5} - 12 \beta_{6} + 459 \beta_{7} + 12 \beta_{8} + 524 \beta_{9} - \beta_{10} + 2 \beta_{11} + 37 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} - 21 \beta_{15} ) q^{47} + ( 128 - 64 \beta_{3} + 64 \beta_{4} - 64 \beta_{7} ) q^{48} + ( 1632 + 21 \beta_{1} + 51 \beta_{2} - 80 \beta_{3} + 40 \beta_{4} - 107 \beta_{5} - 57 \beta_{6} - 40 \beta_{7} + 25 \beta_{9} + 11 \beta_{13} + 11 \beta_{14} ) q^{49} + ( 598 - 8 \beta_{1} - 2 \beta_{2} + 16 \beta_{3} + 60 \beta_{4} - 21 \beta_{5} + 5 \beta_{6} - 1136 \beta_{7} - 10 \beta_{8} - 22 \beta_{9} - \beta_{10} + \beta_{11} + 4 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} - 16 \beta_{15} ) q^{50} + ( 787 + \beta_{1} - 58 \beta_{2} + 80 \beta_{3} - 80 \beta_{4} + 209 \beta_{5} - 33 \beta_{6} - 393 \beta_{7} - 33 \beta_{8} - 8 \beta_{10} + 29 \beta_{12} - 14 \beta_{13} - \beta_{15} ) q^{51} + ( 144 + 8 \beta_{2} + 24 \beta_{3} - 88 \beta_{5} - 32 \beta_{6} + 168 \beta_{7} + 16 \beta_{8} + 72 \beta_{9} + 8 \beta_{11} + 8 \beta_{12} + 8 \beta_{14} ) q^{52} + ( 272 - 20 \beta_{1} + 42 \beta_{2} + 49 \beta_{3} - 49 \beta_{4} + 193 \beta_{5} + 28 \beta_{6} - 146 \beta_{7} + 28 \beta_{8} - 9 \beta_{10} - 21 \beta_{12} - 15 \beta_{13} + 20 \beta_{15} ) q^{53} + ( -161 - 23 \beta_{2} + 102 \beta_{3} - 212 \beta_{4} - 101 \beta_{5} + 51 \beta_{7} - 49 \beta_{8} - 60 \beta_{9} - 2 \beta_{10} + \beta_{11} + 23 \beta_{12} + 12 \beta_{13} - 6 \beta_{14} + 8 \beta_{15} ) q^{54} + ( -1609 - 30 \beta_{2} - 66 \beta_{3} + 113 \beta_{4} + 92 \beta_{5} + 1656 \beta_{7} + 6 \beta_{8} + 67 \beta_{9} + 16 \beta_{10} - 8 \beta_{11} + 30 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} + 19 \beta_{15} ) q^{55} + ( -88 - 8 \beta_{2} + 16 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 192 \beta_{7} + 16 \beta_{8} + 24 \beta_{9} + 8 \beta_{10} - 8 \beta_{11} + 16 \beta_{12} ) q^{56} + ( 800 - 19 \beta_{1} - 38 \beta_{2} - 22 \beta_{3} + 126 \beta_{4} - 151 \beta_{5} - 43 \beta_{6} - 1012 \beta_{7} + 101 \beta_{8} - 264 \beta_{9} + 7 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + 6 \beta_{14} + \beta_{15} ) q^{57} + ( -105 - 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 410 \beta_{5} + 54 \beta_{6} - 2 \beta_{7} - 178 \beta_{9} + \beta_{13} + \beta_{14} ) q^{58} + ( -730 - 2 \beta_{1} - 30 \beta_{2} - 86 \beta_{3} - \beta_{4} - 281 \beta_{5} - 62 \beta_{6} - 815 \beta_{7} + 31 \beta_{8} + 249 \beta_{9} - 9 \beta_{11} - 30 \beta_{12} - 3 \beta_{14} - \beta_{15} ) q^{59} + ( -104 - 16 \beta_{1} - 8 \beta_{2} - 56 \beta_{3} - 8 \beta_{4} - 184 \beta_{5} + 32 \beta_{6} - 152 \beta_{7} - 16 \beta_{8} + 192 \beta_{9} + 8 \beta_{11} - 8 \beta_{12} + 8 \beta_{14} - 8 \beta_{15} ) q^{60} + ( 20 + 20 \beta_{1} + 84 \beta_{3} + 124 \beta_{4} + 430 \beta_{5} - 18 \beta_{6} - 766 \beta_{7} + 18 \beta_{8} - 820 \beta_{9} - 8 \beta_{10} + 16 \beta_{11} + 45 \beta_{12} + 20 \beta_{15} ) q^{61} + ( -717 - 99 \beta_{2} - 50 \beta_{3} + 100 \beta_{4} - 152 \beta_{5} + 767 \beta_{7} + 9 \beta_{8} - 161 \beta_{9} + 2 \beta_{10} - \beta_{11} + 99 \beta_{12} ) q^{62} + ( -20 - 20 \beta_{1} - 90 \beta_{3} - 130 \beta_{4} + 196 \beta_{5} + 74 \beta_{6} - 1980 \beta_{7} - 74 \beta_{8} - 432 \beta_{9} + 8 \beta_{10} - 16 \beta_{11} + 76 \beta_{12} - 2 \beta_{13} + 4 \beta_{14} - 20 \beta_{15} ) q^{63} -512 q^{64} + ( -4 - 20 \beta_{1} - 9 \beta_{2} + 40 \beta_{3} - 187 \beta_{4} + 10 \beta_{5} - 50 \beta_{6} - 179 \beta_{7} + 100 \beta_{8} - 45 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} + 18 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} - 40 \beta_{15} ) q^{65} + ( 8 + 8 \beta_{1} - 146 \beta_{3} - 130 \beta_{4} - 280 \beta_{5} - 11 \beta_{6} + 545 \beta_{7} + 11 \beta_{8} + 576 \beta_{9} - 7 \beta_{12} - 5 \beta_{13} + 10 \beta_{14} + 8 \beta_{15} ) q^{66} + ( -2417 - \beta_{1} - 4 \beta_{2} - 94 \beta_{3} + 94 \beta_{4} + 714 \beta_{5} - 17 \beta_{6} + 1208 \beta_{7} - 17 \beta_{8} + \beta_{10} + 2 \beta_{12} + 7 \beta_{13} + \beta_{15} ) q^{67} + ( 552 - 8 \beta_{1} - 48 \beta_{2} + 48 \beta_{3} - 24 \beta_{4} + 120 \beta_{5} - 8 \beta_{6} + 24 \beta_{7} - 64 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} ) q^{68} + ( 2086 + 20 \beta_{1} + 43 \beta_{2} - 40 \beta_{3} - 292 \beta_{4} + 26 \beta_{5} + 14 \beta_{6} - 4464 \beta_{7} - 28 \beta_{8} - 360 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 86 \beta_{12} - 46 \beta_{13} + 46 \beta_{14} + 40 \beta_{15} ) q^{69} + ( 1602 - 8 \beta_{1} + 146 \beta_{2} - 286 \beta_{3} + 286 \beta_{4} - 242 \beta_{5} - 46 \beta_{6} - 805 \beta_{7} - 46 \beta_{8} - \beta_{10} - 73 \beta_{12} + 8 \beta_{15} ) q^{70} + ( 1566 - 27 \beta_{2} - 153 \beta_{3} + 639 \beta_{5} - 72 \beta_{6} + 1413 \beta_{7} + 36 \beta_{8} - 675 \beta_{9} + 9 \beta_{11} - 27 \beta_{12} - 3 \beta_{14} ) q^{71} + ( 400 - 48 \beta_{2} + 16 \beta_{3} - 16 \beta_{4} + 368 \beta_{5} + 16 \beta_{6} - 200 \beta_{7} + 16 \beta_{8} - 8 \beta_{10} + 24 \beta_{12} - 32 \beta_{13} ) q^{72} + ( -869 - 27 \beta_{2} - 79 \beta_{3} + 179 \beta_{4} - 241 \beta_{5} + 969 \beta_{7} - 63 \beta_{8} - 157 \beta_{9} - 16 \beta_{10} + 8 \beta_{11} + 27 \beta_{12} + 66 \beta_{13} - 33 \beta_{14} - 21 \beta_{15} ) q^{73} + ( -1212 + 34 \beta_{2} + 36 \beta_{3} - 80 \beta_{4} + 264 \beta_{5} + 1168 \beta_{7} + 25 \beta_{8} + 231 \beta_{9} - 34 \beta_{12} - 60 \beta_{13} + 30 \beta_{14} + 8 \beta_{15} ) q^{74} + ( -237 + \beta_{1} - 67 \beta_{2} - 2 \beta_{3} + 32 \beta_{4} + 61 \beta_{5} - 59 \beta_{6} + 506 \beta_{7} + 118 \beta_{8} + 1026 \beta_{9} - 8 \beta_{10} + 8 \beta_{11} + 134 \beta_{12} - 14 \beta_{13} + 14 \beta_{14} + 2 \beta_{15} ) q^{75} + ( 80 - 8 \beta_{1} + 40 \beta_{2} + 80 \beta_{3} - 80 \beta_{4} - 304 \beta_{5} + 8 \beta_{6} - 192 \beta_{7} + 8 \beta_{8} - 120 \beta_{9} - 8 \beta_{11} + 16 \beta_{12} + 32 \beta_{13} - 24 \beta_{14} - 8 \beta_{15} ) q^{76} + ( 2212 - 152 \beta_{2} - 54 \beta_{3} + 27 \beta_{4} + 420 \beta_{5} + 54 \beta_{6} - 27 \beta_{7} - 183 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} + 57 \beta_{13} + 57 \beta_{14} ) q^{77} + ( -2119 + 16 \beta_{1} + 77 \beta_{2} + 418 \beta_{3} + 8 \beta_{4} - 568 \beta_{5} - 30 \beta_{6} - 1709 \beta_{7} + 15 \beta_{8} + 561 \beta_{9} + 77 \beta_{12} + 27 \beta_{14} + 8 \beta_{15} ) q^{78} + ( -1266 + 40 \beta_{1} + 49 \beta_{2} - 5 \beta_{3} + 20 \beta_{4} - 1455 \beta_{5} + 116 \beta_{6} - 1291 \beta_{7} - 58 \beta_{8} + 1533 \beta_{9} + \beta_{11} + 49 \beta_{12} - 41 \beta_{14} + 20 \beta_{15} ) q^{79} + ( -128 \beta_{7} + 64 \beta_{12} ) q^{80} + ( -3877 + 149 \beta_{2} + 49 \beta_{3} - 99 \beta_{4} - 1681 \beta_{5} + 3827 \beta_{7} - 77 \beta_{8} - 1605 \beta_{9} + 16 \beta_{10} - 8 \beta_{11} - 149 \beta_{12} + 50 \beta_{13} - 25 \beta_{14} + \beta_{15} ) q^{81} + ( 246 \beta_{3} + 246 \beta_{4} + 3 \beta_{5} + 12 \beta_{6} + 957 \beta_{7} - 12 \beta_{8} - 6 \beta_{9} + \beta_{10} - 2 \beta_{11} - 27 \beta_{12} - 2 \beta_{13} + 4 \beta_{14} ) q^{82} + ( -515 - 70 \beta_{1} - 35 \beta_{2} + 520 \beta_{3} - 260 \beta_{4} - 124 \beta_{5} + 38 \beta_{6} + 260 \beta_{7} + 81 \beta_{9} + \beta_{10} + \beta_{11} - 33 \beta_{13} - 33 \beta_{14} ) q^{83} + ( 640 + 48 \beta_{2} - 152 \beta_{4} + 16 \beta_{5} - 16 \beta_{6} - 1432 \beta_{7} + 32 \beta_{8} + 216 \beta_{9} - 8 \beta_{10} + 8 \beta_{11} - 96 \beta_{12} - 56 \beta_{13} + 56 \beta_{14} ) q^{84} + ( 50 + 50 \beta_{1} + 159 \beta_{3} + 259 \beta_{4} - 443 \beta_{5} + 30 \beta_{6} + 4194 \beta_{7} - 30 \beta_{8} + 986 \beta_{9} + 9 \beta_{10} - 18 \beta_{11} - 177 \beta_{12} + 17 \beta_{13} - 34 \beta_{14} + 50 \beta_{15} ) q^{85} + ( 366 + 8 \beta_{1} + 86 \beta_{2} - 322 \beta_{3} + 322 \beta_{4} + 1009 \beta_{5} + 5 \beta_{6} - 179 \beta_{7} + 5 \beta_{8} - 43 \beta_{12} - 3 \beta_{13} - 8 \beta_{15} ) q^{86} + ( 7272 - 51 \beta_{1} - 51 \beta_{2} - 344 \beta_{3} + 172 \beta_{4} + 420 \beta_{5} + 48 \beta_{6} - 172 \beta_{7} - 186 \beta_{9} - \beta_{10} - \beta_{11} - 55 \beta_{13} - 55 \beta_{14} ) q^{87} + ( 576 + 16 \beta_{2} + 32 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} - 1120 \beta_{7} - 32 \beta_{8} - 8 \beta_{9} + 8 \beta_{10} - 8 \beta_{11} - 32 \beta_{12} - 24 \beta_{13} + 24 \beta_{14} ) q^{88} + ( 1463 + 19 \beta_{1} - 12 \beta_{2} + 226 \beta_{3} - 226 \beta_{4} - 719 \beta_{5} - 5 \beta_{6} - 722 \beta_{7} - 5 \beta_{8} + 21 \beta_{10} + 6 \beta_{12} + 66 \beta_{13} - 19 \beta_{15} ) q^{89} + ( 1640 - 12 \beta_{2} + 376 \beta_{3} + 958 \beta_{5} + 44 \beta_{6} + 2016 \beta_{7} - 22 \beta_{8} - 936 \beta_{9} - 20 \beta_{11} - 12 \beta_{12} - 32 \beta_{14} ) q^{90} + ( -2970 + 70 \beta_{1} + 200 \beta_{2} + 214 \beta_{3} - 214 \beta_{4} + 2170 \beta_{5} - 56 \beta_{6} + 1520 \beta_{7} - 56 \beta_{8} + 28 \beta_{10} - 100 \beta_{12} + 70 \beta_{13} - 70 \beta_{15} ) q^{91} + ( -200 - 40 \beta_{2} + 152 \beta_{3} - 312 \beta_{4} + 72 \beta_{5} + 40 \beta_{7} + 64 \beta_{8} + 16 \beta_{10} - 8 \beta_{11} + 40 \beta_{12} - 48 \beta_{13} + 24 \beta_{14} + 8 \beta_{15} ) q^{92} + ( -648 + 54 \beta_{2} + 93 \beta_{3} - 96 \beta_{4} + 519 \beta_{5} + 645 \beta_{7} - 30 \beta_{8} + 639 \beta_{9} - 38 \beta_{10} + 19 \beta_{11} - 54 \beta_{12} - 50 \beta_{13} + 25 \beta_{14} - 90 \beta_{15} ) q^{93} + ( -2018 - 8 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} - 56 \beta_{4} - 76 \beta_{5} + 60 \beta_{6} + 3980 \beta_{7} - 120 \beta_{8} + 484 \beta_{9} - 21 \beta_{10} + 21 \beta_{11} + 8 \beta_{12} + 49 \beta_{13} - 49 \beta_{14} - 16 \beta_{15} ) q^{94} + ( 1198 + 89 \beta_{1} - 55 \beta_{2} - 243 \beta_{3} + 349 \beta_{4} - 1267 \beta_{5} + 62 \beta_{6} - 4922 \beta_{7} - 110 \beta_{8} - 300 \beta_{9} - 10 \beta_{10} - \beta_{11} - 103 \beta_{12} + 24 \beta_{13} + 27 \beta_{14} + 20 \beta_{15} ) q^{95} + ( 128 - 64 \beta_{5} - 64 \beta_{6} ) q^{96} + ( -263 - 40 \beta_{1} + 4 \beta_{2} - 358 \beta_{3} - 20 \beta_{4} - 1492 \beta_{5} + 116 \beta_{6} - 601 \beta_{7} - 58 \beta_{8} + 1530 \beta_{9} + 28 \beta_{11} + 4 \beta_{12} - 104 \beta_{14} - 20 \beta_{15} ) q^{97} + ( 629 + 17 \beta_{2} - 246 \beta_{3} - 1449 \beta_{5} - 36 \beta_{6} + 383 \beta_{7} + 18 \beta_{8} + 1431 \beta_{9} - 21 \beta_{11} + 17 \beta_{12} - 84 \beta_{14} ) q^{98} + ( -71 - 71 \beta_{1} + 253 \beta_{3} + 111 \beta_{4} + 1318 \beta_{5} + 11 \beta_{6} + 605 \beta_{7} - 11 \beta_{8} - 2778 \beta_{9} + 19 \beta_{10} - 38 \beta_{11} + 76 \beta_{12} + 57 \beta_{13} - 114 \beta_{14} - 71 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 12q^{3} + 64q^{4} - 18q^{5} - 16q^{6} + 72q^{7} + 352q^{9} + O(q^{10}) \) \( 16q - 12q^{3} + 64q^{4} - 18q^{5} - 16q^{6} + 72q^{7} + 352q^{9} - 84q^{11} + 450q^{13} + 288q^{14} - 390q^{15} - 512q^{16} + 606q^{17} - 306q^{19} - 288q^{20} - 2160q^{21} - 1680q^{22} - 54q^{23} + 128q^{24} - 434q^{25} + 1344q^{26} + 288q^{28} - 4914q^{29} + 2752q^{30} + 7890q^{33} - 1536q^{34} + 2328q^{35} - 2816q^{36} + 1344q^{38} + 7620q^{39} - 1692q^{41} + 2080q^{42} - 7402q^{43} - 336q^{44} - 16720q^{45} + 3198q^{47} + 768q^{48} + 24816q^{49} + 10710q^{51} + 3600q^{52} + 3870q^{53} - 16q^{54} - 13588q^{55} + 3702q^{57} - 1728q^{58} - 18288q^{59} - 3120q^{60} - 6522q^{61} - 6144q^{62} - 15676q^{63} - 8192q^{64} + 4960q^{66} - 30168q^{67} + 9696q^{68} + 15360q^{70} + 35874q^{71} + 5376q^{72} - 8080q^{73} - 9120q^{74} + 480q^{76} + 34560q^{77} - 46560q^{78} - 30738q^{79} - 1152q^{80} - 30920q^{81} + 6720q^{82} - 1476q^{83} + 33626q^{85} + 288q^{86} + 113100q^{87} + 19782q^{89} + 44256q^{90} - 34260q^{91} + 432q^{92} - 4272q^{93} - 23706q^{95} + 2048q^{96} - 9936q^{97} + 12672q^{98} + 3848q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 1024 x^{14} - 7028 x^{13} + 404698 x^{12} - 2337188 x^{11} + 77836288 x^{10} - 367923764 x^{9} + 7565874847 x^{8} - 28081380444 x^{7} + 352870494000 x^{6} - 961846520868 x^{5} + 6856327325898 x^{4} - 12141615583188 x^{3} + 32749209391860 x^{2} - 26834137576128 x + 23840536514409\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(357019603350908106876967 \nu^{14} - 2499137223456356748138769 \nu^{13} + 366562502493643356519312600 \nu^{12} - 2166886231056927501390071603 \nu^{11} + 145454414660531514940941492395 \nu^{10} - 707468512288461449111129112942 \nu^{9} + 28099991932973319797867258073373 \nu^{8} - 108178890299097578339598981590911 \nu^{7} + 2731973206040444172522474313208367 \nu^{6} - 7820241141290547791540282450696826 \nu^{5} + 124535911670026178201426201806653003 \nu^{4} - 236162893736618777973663437518161963 \nu^{3} + 2151960244126931007288971252385198642 \nu^{2} - 2035164351410158705525850713113042333 \nu + 4936421210961035464615317005065064115\)\()/ \)\(32\!\cdots\!22\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-1041888364417866363358490 \nu^{14} + 7293218550925064543509430 \nu^{13} - 1015501228098561436163031345 \nu^{12} + 5998195527429342777912565480 \nu^{11} - 373071580429948555418731061923 \nu^{10} + 1810548264857104182334410434130 \nu^{9} - 63839063412907858713712049058992 \nu^{8} + 244558646233218309021618851323328 \nu^{7} - 5043221924582717574186210404121831 \nu^{6} + 14281249147372562590463481634891458 \nu^{5} - 154779177471667179820242430736641176 \nu^{4} + 286038003187178128066548491997211368 \nu^{3} - 1202549000450983678773376078212012681 \nu^{2} + 1061869985464232518387569383309351244 \nu - 756917815206784790872572091930476567\)\()/ \)\(64\!\cdots\!44\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(57\!\cdots\!39\)\( \nu^{15} + \)\(37\!\cdots\!91\)\( \nu^{14} + \)\(24\!\cdots\!57\)\( \nu^{13} + \)\(37\!\cdots\!99\)\( \nu^{12} - \)\(57\!\cdots\!04\)\( \nu^{11} + \)\(14\!\cdots\!45\)\( \nu^{10} - \)\(46\!\cdots\!89\)\( \nu^{9} + \)\(24\!\cdots\!15\)\( \nu^{8} - \)\(85\!\cdots\!60\)\( \nu^{7} + \)\(20\!\cdots\!91\)\( \nu^{6} - \)\(60\!\cdots\!45\)\( \nu^{5} + \)\(62\!\cdots\!69\)\( \nu^{4} - \)\(13\!\cdots\!55\)\( \nu^{3} + \)\(50\!\cdots\!10\)\( \nu^{2} - \)\(59\!\cdots\!74\)\( \nu + \)\(91\!\cdots\!54\)\(\)\()/ \)\(18\!\cdots\!22\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-89558295647381110 \nu^{15} + 671687217355358325 \nu^{14} - 94593775335736385855 \nu^{13} + 604672283552396906795 \nu^{12} - 39078023355795006991701 \nu^{11} + 208300145301281929602388 \nu^{10} - 8011548492751123260081748 \nu^{9} + 34499646194663013839191836 \nu^{8} - 852625159035523569374752289 \nu^{7} + 2824634573178666372942040002 \nu^{6} - 44607825550312627268934621912 \nu^{5} + 104537445042719007494392512924 \nu^{4} - 930860479536463580670774478107 \nu^{3} + 1293142901226763651089675719373 \nu^{2} - 3973973317925255425531675345005 \nu + 1247009452444923177980948420961\)\()/ \)\(10\!\cdots\!62\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(76\!\cdots\!05\)\( \nu^{15} - \)\(65\!\cdots\!48\)\( \nu^{14} + \)\(81\!\cdots\!26\)\( \nu^{13} - \)\(60\!\cdots\!42\)\( \nu^{12} + \)\(34\!\cdots\!27\)\( \nu^{11} - \)\(21\!\cdots\!47\)\( \nu^{10} + \)\(70\!\cdots\!97\)\( \nu^{9} - \)\(35\!\cdots\!23\)\( \nu^{8} + \)\(75\!\cdots\!95\)\( \nu^{7} - \)\(29\!\cdots\!75\)\( \nu^{6} + \)\(39\!\cdots\!23\)\( \nu^{5} - \)\(10\!\cdots\!55\)\( \nu^{4} + \)\(83\!\cdots\!54\)\( \nu^{3} - \)\(12\!\cdots\!37\)\( \nu^{2} + \)\(25\!\cdots\!91\)\( \nu + \)\(10\!\cdots\!51\)\(\)\()/ \)\(90\!\cdots\!11\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-44779147823690555 \nu^{15} + 3557767810173487540 \nu^{14} - 69850357078338851570 \nu^{13} + 3438632507443563443716 \nu^{12} - 38063594769565574875409 \nu^{11} + 1248963345253702665964456 \nu^{10} - 9560569455404862365575544 \nu^{9} + 209537948851476261642003955 \nu^{8} - 1162339160223851720147779451 \nu^{7} + 15829980096601871549908321914 \nu^{6} - 63003935509730366374163131356 \nu^{5} + 421356934871191569491988004251 \nu^{4} - 1136620982727226713862905418800 \nu^{3} + 1278409137609756088590617435616 \nu^{2} - 1762019439276733470220187759724 \nu - 1943852069712777767973662857191\)\()/ \)\(52\!\cdots\!81\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-1548537014345583712 \nu^{15} + 11614027607591877840 \nu^{14} - 1555362509924491208012 \nu^{13} + 9933710229127382704838 \nu^{12} - 597732711038634090679884 \nu^{11} + 3178646619579926271310072 \nu^{10} - 110094471049799535917720548 \nu^{9} + 471748345893055123590281193 \nu^{8} - 9959611324735244242914950468 \nu^{7} + 32679180503015279808207650256 \nu^{6} - 409646428338517485078786437748 \nu^{5} + 943503135521563095859564968114 \nu^{4} - 6469222386322007236855236837492 \nu^{3} + 8776360253884581356059026952956 \nu^{2} - 18499195969218272608452529182648 \nu + 13036131316404757271795438083023\)\()/ \)\(10\!\cdots\!03\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(14\!\cdots\!75\)\( \nu^{15} + \)\(39\!\cdots\!19\)\( \nu^{14} - \)\(16\!\cdots\!51\)\( \nu^{13} + \)\(36\!\cdots\!26\)\( \nu^{12} - \)\(71\!\cdots\!57\)\( \nu^{11} + \)\(12\!\cdots\!79\)\( \nu^{10} - \)\(14\!\cdots\!25\)\( \nu^{9} + \)\(20\!\cdots\!11\)\( \nu^{8} - \)\(14\!\cdots\!99\)\( \nu^{7} + \)\(15\!\cdots\!93\)\( \nu^{6} - \)\(68\!\cdots\!01\)\( \nu^{5} + \)\(40\!\cdots\!09\)\( \nu^{4} - \)\(11\!\cdots\!94\)\( \nu^{3} + \)\(14\!\cdots\!86\)\( \nu^{2} - \)\(43\!\cdots\!54\)\( \nu - \)\(13\!\cdots\!65\)\(\)\()/ \)\(90\!\cdots\!11\)\( \)
\(\beta_{9}\)\(=\)\((\)\(89558295647381110 \nu^{15} - 671687217355358325 \nu^{14} + 94593775335736385855 \nu^{13} - 604672283552396906795 \nu^{12} + 39078023355795006991701 \nu^{11} - 208300145301281929602388 \nu^{10} + 8011548492751123260081748 \nu^{9} - 34499646194663013839191836 \nu^{8} + 852625159035523569374752289 \nu^{7} - 2824634573178666372942040002 \nu^{6} + 44607825550312627268934621912 \nu^{5} - 104537445042719007494392512924 \nu^{4} + 930860479536463580670774478107 \nu^{3} - 1293142901226763651089675719373 \nu^{2} + 2918229613317180443925693480843 \nu - 1247009452444923177980948420961\)\()/ \)\(52\!\cdots\!81\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(64\!\cdots\!72\)\( \nu^{15} + \)\(52\!\cdots\!33\)\( \nu^{14} + \)\(74\!\cdots\!06\)\( \nu^{13} + \)\(52\!\cdots\!92\)\( \nu^{12} + \)\(35\!\cdots\!48\)\( \nu^{11} + \)\(20\!\cdots\!73\)\( \nu^{10} + \)\(84\!\cdots\!76\)\( \nu^{9} + \)\(40\!\cdots\!07\)\( \nu^{8} + \)\(10\!\cdots\!48\)\( \nu^{7} + \)\(42\!\cdots\!53\)\( \nu^{6} + \)\(67\!\cdots\!56\)\( \nu^{5} + \)\(24\!\cdots\!31\)\( \nu^{4} + \)\(15\!\cdots\!56\)\( \nu^{3} + \)\(63\!\cdots\!68\)\( \nu^{2} + \)\(34\!\cdots\!78\)\( \nu + \)\(19\!\cdots\!55\)\(\)\()/ \)\(68\!\cdots\!36\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(64\!\cdots\!72\)\( \nu^{15} + \)\(14\!\cdots\!13\)\( \nu^{14} - \)\(89\!\cdots\!28\)\( \nu^{13} + \)\(15\!\cdots\!33\)\( \nu^{12} - \)\(47\!\cdots\!12\)\( \nu^{11} + \)\(65\!\cdots\!38\)\( \nu^{10} - \)\(12\!\cdots\!02\)\( \nu^{9} + \)\(13\!\cdots\!77\)\( \nu^{8} - \)\(17\!\cdots\!76\)\( \nu^{7} + \)\(13\!\cdots\!38\)\( \nu^{6} - \)\(11\!\cdots\!90\)\( \nu^{5} + \)\(68\!\cdots\!65\)\( \nu^{4} - \)\(33\!\cdots\!64\)\( \nu^{3} + \)\(13\!\cdots\!45\)\( \nu^{2} - \)\(22\!\cdots\!62\)\( \nu + \)\(31\!\cdots\!52\)\(\)\()/ \)\(68\!\cdots\!36\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(88\!\cdots\!28\)\( \nu^{15} + \)\(10\!\cdots\!55\)\( \nu^{14} - \)\(85\!\cdots\!14\)\( \nu^{13} + \)\(28\!\cdots\!06\)\( \nu^{12} - \)\(31\!\cdots\!44\)\( \nu^{11} - \)\(14\!\cdots\!83\)\( \nu^{10} - \)\(55\!\cdots\!28\)\( \nu^{9} - \)\(62\!\cdots\!19\)\( \nu^{8} - \)\(47\!\cdots\!84\)\( \nu^{7} - \)\(71\!\cdots\!27\)\( \nu^{6} - \)\(18\!\cdots\!80\)\( \nu^{5} - \)\(22\!\cdots\!43\)\( \nu^{4} - \)\(29\!\cdots\!16\)\( \nu^{3} - \)\(33\!\cdots\!94\)\( \nu^{2} - \)\(35\!\cdots\!74\)\( \nu - \)\(40\!\cdots\!25\)\(\)\()/ \)\(68\!\cdots\!36\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(21\!\cdots\!74\)\( \nu^{15} - \)\(91\!\cdots\!53\)\( \nu^{14} + \)\(21\!\cdots\!58\)\( \nu^{13} - \)\(73\!\cdots\!14\)\( \nu^{12} + \)\(83\!\cdots\!18\)\( \nu^{11} - \)\(22\!\cdots\!23\)\( \nu^{10} + \)\(15\!\cdots\!58\)\( \nu^{9} - \)\(34\!\cdots\!59\)\( \nu^{8} + \)\(14\!\cdots\!22\)\( \nu^{7} - \)\(29\!\cdots\!95\)\( \nu^{6} + \)\(65\!\cdots\!46\)\( \nu^{5} - \)\(14\!\cdots\!39\)\( \nu^{4} + \)\(12\!\cdots\!96\)\( \nu^{3} - \)\(30\!\cdots\!22\)\( \nu^{2} + \)\(41\!\cdots\!28\)\( \nu - \)\(11\!\cdots\!49\)\(\)\()/ \)\(68\!\cdots\!36\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(21\!\cdots\!74\)\( \nu^{15} + \)\(23\!\cdots\!57\)\( \nu^{14} - \)\(22\!\cdots\!86\)\( \nu^{13} + \)\(21\!\cdots\!87\)\( \nu^{12} - \)\(92\!\cdots\!92\)\( \nu^{11} + \)\(71\!\cdots\!08\)\( \nu^{10} - \)\(17\!\cdots\!72\)\( \nu^{9} + \)\(10\!\cdots\!45\)\( \nu^{8} - \)\(17\!\cdots\!52\)\( \nu^{7} + \)\(75\!\cdots\!68\)\( \nu^{6} - \)\(78\!\cdots\!36\)\( \nu^{5} + \)\(18\!\cdots\!21\)\( \nu^{4} - \)\(12\!\cdots\!34\)\( \nu^{3} + \)\(29\!\cdots\!67\)\( \nu^{2} - \)\(13\!\cdots\!02\)\( \nu + \)\(98\!\cdots\!46\)\(\)\()/ \)\(68\!\cdots\!36\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(20\!\cdots\!84\)\( \nu^{15} + \)\(13\!\cdots\!04\)\( \nu^{14} - \)\(19\!\cdots\!81\)\( \nu^{13} + \)\(11\!\cdots\!63\)\( \nu^{12} - \)\(75\!\cdots\!11\)\( \nu^{11} + \)\(35\!\cdots\!70\)\( \nu^{10} - \)\(13\!\cdots\!52\)\( \nu^{9} + \)\(49\!\cdots\!25\)\( \nu^{8} - \)\(11\!\cdots\!83\)\( \nu^{7} + \)\(29\!\cdots\!80\)\( \nu^{6} - \)\(45\!\cdots\!28\)\( \nu^{5} + \)\(56\!\cdots\!67\)\( \nu^{4} - \)\(66\!\cdots\!47\)\( \nu^{3} - \)\(17\!\cdots\!78\)\( \nu^{2} - \)\(27\!\cdots\!87\)\( \nu - \)\(88\!\cdots\!42\)\(\)\()/ \)\(34\!\cdots\!18\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} - 2 \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{14} + \beta_{13} + 2 \beta_{9} - \beta_{7} - 2 \beta_{6} - 7 \beta_{5} - 2 \beta_{3} + \beta_{2} + \beta_{1} - 122\)
\(\nu^{3}\)\(=\)\((\)\(-10 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 44 \beta_{12} + \beta_{11} - \beta_{10} + 335 \beta_{9} + 60 \beta_{8} + 46 \beta_{7} - 36 \beta_{6} - \beta_{5} + 432 \beta_{4} + 4 \beta_{3} - 19 \beta_{2} - 2 \beta_{1} - 175\)\()/2\)
\(\nu^{4}\)\(=\)\(-10 \beta_{15} - 284 \beta_{14} - 286 \beta_{13} + 44 \beta_{12} - 4 \beta_{11} - 6 \beta_{10} - 1818 \beta_{9} + 60 \beta_{8} + 215 \beta_{7} + 626 \beta_{6} + 4968 \beta_{5} + 264 \beta_{4} + 342 \beta_{3} - 308 \beta_{2} - 252 \beta_{1} + 27478\)
\(\nu^{5}\)\(=\)\((\)\(2416 \beta_{15} - 3524 \beta_{14} + 664 \beta_{13} - 12266 \beta_{12} - 502 \beta_{11} + 452 \beta_{10} - 117002 \beta_{9} - 24736 \beta_{8} + 61072 \beta_{7} + 15658 \beta_{6} + 14823 \beta_{5} - 106130 \beta_{4} - 746 \beta_{3} + 4698 \beta_{2} - 32 \beta_{1} + 54280\)\()/2\)
\(\nu^{6}\)\(=\)\(3649 \beta_{15} + 76808 \beta_{14} + 83095 \beta_{13} - 18509 \beta_{12} + 2576 \beta_{11} + 4012 \beta_{10} + 641137 \beta_{9} - 37254 \beta_{8} + 95804 \beta_{7} - 180840 \beta_{6} - 1817138 \beta_{5} - 164589 \beta_{4} + 7493 \beta_{3} + 97162 \beta_{2} + 60761 \beta_{1} - 7129316\)
\(\nu^{7}\)\(=\)\((\)\(-450806 \beta_{15} + 1700462 \beta_{14} - 571124 \beta_{13} + 3275725 \beta_{12} + 176728 \beta_{11} - 130437 \beta_{10} + 38873979 \beta_{9} + 8424690 \beta_{8} - 43762745 \beta_{7} - 5620136 \beta_{6} - 8940318 \beta_{5} + 27458742 \beta_{4} + 522948 \beta_{3} - 1017484 \beta_{2} + 191496 \beta_{1} - 13866106\)\()/2\)
\(\nu^{8}\)\(=\)\(-918664 \beta_{15} - 20280184 \beta_{14} - 24852700 \beta_{13} + 6637928 \beta_{12} - 988892 \beta_{11} - 1609928 \beta_{10} - 194060764 \beta_{9} + 17023372 \beta_{8} - 105073903 \beta_{7} + 51636104 \beta_{6} + 590266944 \beta_{5} + 72788012 \beta_{4} - 33191932 \beta_{3} - 30730932 \beta_{2} - 14988760 \beta_{1} + 1970744963\)
\(\nu^{9}\)\(=\)\((\)\(57753080 \beta_{15} - 671360072 \beta_{14} + 258376064 \beta_{13} - 921077488 \beta_{12} - 58567136 \beta_{11} + 34899800 \beta_{10} - 12719024191 \beta_{9} - 2696381628 \beta_{8} + 20015162890 \beta_{7} + 1897981500 \beta_{6} + 4029942803 \beta_{5} - 7293707688 \beta_{4} - 365174270 \beta_{3} + 210102728 \beta_{2} - 104394920 \beta_{1} + 3579331694\)\()/2\)
\(\nu^{10}\)\(=\)\(151298273 \beta_{15} + 5242407157 \beta_{14} + 7601085386 \beta_{13} - 2352607963 \beta_{12} + 317235800 \beta_{11} + 555570972 \beta_{10} + 54806086431 \beta_{9} - 6868890438 \beta_{8} + 60074468765 \beta_{7} - 14773001236 \beta_{6} - 183781693335 \beta_{5} - 28029170469 \beta_{4} + 17831729985 \beta_{3} + 9659123515 \beta_{2} + 3790132296 \beta_{1} - 565056048674\)
\(\nu^{11}\)\(=\)\((\)\(2162257528 \beta_{15} + 241933383058 \beta_{14} - 96856824134 \beta_{13} + 270375367717 \beta_{12} + 19215560351 \beta_{11} - 9397225528 \beta_{10} + 4136484665500 \beta_{9} + 838268484296 \beta_{8} - 7725565540847 \beta_{7} - 624471423032 \beta_{6} - 1613097658593 \beta_{5} + 1966462311274 \beta_{4} + 198469856044 \beta_{3} - 39574165545 \beta_{2} + 43166698114 \beta_{1} - 1043554554971\)\()/2\)
\(\nu^{12}\)\(=\)\(4807253182 \beta_{15} - 1320323145370 \beta_{14} - 2362714812326 \beta_{13} + 837114724480 \beta_{12} - 92663945242 \beta_{11} - 181134268490 \beta_{10} - 14672149786886 \beta_{9} + 2590644953922 \beta_{8} - 27678031003288 \beta_{7} + 4240045304880 \beta_{6} + 56093227080406 \beta_{5} + 10047691837290 \beta_{4} - 7278846089368 \beta_{3} - 3013013489528 \beta_{2} - 976035949774 \beta_{1} + 165852806135551\)
\(\nu^{13}\)\(=\)\((\)\(-5327342762360 \beta_{15} - 82924460318752 \beta_{14} + 33149082552820 \beta_{13} - 81356901327878 \beta_{12} - 6292241798612 \beta_{11} + 2604660702426 \beta_{10} - 1338100300955223 \beta_{9} - 256413629835966 \beta_{8} + 2731583017652146 \beta_{7} + 202849140437680 \beta_{6} + 605350554358260 \beta_{5} - 534729555425850 \beta_{4} - 91850709331932 \beta_{3} + 5701551137552 \beta_{2} - 15933691665168 \beta_{1} + 354690150531692\)\()/2\)
\(\nu^{14}\)\(=\)\(-18713495305312 \beta_{15} + 320402143333631 \beta_{14} + 742548179127581 \beta_{13} - 297524369789158 \beta_{12} + 25280726004856 \beta_{11} + 57769681659292 \beta_{10} + 3709249409270924 \beta_{9} - 936969565776086 \beta_{8} + 11411464739486201 \beta_{7} - 1217758271053906 \beta_{6} - 16937581621719267 \beta_{5} - 3454012598037238 \beta_{4} + 2647773109741316 \beta_{3} + 933997388659551 \beta_{2} + 254505864635039 \beta_{1} - 49438344520598908\)
\(\nu^{15}\)\(=\)\((\)\(2696470653762998 \beta_{15} + 27552930629391148 \beta_{14} - 10730956241316154 \beta_{13} + 24764208528593996 \beta_{12} + 2052702173081947 \beta_{11} - 741963824403691 \beta_{10} + 430158188049709967 \beta_{9} + 77639953128210642 \beta_{8} - 916487537657806334 \beta_{7} - 65429271731003226 \beta_{6} - 218262987072098203 \beta_{5} + 146030887715084772 \beta_{4} + 38449204885047016 \beta_{3} + 12899925995855 \beta_{2} + 5540330730795070 \beta_{1} - 133882227440496301\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
0.500000 + 17.6194i
0.500000 + 2.12625i
0.500000 0.961202i
0.500000 15.9560i
0.500000 + 11.2515i
0.500000 + 6.86019i
0.500000 6.55779i
0.500000 14.3823i
0.500000 17.6194i
0.500000 2.12625i
0.500000 + 0.961202i
0.500000 + 15.9560i
0.500000 11.2515i
0.500000 6.86019i
0.500000 + 6.55779i
0.500000 + 14.3823i
−2.44949 1.41421i −14.7841 8.53562i 4.00000 + 6.92820i −3.60911 + 6.25116i 24.1424 + 41.8158i 27.7970 22.6274i 105.214 + 182.235i 17.6809 10.2081i
27.2 −2.44949 1.41421i −1.36664 0.789030i 4.00000 + 6.92820i 11.3097 19.5890i 2.23171 + 3.86544i −73.1169 22.6274i −39.2549 67.9914i −55.4061 + 31.9887i
27.3 −2.44949 1.41421i 1.30717 + 0.754695i 4.00000 + 6.92820i 1.35492 2.34679i −2.13460 3.69723i 79.4947 22.6274i −39.3609 68.1750i −6.63772 + 3.83229i
27.4 −2.44949 1.41421i 14.2931 + 8.25212i 4.00000 + 6.92820i −13.5555 + 23.4789i −23.3405 40.4270i −45.5687 22.6274i 95.6949 + 165.748i 66.4083 38.3408i
27.5 2.44949 + 1.41421i −11.7188 6.76588i 4.00000 + 6.92820i −15.6059 + 27.0302i −19.1368 33.1459i −61.1277 22.6274i 51.0542 + 88.4285i −76.4529 + 44.1401i
27.6 2.44949 + 1.41421i −7.91584 4.57021i 4.00000 + 6.92820i 19.7357 34.1832i −12.9265 22.3894i 91.5785 22.6274i 1.27372 + 2.20614i 96.6847 55.8209i
27.7 2.44949 + 1.41421i 3.70447 + 2.13878i 4.00000 + 6.92820i −17.7629 + 30.7663i 6.04937 + 10.4778i 57.6374 22.6274i −31.3513 54.3020i −87.0203 + 50.2412i
27.8 2.44949 + 1.41421i 10.4807 + 6.05105i 4.00000 + 6.92820i 9.13314 15.8191i 17.1150 + 29.6440i −40.6944 22.6274i 32.7305 + 56.6909i 44.7431 25.8324i
31.1 −2.44949 + 1.41421i −14.7841 + 8.53562i 4.00000 6.92820i −3.60911 6.25116i 24.1424 41.8158i 27.7970 22.6274i 105.214 182.235i 17.6809 + 10.2081i
31.2 −2.44949 + 1.41421i −1.36664 + 0.789030i 4.00000 6.92820i 11.3097 + 19.5890i 2.23171 3.86544i −73.1169 22.6274i −39.2549 + 67.9914i −55.4061 31.9887i
31.3 −2.44949 + 1.41421i 1.30717 0.754695i 4.00000 6.92820i 1.35492 + 2.34679i −2.13460 + 3.69723i 79.4947 22.6274i −39.3609 + 68.1750i −6.63772 3.83229i
31.4 −2.44949 + 1.41421i 14.2931 8.25212i 4.00000 6.92820i −13.5555 23.4789i −23.3405 + 40.4270i −45.5687 22.6274i 95.6949 165.748i 66.4083 + 38.3408i
31.5 2.44949 1.41421i −11.7188 + 6.76588i 4.00000 6.92820i −15.6059 27.0302i −19.1368 + 33.1459i −61.1277 22.6274i 51.0542 88.4285i −76.4529 44.1401i
31.6 2.44949 1.41421i −7.91584 + 4.57021i 4.00000 6.92820i 19.7357 + 34.1832i −12.9265 + 22.3894i 91.5785 22.6274i 1.27372 2.20614i 96.6847 + 55.8209i
31.7 2.44949 1.41421i 3.70447 2.13878i 4.00000 6.92820i −17.7629 30.7663i 6.04937 10.4778i 57.6374 22.6274i −31.3513 + 54.3020i −87.0203 50.2412i
31.8 2.44949 1.41421i 10.4807 6.05105i 4.00000 6.92820i 9.13314 + 15.8191i 17.1150 29.6440i −40.6944 22.6274i 32.7305 56.6909i 44.7431 + 25.8324i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.5.d.a 16
3.b odd 2 1 342.5.m.c 16
4.b odd 2 1 304.5.r.c 16
19.d odd 6 1 inner 38.5.d.a 16
57.f even 6 1 342.5.m.c 16
76.f even 6 1 304.5.r.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.5.d.a 16 1.a even 1 1 trivial
38.5.d.a 16 19.d odd 6 1 inner
304.5.r.c 16 4.b odd 2 1
304.5.r.c 16 76.f even 6 1
342.5.m.c 16 3.b odd 2 1
342.5.m.c 16 57.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T^{2} + 64 T^{4} )^{4} \)
$3$ \( 1 + 12 T + 220 T^{2} + 2064 T^{3} + 23074 T^{4} + 200160 T^{5} + 1048944 T^{6} + 2917944 T^{7} - 75215223 T^{8} - 1195754724 T^{9} - 13258495260 T^{10} - 122243886180 T^{11} - 530012067822 T^{12} - 793793505420 T^{13} + 56218291157088 T^{14} + 799179948504864 T^{15} + 8820827367994980 T^{16} + 64733575828893984 T^{17} + 368848208281654368 T^{18} - 421854414313910220 T^{19} - 22815281610166711662 T^{20} - \)\(42\!\cdots\!80\)\( T^{21} - \)\(37\!\cdots\!60\)\( T^{22} - \)\(27\!\cdots\!64\)\( T^{23} - \)\(13\!\cdots\!43\)\( T^{24} + \)\(43\!\cdots\!24\)\( T^{25} + \)\(12\!\cdots\!44\)\( T^{26} + \)\(19\!\cdots\!60\)\( T^{27} + \)\(18\!\cdots\!14\)\( T^{28} + \)\(13\!\cdots\!24\)\( T^{29} + \)\(11\!\cdots\!20\)\( T^{30} + \)\(50\!\cdots\!12\)\( T^{31} + \)\(34\!\cdots\!81\)\( T^{32} \)
$5$ \( 1 + 18 T - 2121 T^{2} - 28530 T^{3} + 2086807 T^{4} + 9822204 T^{5} - 1405623192 T^{6} + 14052148560 T^{7} + 909692080087 T^{8} - 20542929979446 T^{9} - 599034946608297 T^{10} + 13275141996377250 T^{11} + 253935491820640198 T^{12} - 5183275443247796166 T^{13} + 4077825853380152259 T^{14} + \)\(10\!\cdots\!14\)\( T^{15} - \)\(57\!\cdots\!24\)\( T^{16} + \)\(64\!\cdots\!50\)\( T^{17} + \)\(15\!\cdots\!75\)\( T^{18} - \)\(12\!\cdots\!50\)\( T^{19} + \)\(38\!\cdots\!50\)\( T^{20} + \)\(12\!\cdots\!50\)\( T^{21} - \)\(35\!\cdots\!25\)\( T^{22} - \)\(76\!\cdots\!50\)\( T^{23} + \)\(21\!\cdots\!75\)\( T^{24} + \)\(20\!\cdots\!00\)\( T^{25} - \)\(12\!\cdots\!00\)\( T^{26} + \)\(55\!\cdots\!00\)\( T^{27} + \)\(74\!\cdots\!75\)\( T^{28} - \)\(63\!\cdots\!50\)\( T^{29} - \)\(29\!\cdots\!25\)\( T^{30} + \)\(15\!\cdots\!50\)\( T^{31} + \)\(54\!\cdots\!25\)\( T^{32} \)
$7$ \( ( 1 - 36 T + 4048 T^{2} - 277752 T^{3} + 23341552 T^{4} - 1116927864 T^{5} + 70654987156 T^{6} - 3553393594812 T^{7} + 212429474064886 T^{8} - 8531698021143612 T^{9} + 407311940611895956 T^{10} - 15459719348423468664 T^{11} + \)\(77\!\cdots\!52\)\( T^{12} - \)\(22\!\cdots\!52\)\( T^{13} + \)\(77\!\cdots\!48\)\( T^{14} - \)\(16\!\cdots\!36\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \)
$11$ \( ( 1 + 42 T + 66903 T^{2} + 2734506 T^{3} + 2441731097 T^{4} + 91423744404 T^{5} + 58701499839330 T^{6} + 1953995756236260 T^{7} + 1008748232239081890 T^{8} + 28608451867055082660 T^{9} + \)\(12\!\cdots\!30\)\( T^{10} + \)\(28\!\cdots\!84\)\( T^{11} + \)\(11\!\cdots\!17\)\( T^{12} + \)\(18\!\cdots\!06\)\( T^{13} + \)\(65\!\cdots\!23\)\( T^{14} + \)\(60\!\cdots\!02\)\( T^{15} + \)\(21\!\cdots\!21\)\( T^{16} )^{2} \)
$13$ \( 1 - 450 T + 156659 T^{2} - 40121550 T^{3} + 9419939307 T^{4} - 1979888920272 T^{5} + 350012323407748 T^{6} - 56951468985831300 T^{7} + 7541316836900984567 T^{8} - \)\(92\!\cdots\!22\)\( T^{9} + \)\(80\!\cdots\!51\)\( T^{10} - \)\(68\!\cdots\!46\)\( T^{11} + \)\(92\!\cdots\!66\)\( T^{12} - \)\(19\!\cdots\!30\)\( T^{13} + \)\(62\!\cdots\!47\)\( T^{14} - \)\(14\!\cdots\!06\)\( T^{15} + \)\(30\!\cdots\!76\)\( T^{16} - \)\(40\!\cdots\!66\)\( T^{17} + \)\(50\!\cdots\!87\)\( T^{18} - \)\(46\!\cdots\!30\)\( T^{19} + \)\(61\!\cdots\!06\)\( T^{20} - \)\(13\!\cdots\!46\)\( T^{21} + \)\(43\!\cdots\!11\)\( T^{22} - \)\(14\!\cdots\!62\)\( T^{23} + \)\(33\!\cdots\!27\)\( T^{24} - \)\(72\!\cdots\!00\)\( T^{25} + \)\(12\!\cdots\!48\)\( T^{26} - \)\(20\!\cdots\!92\)\( T^{27} + \)\(27\!\cdots\!47\)\( T^{28} - \)\(33\!\cdots\!50\)\( T^{29} + \)\(37\!\cdots\!19\)\( T^{30} - \)\(30\!\cdots\!50\)\( T^{31} + \)\(19\!\cdots\!61\)\( T^{32} \)
$17$ \( 1 - 606 T - 27129 T^{2} + 135883638 T^{3} - 37104496145 T^{4} - 7553214775368 T^{5} + 6820848444642240 T^{6} - 1128094034581979364 T^{7} - \)\(39\!\cdots\!01\)\( T^{8} + \)\(21\!\cdots\!94\)\( T^{9} - \)\(17\!\cdots\!29\)\( T^{10} - \)\(12\!\cdots\!82\)\( T^{11} + \)\(37\!\cdots\!62\)\( T^{12} + \)\(28\!\cdots\!58\)\( T^{13} - \)\(18\!\cdots\!05\)\( T^{14} + \)\(17\!\cdots\!78\)\( T^{15} + \)\(44\!\cdots\!32\)\( T^{16} + \)\(14\!\cdots\!38\)\( T^{17} - \)\(12\!\cdots\!05\)\( T^{18} + \)\(16\!\cdots\!38\)\( T^{19} + \)\(18\!\cdots\!22\)\( T^{20} - \)\(50\!\cdots\!82\)\( T^{21} - \)\(59\!\cdots\!09\)\( T^{22} + \)\(60\!\cdots\!54\)\( T^{23} - \)\(93\!\cdots\!61\)\( T^{24} - \)\(22\!\cdots\!84\)\( T^{25} + \)\(11\!\cdots\!40\)\( T^{26} - \)\(10\!\cdots\!28\)\( T^{27} - \)\(42\!\cdots\!45\)\( T^{28} + \)\(13\!\cdots\!18\)\( T^{29} - \)\(21\!\cdots\!49\)\( T^{30} - \)\(40\!\cdots\!06\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$19$ \( 1 + 306 T + 265315 T^{2} + 81446994 T^{3} + 44230967749 T^{4} + 7637163561252 T^{5} + 3726453461291350 T^{6} + 152860155109547508 T^{7} + \)\(25\!\cdots\!46\)\( T^{8} + \)\(19\!\cdots\!68\)\( T^{9} + \)\(63\!\cdots\!50\)\( T^{10} + \)\(16\!\cdots\!72\)\( T^{11} + \)\(12\!\cdots\!69\)\( T^{12} + \)\(30\!\cdots\!94\)\( T^{13} + \)\(12\!\cdots\!15\)\( T^{14} + \)\(19\!\cdots\!46\)\( T^{15} + \)\(83\!\cdots\!61\)\( T^{16} \)
$23$ \( 1 + 54 T - 895569 T^{2} - 110020446 T^{3} + 387099204235 T^{4} + 67734152183124 T^{5} - 64335376978182492 T^{6} - 19898812923522769512 T^{7} - \)\(13\!\cdots\!41\)\( T^{8} + \)\(12\!\cdots\!70\)\( T^{9} + \)\(10\!\cdots\!23\)\( T^{10} + \)\(19\!\cdots\!42\)\( T^{11} - \)\(22\!\cdots\!82\)\( T^{12} - \)\(78\!\cdots\!66\)\( T^{13} - \)\(10\!\cdots\!25\)\( T^{14} + \)\(97\!\cdots\!34\)\( T^{15} + \)\(14\!\cdots\!04\)\( T^{16} + \)\(27\!\cdots\!94\)\( T^{17} - \)\(83\!\cdots\!25\)\( T^{18} - \)\(17\!\cdots\!86\)\( T^{19} - \)\(13\!\cdots\!02\)\( T^{20} + \)\(34\!\cdots\!42\)\( T^{21} + \)\(52\!\cdots\!43\)\( T^{22} + \)\(17\!\cdots\!70\)\( T^{23} - \)\(51\!\cdots\!61\)\( T^{24} - \)\(20\!\cdots\!32\)\( T^{25} - \)\(18\!\cdots\!92\)\( T^{26} + \)\(55\!\cdots\!84\)\( T^{27} + \)\(89\!\cdots\!35\)\( T^{28} - \)\(71\!\cdots\!66\)\( T^{29} - \)\(16\!\cdots\!09\)\( T^{30} + \)\(27\!\cdots\!54\)\( T^{31} + \)\(14\!\cdots\!41\)\( T^{32} \)
$29$ \( 1 + 4914 T + 15749015 T^{2} + 37837225062 T^{3} + 75611135966079 T^{4} + 130807785035806812 T^{5} + \)\(20\!\cdots\!28\)\( T^{6} + \)\(28\!\cdots\!52\)\( T^{7} + \)\(37\!\cdots\!59\)\( T^{8} + \)\(45\!\cdots\!06\)\( T^{9} + \)\(51\!\cdots\!11\)\( T^{10} + \)\(55\!\cdots\!62\)\( T^{11} + \)\(57\!\cdots\!14\)\( T^{12} + \)\(55\!\cdots\!70\)\( T^{13} + \)\(52\!\cdots\!51\)\( T^{14} + \)\(46\!\cdots\!98\)\( T^{15} + \)\(40\!\cdots\!16\)\( T^{16} + \)\(33\!\cdots\!38\)\( T^{17} + \)\(26\!\cdots\!11\)\( T^{18} + \)\(19\!\cdots\!70\)\( T^{19} + \)\(14\!\cdots\!94\)\( T^{20} + \)\(98\!\cdots\!62\)\( T^{21} + \)\(64\!\cdots\!91\)\( T^{22} + \)\(39\!\cdots\!66\)\( T^{23} + \)\(23\!\cdots\!19\)\( T^{24} + \)\(12\!\cdots\!92\)\( T^{25} + \)\(63\!\cdots\!28\)\( T^{26} + \)\(28\!\cdots\!72\)\( T^{27} + \)\(11\!\cdots\!19\)\( T^{28} + \)\(41\!\cdots\!42\)\( T^{29} + \)\(12\!\cdots\!15\)\( T^{30} + \)\(27\!\cdots\!14\)\( T^{31} + \)\(39\!\cdots\!81\)\( T^{32} \)
$31$ \( 1 - 6437968 T^{2} + 22425565747056 T^{4} - 55085751421175590712 T^{6} + \)\(10\!\cdots\!40\)\( T^{8} - \)\(16\!\cdots\!92\)\( T^{10} + \)\(22\!\cdots\!48\)\( T^{12} - \)\(25\!\cdots\!76\)\( T^{14} + \)\(25\!\cdots\!10\)\( T^{16} - \)\(21\!\cdots\!16\)\( T^{18} + \)\(16\!\cdots\!88\)\( T^{20} - \)\(10\!\cdots\!32\)\( T^{22} + \)\(55\!\cdots\!40\)\( T^{24} - \)\(24\!\cdots\!12\)\( T^{26} + \)\(86\!\cdots\!96\)\( T^{28} - \)\(21\!\cdots\!08\)\( T^{30} + \)\(27\!\cdots\!21\)\( T^{32} \)
$37$ \( 1 - 18000880 T^{2} + 162878051927064 T^{4} - \)\(98\!\cdots\!16\)\( T^{6} + \)\(44\!\cdots\!20\)\( T^{8} - \)\(15\!\cdots\!92\)\( T^{10} + \)\(45\!\cdots\!84\)\( T^{12} - \)\(11\!\cdots\!72\)\( T^{14} + \)\(22\!\cdots\!58\)\( T^{16} - \)\(39\!\cdots\!12\)\( T^{18} + \)\(56\!\cdots\!44\)\( T^{20} - \)\(68\!\cdots\!12\)\( T^{22} + \)\(67\!\cdots\!20\)\( T^{24} - \)\(52\!\cdots\!16\)\( T^{26} + \)\(30\!\cdots\!44\)\( T^{28} - \)\(11\!\cdots\!80\)\( T^{30} + \)\(23\!\cdots\!61\)\( T^{32} \)
$41$ \( 1 + 1692 T + 11942060 T^{2} + 18591310224 T^{3} + 64846251000198 T^{4} + 82077789005033724 T^{5} + \)\(21\!\cdots\!64\)\( T^{6} + \)\(26\!\cdots\!00\)\( T^{7} + \)\(65\!\cdots\!49\)\( T^{8} + \)\(11\!\cdots\!04\)\( T^{9} + \)\(27\!\cdots\!76\)\( T^{10} + \)\(56\!\cdots\!88\)\( T^{11} + \)\(97\!\cdots\!10\)\( T^{12} + \)\(19\!\cdots\!52\)\( T^{13} + \)\(24\!\cdots\!88\)\( T^{14} + \)\(49\!\cdots\!60\)\( T^{15} + \)\(57\!\cdots\!36\)\( T^{16} + \)\(14\!\cdots\!60\)\( T^{17} + \)\(19\!\cdots\!48\)\( T^{18} + \)\(43\!\cdots\!12\)\( T^{19} + \)\(62\!\cdots\!10\)\( T^{20} + \)\(10\!\cdots\!88\)\( T^{21} + \)\(13\!\cdots\!36\)\( T^{22} + \)\(17\!\cdots\!84\)\( T^{23} + \)\(26\!\cdots\!69\)\( T^{24} + \)\(30\!\cdots\!00\)\( T^{25} + \)\(70\!\cdots\!64\)\( T^{26} + \)\(75\!\cdots\!64\)\( T^{27} + \)\(16\!\cdots\!58\)\( T^{28} + \)\(13\!\cdots\!44\)\( T^{29} + \)\(24\!\cdots\!60\)\( T^{30} + \)\(98\!\cdots\!92\)\( T^{31} + \)\(16\!\cdots\!61\)\( T^{32} \)
$43$ \( 1 + 7402 T + 13485727 T^{2} - 12515487826 T^{3} - 488538342193 T^{4} + 197550457232213648 T^{5} - 49592023197193748448 T^{6} - \)\(13\!\cdots\!48\)\( T^{7} - \)\(53\!\cdots\!61\)\( T^{8} + \)\(20\!\cdots\!10\)\( T^{9} - \)\(72\!\cdots\!01\)\( T^{10} - \)\(18\!\cdots\!82\)\( T^{11} + \)\(19\!\cdots\!94\)\( T^{12} + \)\(42\!\cdots\!10\)\( T^{13} - \)\(66\!\cdots\!65\)\( T^{14} + \)\(51\!\cdots\!34\)\( T^{15} + \)\(58\!\cdots\!32\)\( T^{16} + \)\(17\!\cdots\!34\)\( T^{17} - \)\(77\!\cdots\!65\)\( T^{18} + \)\(17\!\cdots\!10\)\( T^{19} + \)\(26\!\cdots\!94\)\( T^{20} - \)\(84\!\cdots\!82\)\( T^{21} - \)\(11\!\cdots\!01\)\( T^{22} + \)\(11\!\cdots\!10\)\( T^{23} - \)\(10\!\cdots\!61\)\( T^{24} - \)\(83\!\cdots\!48\)\( T^{25} - \)\(10\!\cdots\!48\)\( T^{26} + \)\(14\!\cdots\!48\)\( T^{27} - \)\(12\!\cdots\!93\)\( T^{28} - \)\(10\!\cdots\!26\)\( T^{29} + \)\(40\!\cdots\!27\)\( T^{30} + \)\(75\!\cdots\!02\)\( T^{31} + \)\(34\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 3198 T - 13943157 T^{2} + 47488086342 T^{3} + 127200203812879 T^{4} - 435875751219740292 T^{5} - \)\(72\!\cdots\!76\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!71\)\( T^{8} - \)\(12\!\cdots\!22\)\( T^{9} + \)\(57\!\cdots\!31\)\( T^{10} + \)\(48\!\cdots\!22\)\( T^{11} - \)\(14\!\cdots\!30\)\( T^{12} - \)\(15\!\cdots\!74\)\( T^{13} + \)\(12\!\cdots\!35\)\( T^{14} + \)\(24\!\cdots\!02\)\( T^{15} - \)\(72\!\cdots\!52\)\( T^{16} + \)\(11\!\cdots\!62\)\( T^{17} + \)\(30\!\cdots\!35\)\( T^{18} - \)\(17\!\cdots\!34\)\( T^{19} - \)\(83\!\cdots\!30\)\( T^{20} + \)\(13\!\cdots\!22\)\( T^{21} + \)\(77\!\cdots\!11\)\( T^{22} - \)\(84\!\cdots\!42\)\( T^{23} + \)\(77\!\cdots\!11\)\( T^{24} + \)\(41\!\cdots\!60\)\( T^{25} - \)\(55\!\cdots\!76\)\( T^{26} - \)\(16\!\cdots\!52\)\( T^{27} + \)\(23\!\cdots\!19\)\( T^{28} + \)\(42\!\cdots\!22\)\( T^{29} - \)\(60\!\cdots\!97\)\( T^{30} - \)\(67\!\cdots\!98\)\( T^{31} + \)\(10\!\cdots\!81\)\( T^{32} \)
$53$ \( 1 - 3870 T + 38256023 T^{2} - 128730608010 T^{3} + 698551543483743 T^{4} - 2141411986130252256 T^{5} + \)\(82\!\cdots\!28\)\( T^{6} - \)\(21\!\cdots\!64\)\( T^{7} + \)\(64\!\cdots\!83\)\( T^{8} - \)\(11\!\cdots\!46\)\( T^{9} + \)\(22\!\cdots\!87\)\( T^{10} + \)\(27\!\cdots\!26\)\( T^{11} - \)\(19\!\cdots\!94\)\( T^{12} + \)\(12\!\cdots\!70\)\( T^{13} - \)\(43\!\cdots\!01\)\( T^{14} + \)\(15\!\cdots\!38\)\( T^{15} - \)\(42\!\cdots\!20\)\( T^{16} + \)\(12\!\cdots\!78\)\( T^{17} - \)\(26\!\cdots\!61\)\( T^{18} + \)\(62\!\cdots\!70\)\( T^{19} - \)\(75\!\cdots\!74\)\( T^{20} + \)\(83\!\cdots\!26\)\( T^{21} + \)\(54\!\cdots\!47\)\( T^{22} - \)\(21\!\cdots\!06\)\( T^{23} + \)\(96\!\cdots\!03\)\( T^{24} - \)\(25\!\cdots\!44\)\( T^{25} + \)\(77\!\cdots\!28\)\( T^{26} - \)\(15\!\cdots\!36\)\( T^{27} + \)\(40\!\cdots\!23\)\( T^{28} - \)\(59\!\cdots\!10\)\( T^{29} + \)\(13\!\cdots\!83\)\( T^{30} - \)\(11\!\cdots\!70\)\( T^{31} + \)\(22\!\cdots\!81\)\( T^{32} \)
$59$ \( 1 + 18288 T + 223785800 T^{2} + 2053781755776 T^{3} + 15822384632669082 T^{4} + \)\(10\!\cdots\!48\)\( T^{5} + \)\(65\!\cdots\!36\)\( T^{6} + \)\(36\!\cdots\!96\)\( T^{7} + \)\(19\!\cdots\!37\)\( T^{8} + \)\(94\!\cdots\!16\)\( T^{9} + \)\(43\!\cdots\!48\)\( T^{10} + \)\(19\!\cdots\!08\)\( T^{11} + \)\(81\!\cdots\!82\)\( T^{12} + \)\(32\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!12\)\( T^{14} + \)\(47\!\cdots\!56\)\( T^{15} + \)\(16\!\cdots\!92\)\( T^{16} + \)\(57\!\cdots\!16\)\( T^{17} + \)\(18\!\cdots\!52\)\( T^{18} + \)\(58\!\cdots\!36\)\( T^{19} + \)\(17\!\cdots\!62\)\( T^{20} + \)\(50\!\cdots\!08\)\( T^{21} + \)\(13\!\cdots\!28\)\( T^{22} + \)\(36\!\cdots\!36\)\( T^{23} + \)\(88\!\cdots\!97\)\( T^{24} + \)\(20\!\cdots\!36\)\( T^{25} + \)\(44\!\cdots\!36\)\( T^{26} + \)\(88\!\cdots\!28\)\( T^{27} + \)\(15\!\cdots\!22\)\( T^{28} + \)\(24\!\cdots\!56\)\( T^{29} + \)\(32\!\cdots\!00\)\( T^{30} + \)\(32\!\cdots\!88\)\( T^{31} + \)\(21\!\cdots\!61\)\( T^{32} \)
$61$ \( 1 + 6522 T - 22066825 T^{2} - 21555809418 T^{3} + 1450237788720423 T^{4} - 883667360510693220 T^{5} - \)\(26\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!84\)\( T^{7} + \)\(32\!\cdots\!63\)\( T^{8} - \)\(30\!\cdots\!70\)\( T^{9} + \)\(40\!\cdots\!59\)\( T^{10} + \)\(50\!\cdots\!94\)\( T^{11} - \)\(12\!\cdots\!98\)\( T^{12} - \)\(60\!\cdots\!18\)\( T^{13} + \)\(29\!\cdots\!51\)\( T^{14} + \)\(32\!\cdots\!06\)\( T^{15} - \)\(48\!\cdots\!28\)\( T^{16} + \)\(44\!\cdots\!46\)\( T^{17} + \)\(56\!\cdots\!31\)\( T^{18} - \)\(15\!\cdots\!78\)\( T^{19} - \)\(44\!\cdots\!78\)\( T^{20} + \)\(25\!\cdots\!94\)\( T^{21} + \)\(28\!\cdots\!19\)\( T^{22} - \)\(29\!\cdots\!70\)\( T^{23} + \)\(44\!\cdots\!23\)\( T^{24} + \)\(21\!\cdots\!24\)\( T^{25} - \)\(68\!\cdots\!12\)\( T^{26} - \)\(31\!\cdots\!20\)\( T^{27} + \)\(71\!\cdots\!63\)\( T^{28} - \)\(14\!\cdots\!78\)\( T^{29} - \)\(20\!\cdots\!25\)\( T^{30} + \)\(85\!\cdots\!22\)\( T^{31} + \)\(18\!\cdots\!41\)\( T^{32} \)
$67$ \( 1 + 30168 T + 574581992 T^{2} + 8181941234112 T^{3} + 96622024838601042 T^{4} + \)\(98\!\cdots\!12\)\( T^{5} + \)\(89\!\cdots\!00\)\( T^{6} + \)\(73\!\cdots\!56\)\( T^{7} + \)\(55\!\cdots\!21\)\( T^{8} + \)\(39\!\cdots\!04\)\( T^{9} + \)\(25\!\cdots\!40\)\( T^{10} + \)\(15\!\cdots\!64\)\( T^{11} + \)\(90\!\cdots\!46\)\( T^{12} + \)\(49\!\cdots\!56\)\( T^{13} + \)\(25\!\cdots\!60\)\( T^{14} + \)\(12\!\cdots\!28\)\( T^{15} + \)\(56\!\cdots\!92\)\( T^{16} + \)\(24\!\cdots\!88\)\( T^{17} + \)\(10\!\cdots\!60\)\( T^{18} + \)\(40\!\cdots\!16\)\( T^{19} + \)\(14\!\cdots\!26\)\( T^{20} + \)\(52\!\cdots\!64\)\( T^{21} + \)\(17\!\cdots\!40\)\( T^{22} + \)\(52\!\cdots\!64\)\( T^{23} + \)\(15\!\cdots\!81\)\( T^{24} + \)\(40\!\cdots\!36\)\( T^{25} + \)\(99\!\cdots\!00\)\( T^{26} + \)\(21\!\cdots\!52\)\( T^{27} + \)\(43\!\cdots\!22\)\( T^{28} + \)\(73\!\cdots\!32\)\( T^{29} + \)\(10\!\cdots\!52\)\( T^{30} + \)\(11\!\cdots\!68\)\( T^{31} + \)\(73\!\cdots\!21\)\( T^{32} \)
$71$ \( 1 - 35874 T + 765398411 T^{2} - 12068627727006 T^{3} + 155044868619458331 T^{4} - \)\(17\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!08\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!99\)\( T^{8} - \)\(89\!\cdots\!30\)\( T^{9} + \)\(62\!\cdots\!95\)\( T^{10} - \)\(41\!\cdots\!54\)\( T^{11} + \)\(26\!\cdots\!34\)\( T^{12} - \)\(15\!\cdots\!54\)\( T^{13} + \)\(87\!\cdots\!59\)\( T^{14} - \)\(47\!\cdots\!42\)\( T^{15} + \)\(24\!\cdots\!64\)\( T^{16} - \)\(12\!\cdots\!02\)\( T^{17} + \)\(56\!\cdots\!99\)\( T^{18} - \)\(25\!\cdots\!14\)\( T^{19} + \)\(10\!\cdots\!14\)\( T^{20} - \)\(44\!\cdots\!54\)\( T^{21} + \)\(16\!\cdots\!95\)\( T^{22} - \)\(61\!\cdots\!30\)\( T^{23} + \)\(20\!\cdots\!59\)\( T^{24} - \)\(64\!\cdots\!40\)\( T^{25} + \)\(18\!\cdots\!08\)\( T^{26} - \)\(48\!\cdots\!92\)\( T^{27} + \)\(11\!\cdots\!91\)\( T^{28} - \)\(22\!\cdots\!46\)\( T^{29} + \)\(35\!\cdots\!31\)\( T^{30} - \)\(42\!\cdots\!74\)\( T^{31} + \)\(30\!\cdots\!81\)\( T^{32} \)
$73$ \( 1 + 8080 T - 109749380 T^{2} - 914006920432 T^{3} + 6832868105965790 T^{4} + 51636721949355226616 T^{5} - \)\(35\!\cdots\!12\)\( T^{6} - \)\(20\!\cdots\!08\)\( T^{7} + \)\(16\!\cdots\!69\)\( T^{8} + \)\(70\!\cdots\!04\)\( T^{9} - \)\(67\!\cdots\!52\)\( T^{10} - \)\(19\!\cdots\!28\)\( T^{11} + \)\(24\!\cdots\!78\)\( T^{12} + \)\(41\!\cdots\!04\)\( T^{13} - \)\(78\!\cdots\!04\)\( T^{14} - \)\(42\!\cdots\!88\)\( T^{15} + \)\(23\!\cdots\!40\)\( T^{16} - \)\(12\!\cdots\!08\)\( T^{17} - \)\(63\!\cdots\!24\)\( T^{18} + \)\(95\!\cdots\!84\)\( T^{19} + \)\(15\!\cdots\!58\)\( T^{20} - \)\(36\!\cdots\!28\)\( T^{21} - \)\(35\!\cdots\!32\)\( T^{22} + \)\(10\!\cdots\!24\)\( T^{23} + \)\(70\!\cdots\!49\)\( T^{24} - \)\(25\!\cdots\!88\)\( T^{25} - \)\(12\!\cdots\!12\)\( T^{26} + \)\(50\!\cdots\!56\)\( T^{27} + \)\(18\!\cdots\!90\)\( T^{28} - \)\(71\!\cdots\!72\)\( T^{29} - \)\(24\!\cdots\!80\)\( T^{30} + \)\(50\!\cdots\!80\)\( T^{31} + \)\(17\!\cdots\!41\)\( T^{32} \)
$79$ \( 1 + 30738 T + 579590639 T^{2} + 8134783759158 T^{3} + 92552899346028159 T^{4} + \)\(89\!\cdots\!68\)\( T^{5} + \)\(76\!\cdots\!16\)\( T^{6} + \)\(58\!\cdots\!40\)\( T^{7} + \)\(41\!\cdots\!63\)\( T^{8} + \)\(28\!\cdots\!82\)\( T^{9} + \)\(18\!\cdots\!03\)\( T^{10} + \)\(12\!\cdots\!02\)\( T^{11} + \)\(84\!\cdots\!06\)\( T^{12} + \)\(59\!\cdots\!02\)\( T^{13} + \)\(41\!\cdots\!47\)\( T^{14} + \)\(27\!\cdots\!30\)\( T^{15} + \)\(17\!\cdots\!28\)\( T^{16} + \)\(10\!\cdots\!30\)\( T^{17} + \)\(62\!\cdots\!67\)\( T^{18} + \)\(34\!\cdots\!82\)\( T^{19} + \)\(19\!\cdots\!26\)\( T^{20} + \)\(11\!\cdots\!02\)\( T^{21} + \)\(64\!\cdots\!43\)\( T^{22} + \)\(38\!\cdots\!02\)\( T^{23} + \)\(22\!\cdots\!83\)\( T^{24} + \)\(12\!\cdots\!40\)\( T^{25} + \)\(61\!\cdots\!16\)\( T^{26} + \)\(28\!\cdots\!08\)\( T^{27} + \)\(11\!\cdots\!99\)\( T^{28} + \)\(38\!\cdots\!78\)\( T^{29} + \)\(10\!\cdots\!19\)\( T^{30} + \)\(22\!\cdots\!38\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$83$ \( ( 1 + 738 T + 209637123 T^{2} - 436284340398 T^{3} + 20800218550032725 T^{4} - 81050996008981429308 T^{5} + \)\(15\!\cdots\!66\)\( T^{6} - \)\(60\!\cdots\!28\)\( T^{7} + \)\(83\!\cdots\!14\)\( T^{8} - \)\(28\!\cdots\!88\)\( T^{9} + \)\(33\!\cdots\!06\)\( T^{10} - \)\(86\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!25\)\( T^{12} - \)\(10\!\cdots\!98\)\( T^{13} + \)\(23\!\cdots\!83\)\( T^{14} + \)\(40\!\cdots\!58\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} )^{2} \)
$89$ \( 1 - 19782 T + 552306071 T^{2} - 8345305003266 T^{3} + 142273060053685167 T^{4} - \)\(17\!\cdots\!12\)\( T^{5} + \)\(23\!\cdots\!04\)\( T^{6} - \)\(25\!\cdots\!48\)\( T^{7} + \)\(29\!\cdots\!19\)\( T^{8} - \)\(28\!\cdots\!62\)\( T^{9} + \)\(29\!\cdots\!39\)\( T^{10} - \)\(26\!\cdots\!30\)\( T^{11} + \)\(24\!\cdots\!66\)\( T^{12} - \)\(20\!\cdots\!34\)\( T^{13} + \)\(17\!\cdots\!19\)\( T^{14} - \)\(14\!\cdots\!18\)\( T^{15} + \)\(11\!\cdots\!08\)\( T^{16} - \)\(89\!\cdots\!38\)\( T^{17} + \)\(70\!\cdots\!39\)\( T^{18} - \)\(50\!\cdots\!14\)\( T^{19} + \)\(37\!\cdots\!26\)\( T^{20} - \)\(25\!\cdots\!30\)\( T^{21} + \)\(17\!\cdots\!99\)\( T^{22} - \)\(11\!\cdots\!22\)\( T^{23} + \)\(70\!\cdots\!99\)\( T^{24} - \)\(39\!\cdots\!28\)\( T^{25} + \)\(22\!\cdots\!04\)\( T^{26} - \)\(10\!\cdots\!92\)\( T^{27} + \)\(52\!\cdots\!27\)\( T^{28} - \)\(19\!\cdots\!86\)\( T^{29} + \)\(80\!\cdots\!31\)\( T^{30} - \)\(18\!\cdots\!82\)\( T^{31} + \)\(57\!\cdots\!41\)\( T^{32} \)
$97$ \( 1 + 9936 T + 333432212 T^{2} + 2986008252480 T^{3} + 48744620267900718 T^{4} + \)\(38\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!16\)\( T^{6} + \)\(43\!\cdots\!16\)\( T^{7} + \)\(64\!\cdots\!45\)\( T^{8} + \)\(60\!\cdots\!44\)\( T^{9} + \)\(70\!\cdots\!52\)\( T^{10} + \)\(63\!\cdots\!00\)\( T^{11} + \)\(65\!\cdots\!26\)\( T^{12} + \)\(61\!\cdots\!32\)\( T^{13} + \)\(64\!\cdots\!12\)\( T^{14} + \)\(66\!\cdots\!72\)\( T^{15} + \)\(63\!\cdots\!52\)\( T^{16} + \)\(59\!\cdots\!32\)\( T^{17} + \)\(50\!\cdots\!32\)\( T^{18} + \)\(42\!\cdots\!12\)\( T^{19} + \)\(40\!\cdots\!46\)\( T^{20} + \)\(34\!\cdots\!00\)\( T^{21} + \)\(34\!\cdots\!12\)\( T^{22} + \)\(25\!\cdots\!84\)\( T^{23} + \)\(24\!\cdots\!45\)\( T^{24} + \)\(14\!\cdots\!36\)\( T^{25} + \)\(15\!\cdots\!16\)\( T^{26} + \)\(99\!\cdots\!00\)\( T^{27} + \)\(11\!\cdots\!98\)\( T^{28} + \)\(61\!\cdots\!80\)\( T^{29} + \)\(60\!\cdots\!52\)\( T^{30} + \)\(15\!\cdots\!36\)\( T^{31} + \)\(14\!\cdots\!81\)\( T^{32} \)
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