Properties

Label 40.6.d.a
Level 40
Weight 6
Character orbit 40.d
Analytic conductor 6.415
Analytic rank 0
Dimension 20
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 40.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.41535279252\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} - 109104 x^{12} - 96128 x^{11} + 3580672 x^{10} - 1538048 x^{9} - 27930624 x^{8} + 79364096 x^{7} + 157024256 x^{6} - 926941184 x^{5} + 4244635648 x^{4} + 20937965568 x^{3} - 73014444032 x^{2} - 137438953472 x + 1099511627776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{4}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{2} q^{3} + ( -2 + \beta_{3} ) q^{4} + \beta_{5} q^{5} + ( 10 - \beta_{2} + \beta_{9} ) q^{6} + ( -10 - 4 \beta_{1} - \beta_{8} ) q^{7} + ( 13 + 2 \beta_{1} - \beta_{5} - \beta_{6} ) q^{8} + ( -80 + 8 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{2} q^{3} + ( -2 + \beta_{3} ) q^{4} + \beta_{5} q^{5} + ( 10 - \beta_{2} + \beta_{9} ) q^{6} + ( -10 - 4 \beta_{1} - \beta_{8} ) q^{7} + ( 13 + 2 \beta_{1} - \beta_{5} - \beta_{6} ) q^{8} + ( -80 + 8 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{9} + ( -3 + \beta_{2} - \beta_{11} ) q^{10} + ( 4 + 22 \beta_{1} + \beta_{2} - 4 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} + \beta_{9} - \beta_{11} + \beta_{17} ) q^{11} + ( -95 - 9 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} - \beta_{18} ) q^{12} + ( -1 - 15 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} - \beta_{18} ) q^{13} + ( 135 + 10 \beta_{1} - \beta_{2} + 5 \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{14} + ( 45 - 9 \beta_{1} - \beta_{3} - \beta_{7} + \beta_{12} - \beta_{19} ) q^{15} + ( 154 - 18 \beta_{1} - 16 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{11} - 4 \beta_{12} + \beta_{13} - \beta_{14} + 4 \beta_{15} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{16} + ( -1 - 32 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} - 3 \beta_{16} - \beta_{17} - 2 \beta_{19} ) q^{17} + ( -255 + 71 \beta_{1} - 24 \beta_{2} - 11 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{14} + 3 \beta_{16} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{18} + ( -8 - 30 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} - 13 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{19} + ( -92 + \beta_{1} - 6 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{11} - 4 \beta_{12} - \beta_{19} ) q^{20} + ( -8 - 6 \beta_{1} + 50 \beta_{2} + 21 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 2 \beta_{10} + 4 \beta_{11} + 7 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - 3 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} ) q^{21} + ( 698 - 5 \beta_{1} - 12 \beta_{2} - 26 \beta_{3} - \beta_{4} + 24 \beta_{5} + 4 \beta_{6} - 14 \beta_{7} + 7 \beta_{8} - \beta_{10} - 4 \beta_{11} + \beta_{12} - 2 \beta_{13} - 3 \beta_{15} + 2 \beta_{16} - 3 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{22} + ( -227 + 50 \beta_{1} + 9 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 10 \beta_{7} - 6 \beta_{8} - 3 \beta_{9} - \beta_{10} + 13 \beta_{11} + 3 \beta_{12} + 2 \beta_{13} + \beta_{14} - 7 \beta_{15} - 3 \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{23} + ( 53 + 102 \beta_{1} + 30 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} + 17 \beta_{5} - 2 \beta_{6} + 10 \beta_{7} + \beta_{8} - 7 \beta_{9} + 2 \beta_{10} + \beta_{11} + 5 \beta_{13} - 6 \beta_{15} + 4 \beta_{16} + \beta_{18} + 5 \beta_{19} ) q^{24} -625 q^{25} + ( -417 + 6 \beta_{1} + 22 \beta_{2} + 17 \beta_{3} - 4 \beta_{4} - 23 \beta_{5} + 5 \beta_{6} - \beta_{7} + 2 \beta_{8} + 11 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 8 \beta_{12} - 6 \beta_{13} + \beta_{14} + 4 \beta_{15} + 3 \beta_{16} - 5 \beta_{17} + \beta_{18} + 6 \beta_{19} ) q^{26} + ( -23 - 140 \beta_{1} + 70 \beta_{2} - 11 \beta_{3} - 3 \beta_{4} + 15 \beta_{5} - 3 \beta_{6} + 10 \beta_{7} - 5 \beta_{8} - 9 \beta_{9} + 3 \beta_{10} - 5 \beta_{11} + 21 \beta_{12} + 6 \beta_{13} - 3 \beta_{14} - 11 \beta_{15} - 5 \beta_{16} - 4 \beta_{17} + 3 \beta_{18} ) q^{27} + ( 91 - 161 \beta_{1} - 48 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} - 48 \beta_{5} - 4 \beta_{6} + 7 \beta_{7} + 9 \beta_{8} - 7 \beta_{9} + 2 \beta_{10} + 6 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} + 6 \beta_{16} + 3 \beta_{18} - 2 \beta_{19} ) q^{28} + ( 16 + 72 \beta_{1} - 54 \beta_{2} - 33 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 7 \beta_{8} - 16 \beta_{9} - 4 \beta_{10} + 10 \beta_{11} + 19 \beta_{12} - 7 \beta_{13} - \beta_{14} - 5 \beta_{15} - \beta_{16} - 8 \beta_{17} + 10 \beta_{18} ) q^{29} + ( 280 - 39 \beta_{1} + 11 \beta_{2} + 16 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + \beta_{10} + 5 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} + 2 \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{30} + ( 374 - 94 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} + 12 \beta_{6} - 10 \beta_{7} + 10 \beta_{10} + 8 \beta_{11} - 22 \beta_{12} + 2 \beta_{13} - 6 \beta_{14} + 16 \beta_{15} - 2 \beta_{16} + 10 \beta_{17} - 2 \beta_{19} ) q^{31} + ( 304 - 128 \beta_{1} + 60 \beta_{2} + 20 \beta_{3} + 8 \beta_{4} + 38 \beta_{5} + 16 \beta_{6} + 16 \beta_{8} + 16 \beta_{9} - 4 \beta_{10} + 14 \beta_{11} - 8 \beta_{12} + 12 \beta_{14} - 8 \beta_{15} - 8 \beta_{16} - 8 \beta_{17} + 6 \beta_{19} ) q^{32} + ( 295 + 80 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} + 17 \beta_{4} - 25 \beta_{5} - 7 \beta_{6} + 9 \beta_{7} - 9 \beta_{8} - 41 \beta_{9} + 3 \beta_{10} - 18 \beta_{11} - 11 \beta_{13} - \beta_{14} - 10 \beta_{15} + \beta_{16} + \beta_{17} - 2 \beta_{18} - 6 \beta_{19} ) q^{33} + ( 1043 + 4 \beta_{1} - 4 \beta_{2} + 43 \beta_{3} + 8 \beta_{4} - 37 \beta_{5} + 11 \beta_{6} - 15 \beta_{7} + 34 \beta_{8} + 11 \beta_{9} - 4 \beta_{10} + \beta_{11} - 24 \beta_{12} - 4 \beta_{13} - 5 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} + \beta_{17} - 9 \beta_{18} + 2 \beta_{19} ) q^{34} + ( 17 + 34 \beta_{1} - 20 \beta_{2} - 31 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} - 7 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} - 8 \beta_{9} - \beta_{10} - 6 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} + 5 \beta_{15} + 3 \beta_{16} + \beta_{17} + 7 \beta_{18} ) q^{35} + ( 968 + 302 \beta_{1} + 128 \beta_{2} - 55 \beta_{3} - 16 \beta_{5} + 16 \beta_{6} - 42 \beta_{7} + 14 \beta_{8} + 38 \beta_{9} + 4 \beta_{10} - 8 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} + 12 \beta_{15} + 4 \beta_{16} + 8 \beta_{17} + 10 \beta_{18} - 4 \beta_{19} ) q^{36} + ( 18 + 116 \beta_{1} - 20 \beta_{2} - 12 \beta_{3} + 4 \beta_{5} + 22 \beta_{6} - 14 \beta_{7} + 8 \beta_{8} - 6 \beta_{9} - 14 \beta_{10} + 26 \beta_{11} - 8 \beta_{12} - 14 \beta_{13} + 8 \beta_{14} - 8 \beta_{15} - 6 \beta_{18} ) q^{37} + ( -1006 - 21 \beta_{1} - 58 \beta_{2} + 26 \beta_{3} - \beta_{4} + 28 \beta_{5} - 8 \beta_{6} + 22 \beta_{7} + 15 \beta_{8} - 14 \beta_{9} - 17 \beta_{10} + 16 \beta_{11} + 17 \beta_{12} - 10 \beta_{13} + 4 \beta_{14} - 3 \beta_{15} - 2 \beta_{16} - 23 \beta_{17} + 6 \beta_{18} - 10 \beta_{19} ) q^{38} + ( -2210 - 164 \beta_{1} - 36 \beta_{2} - 104 \beta_{3} - 22 \beta_{4} + 14 \beta_{5} - 14 \beta_{6} + 16 \beta_{8} + 6 \beta_{9} - 4 \beta_{10} + 46 \beta_{11} + 6 \beta_{12} + 14 \beta_{13} + 8 \beta_{14} - 2 \beta_{15} - 12 \beta_{16} - 6 \beta_{17} - 2 \beta_{18} ) q^{39} + ( 330 + 80 \beta_{1} - 54 \beta_{2} + 2 \beta_{3} + 10 \beta_{4} + 22 \beta_{5} + 7 \beta_{6} - 8 \beta_{7} + 15 \beta_{8} + 13 \beta_{9} + 6 \beta_{10} - 5 \beta_{11} - 16 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} + 10 \beta_{15} + 2 \beta_{16} + 14 \beta_{17} - 7 \beta_{18} + \beta_{19} ) q^{40} + ( 565 + 352 \beta_{1} + 16 \beta_{2} + 32 \beta_{3} + 9 \beta_{4} + 49 \beta_{5} + 49 \beta_{6} - \beta_{7} - 15 \beta_{8} + 65 \beta_{9} - 48 \beta_{11} - 16 \beta_{12} - 8 \beta_{13} - 8 \beta_{14} + 24 \beta_{15} + 8 \beta_{16} + 8 \beta_{19} ) q^{41} + ( -868 - 46 \beta_{1} - 66 \beta_{2} - 6 \beta_{3} - 30 \beta_{4} - 146 \beta_{5} - 34 \beta_{6} + 58 \beta_{7} - 34 \beta_{8} - 74 \beta_{9} - 6 \beta_{10} + 22 \beta_{12} - 8 \beta_{13} - 2 \beta_{14} - 10 \beta_{15} - 2 \beta_{16} - 16 \beta_{17} - 6 \beta_{18} + 8 \beta_{19} ) q^{42} + ( -35 + 80 \beta_{1} + 159 \beta_{2} + 85 \beta_{3} - 3 \beta_{4} + 65 \beta_{5} + 17 \beta_{6} + 10 \beta_{7} + 3 \beta_{8} - 43 \beta_{9} - 9 \beta_{10} - 19 \beta_{11} - 3 \beta_{12} + 34 \beta_{13} + 5 \beta_{14} - 3 \beta_{15} - 5 \beta_{16} + 6 \beta_{17} - 17 \beta_{18} ) q^{43} + ( 3542 - 650 \beta_{1} + 88 \beta_{2} + 36 \beta_{3} + 24 \beta_{4} - 92 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} - 32 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} + 24 \beta_{12} + 24 \beta_{13} - 10 \beta_{14} - 20 \beta_{15} - 6 \beta_{16} + 10 \beta_{17} - 2 \beta_{18} - 10 \beta_{19} ) q^{44} + ( 33 + 61 \beta_{1} - 90 \beta_{2} - 19 \beta_{3} + 5 \beta_{4} - 64 \beta_{5} - 13 \beta_{6} - 9 \beta_{7} + 33 \beta_{9} + 6 \beta_{10} - 4 \beta_{11} - 20 \beta_{12} - 12 \beta_{13} - 2 \beta_{14} + 20 \beta_{15} + 7 \beta_{16} + 9 \beta_{17} + 3 \beta_{18} ) q^{45} + ( -1345 + 248 \beta_{1} + 31 \beta_{2} - 47 \beta_{3} - 18 \beta_{4} + 149 \beta_{5} - 7 \beta_{6} - 49 \beta_{7} + 14 \beta_{8} + 41 \beta_{9} + 22 \beta_{10} - 53 \beta_{11} - 24 \beta_{12} - 17 \beta_{13} - 23 \beta_{14} + 55 \beta_{15} + 13 \beta_{16} + 35 \beta_{17} - 13 \beta_{18} - 4 \beta_{19} ) q^{46} + ( 2168 - 198 \beta_{1} + 40 \beta_{2} + 130 \beta_{3} - 8 \beta_{4} - 80 \beta_{5} - 56 \beta_{6} + 14 \beta_{7} + 3 \beta_{8} - 64 \beta_{9} + 4 \beta_{10} + 40 \beta_{11} + 18 \beta_{12} + 12 \beta_{13} + 4 \beta_{14} - 24 \beta_{15} + 12 \beta_{16} + 12 \beta_{17} + 8 \beta_{18} + 6 \beta_{19} ) q^{47} + ( -4350 - 80 \beta_{1} - 76 \beta_{2} - 174 \beta_{3} - 24 \beta_{4} + 180 \beta_{5} - 40 \beta_{6} - 34 \beta_{7} - 54 \beta_{8} - 12 \beta_{9} + 16 \beta_{10} - 30 \beta_{11} - 12 \beta_{12} + 6 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} + 6 \beta_{16} + 34 \beta_{17} + 16 \beta_{18} + 4 \beta_{19} ) q^{48} + ( 922 - 384 \beta_{1} - 44 \beta_{2} - 194 \beta_{3} - 33 \beta_{4} + 19 \beta_{5} - 9 \beta_{6} - 43 \beta_{7} + 19 \beta_{8} + 3 \beta_{9} - 18 \beta_{10} - 60 \beta_{11} + 40 \beta_{12} - 14 \beta_{13} + 14 \beta_{14} - 4 \beta_{15} + 2 \beta_{16} - 30 \beta_{17} - 12 \beta_{18} - 4 \beta_{19} ) q^{49} + 625 \beta_{1} q^{50} + ( 10 - 64 \beta_{1} + 32 \beta_{2} + 64 \beta_{3} - 12 \beta_{4} - 124 \beta_{5} - 50 \beta_{6} - 30 \beta_{7} + 12 \beta_{8} + 136 \beta_{9} + 50 \beta_{10} - 76 \beta_{11} - 76 \beta_{12} + 14 \beta_{13} - 12 \beta_{14} + 52 \beta_{15} + 12 \beta_{16} + 38 \beta_{17} - 14 \beta_{18} ) q^{51} + ( -1934 + 292 \beta_{1} - 276 \beta_{2} - 42 \beta_{3} - 28 \beta_{4} - 254 \beta_{5} - 44 \beta_{6} + 84 \beta_{7} - 42 \beta_{8} - 78 \beta_{9} - 24 \beta_{10} + 66 \beta_{11} + 40 \beta_{12} + 22 \beta_{13} + 28 \beta_{14} - 32 \beta_{15} - 36 \beta_{16} - 4 \beta_{17} - 10 \beta_{18} - 6 \beta_{19} ) q^{52} + ( -45 - 489 \beta_{1} - 280 \beta_{2} + 44 \beta_{3} - 34 \beta_{4} + 27 \beta_{5} + 3 \beta_{6} - 26 \beta_{7} + 35 \beta_{8} + 91 \beta_{9} + 22 \beta_{10} + 34 \beta_{11} - 59 \beta_{12} - 17 \beta_{13} - 9 \beta_{14} - 11 \beta_{15} + 10 \beta_{16} - 5 \beta_{17} - 25 \beta_{18} ) q^{53} + ( -5018 + 58 \beta_{1} + 20 \beta_{2} + 62 \beta_{3} - 34 \beta_{4} + 282 \beta_{5} - 38 \beta_{6} - 6 \beta_{7} - 118 \beta_{8} - 54 \beta_{9} - 10 \beta_{10} + 6 \beta_{11} - 6 \beta_{12} - 16 \beta_{13} - 14 \beta_{14} - 46 \beta_{15} - 6 \beta_{16} - 8 \beta_{17} - 2 \beta_{18} - 8 \beta_{19} ) q^{54} + ( -1236 + 345 \beta_{1} + 36 \beta_{2} + 88 \beta_{3} - 5 \beta_{4} + 27 \beta_{5} + 23 \beta_{6} + 13 \beta_{7} - 10 \beta_{8} + 67 \beta_{9} - \beta_{10} + 3 \beta_{11} + 6 \beta_{12} + 12 \beta_{13} - 3 \beta_{14} + 15 \beta_{15} + 13 \beta_{16} + 6 \beta_{17} + 7 \beta_{18} + 9 \beta_{19} ) q^{55} + ( -2731 - 62 \beta_{1} - 58 \beta_{2} + 166 \beta_{3} + 26 \beta_{4} + 301 \beta_{5} + 84 \beta_{7} - 97 \beta_{8} + 37 \beta_{9} + 10 \beta_{10} + 49 \beta_{11} + 48 \beta_{12} + 43 \beta_{13} - 14 \beta_{14} + 34 \beta_{15} - 6 \beta_{16} - 2 \beta_{17} + 9 \beta_{18} - \beta_{19} ) q^{56} + ( 67 + 528 \beta_{1} + 106 \beta_{2} + 341 \beta_{3} + 9 \beta_{4} - 81 \beta_{5} - 71 \beta_{6} + 43 \beta_{7} + 15 \beta_{8} + 15 \beta_{9} - 43 \beta_{10} - 48 \beta_{11} + 58 \beta_{12} - 11 \beta_{13} + 5 \beta_{14} - 64 \beta_{15} + 15 \beta_{16} - 27 \beta_{17} + 16 \beta_{18} + 26 \beta_{19} ) q^{57} + ( 2940 + 126 \beta_{1} + 332 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} - 272 \beta_{5} + 24 \beta_{6} - 24 \beta_{7} - 94 \beta_{8} + 52 \beta_{9} - 46 \beta_{10} + 102 \beta_{12} - 20 \beta_{13} - 18 \beta_{15} - 28 \beta_{16} - 26 \beta_{17} + 12 \beta_{18} - 12 \beta_{19} ) q^{58} + ( -118 - 502 \beta_{1} - 225 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 145 \beta_{5} + 42 \beta_{6} + 16 \beta_{7} + 28 \beta_{8} - 17 \beta_{9} - 10 \beta_{10} - 27 \beta_{11} - 28 \beta_{12} + 2 \beta_{13} + 36 \beta_{14} - 28 \beta_{15} - 4 \beta_{16} - 23 \beta_{17} + 22 \beta_{18} ) q^{59} + ( -1587 - 241 \beta_{1} + 164 \beta_{2} + 66 \beta_{3} - 20 \beta_{4} - 138 \beta_{5} - 10 \beta_{6} + 41 \beta_{7} - 55 \beta_{8} - 5 \beta_{9} + 30 \beta_{10} - 14 \beta_{11} - 26 \beta_{12} - 5 \beta_{13} + 30 \beta_{15} + 10 \beta_{17} - 15 \beta_{18} + 6 \beta_{19} ) q^{60} + ( 248 + 1426 \beta_{1} - 136 \beta_{2} - 26 \beta_{3} + 26 \beta_{4} + 58 \beta_{5} - 16 \beta_{6} - 48 \beta_{7} + 4 \beta_{8} - 48 \beta_{9} - 26 \beta_{10} + 26 \beta_{11} - 76 \beta_{12} + 6 \beta_{13} - 16 \beta_{14} + 68 \beta_{15} + 46 \beta_{16} + 6 \beta_{17} + 4 \beta_{18} ) q^{61} + ( 2888 - 478 \beta_{1} - 494 \beta_{2} + 16 \beta_{3} + 74 \beta_{4} + 476 \beta_{5} + 36 \beta_{6} + 36 \beta_{7} + 46 \beta_{8} - 52 \beta_{9} - 14 \beta_{10} + 48 \beta_{11} + 66 \beta_{12} - 46 \beta_{13} + 44 \beta_{14} - 68 \beta_{15} - 52 \beta_{16} - 42 \beta_{17} - 28 \beta_{18} + 8 \beta_{19} ) q^{62} + ( 11969 - 302 \beta_{1} - 32 \beta_{2} - 123 \beta_{3} + 21 \beta_{4} + 169 \beta_{5} + 73 \beta_{6} - 30 \beta_{7} + 18 \beta_{8} + 233 \beta_{9} - 37 \beta_{10} + 81 \beta_{11} + 55 \beta_{12} + 42 \beta_{13} + 21 \beta_{14} - 3 \beta_{15} - 15 \beta_{16} - 32 \beta_{17} + 5 \beta_{18} + 14 \beta_{19} ) q^{63} + ( -638 - 180 \beta_{1} + 484 \beta_{2} + 124 \beta_{3} - 44 \beta_{4} + 374 \beta_{5} - 80 \beta_{6} + 80 \beta_{7} + 86 \beta_{8} - 70 \beta_{9} - 12 \beta_{10} - 2 \beta_{11} + 32 \beta_{12} + 14 \beta_{13} - 12 \beta_{14} - 20 \beta_{15} - 36 \beta_{16} + 20 \beta_{17} + 10 \beta_{18} - 38 \beta_{19} ) q^{64} + ( -61 - 504 \beta_{1} + 46 \beta_{2} + 117 \beta_{3} - 25 \beta_{4} - 23 \beta_{5} + 23 \beta_{6} - 73 \beta_{7} + 25 \beta_{8} - 23 \beta_{9} + 29 \beta_{10} - 62 \beta_{11} - 48 \beta_{12} - 13 \beta_{13} - 23 \beta_{14} + 50 \beta_{15} + 23 \beta_{16} + 31 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{65} + ( -2697 - 714 \beta_{1} - 1188 \beta_{2} - 105 \beta_{3} + 36 \beta_{4} - 769 \beta_{5} - 17 \beta_{6} - 59 \beta_{7} - 58 \beta_{8} + 31 \beta_{9} + 56 \beta_{10} - 75 \beta_{11} - 132 \beta_{12} - 52 \beta_{13} - 49 \beta_{14} + 114 \beta_{15} + 33 \beta_{16} + 65 \beta_{17} + 3 \beta_{18} - 46 \beta_{19} ) q^{66} + ( 131 - 496 \beta_{1} - 409 \beta_{2} - 537 \beta_{3} - 9 \beta_{4} - 161 \beta_{5} + 39 \beta_{6} - 78 \beta_{7} + 65 \beta_{8} + 75 \beta_{9} - 23 \beta_{10} - 69 \beta_{11} - 113 \beta_{12} + 2 \beta_{13} + 7 \beta_{14} - 17 \beta_{15} + 49 \beta_{16} + 58 \beta_{17} + 25 \beta_{18} ) q^{67} + ( 550 - 1094 \beta_{1} + 8 \beta_{2} + 18 \beta_{3} - 48 \beta_{4} - 740 \beta_{5} - 44 \beta_{6} - 106 \beta_{7} + 178 \beta_{8} + 14 \beta_{9} + 68 \beta_{10} + 28 \beta_{11} - 92 \beta_{12} - 18 \beta_{13} - 40 \beta_{14} + 12 \beta_{15} - 32 \beta_{16} + 28 \beta_{17} + 2 \beta_{18} + 4 \beta_{19} ) q^{68} + ( -140 + 672 \beta_{1} - 146 \beta_{2} + 553 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} + 78 \beta_{6} + \beta_{7} + 23 \beta_{8} - 60 \beta_{9} - 24 \beta_{10} + 106 \beta_{11} - 35 \beta_{12} - 21 \beta_{13} + 49 \beta_{14} - 11 \beta_{15} - 31 \beta_{16} - 56 \beta_{17} - 54 \beta_{18} ) q^{69} + ( 1510 + 97 \beta_{1} + 185 \beta_{2} - 12 \beta_{3} + 25 \beta_{4} + 202 \beta_{5} + 70 \beta_{6} - 52 \beta_{7} - 35 \beta_{8} + 65 \beta_{9} - 15 \beta_{10} + 10 \beta_{11} - 33 \beta_{12} - 10 \beta_{13} + 10 \beta_{14} + 35 \beta_{15} - 20 \beta_{16} - 15 \beta_{17} - 2 \beta_{19} ) q^{70} + ( -9932 - 410 \beta_{1} + 8 \beta_{2} - 88 \beta_{3} + 58 \beta_{4} - 22 \beta_{5} + 98 \beta_{6} - 42 \beta_{7} + 18 \beta_{8} - 134 \beta_{9} + 90 \beta_{10} - 38 \beta_{11} - 60 \beta_{12} - 2 \beta_{14} + 66 \beta_{15} + 78 \beta_{16} + 76 \beta_{17} - 14 \beta_{18} + 30 \beta_{19} ) q^{71} + ( 11823 - 730 \beta_{1} + 404 \beta_{2} - 300 \beta_{3} + 60 \beta_{4} + 1097 \beta_{5} + 51 \beta_{6} + 8 \beta_{7} + 2 \beta_{8} - 186 \beta_{9} - 52 \beta_{10} + 126 \beta_{11} + 176 \beta_{12} + 58 \beta_{13} + 28 \beta_{14} - 100 \beta_{15} - 52 \beta_{16} - 76 \beta_{17} - 18 \beta_{18} - 46 \beta_{19} ) q^{72} + ( -5035 + 32 \beta_{1} - 138 \beta_{2} - 423 \beta_{3} - 127 \beta_{4} - 45 \beta_{5} - 7 \beta_{6} - 19 \beta_{7} + 99 \beta_{8} - 221 \beta_{9} + 65 \beta_{10} - 118 \beta_{11} - 176 \beta_{12} - 41 \beta_{13} - 11 \beta_{14} + 98 \beta_{15} + 11 \beta_{16} + 27 \beta_{17} - 38 \beta_{18} + 62 \beta_{19} ) q^{73} + ( 3842 + 92 \beta_{1} + 682 \beta_{2} + 48 \beta_{3} + 68 \beta_{4} - 952 \beta_{5} - 8 \beta_{6} - 88 \beta_{7} + 164 \beta_{8} - 40 \beta_{9} + 4 \beta_{10} - 34 \beta_{11} + 188 \beta_{12} - 24 \beta_{13} - 8 \beta_{14} - 68 \beta_{15} + 48 \beta_{16} + 4 \beta_{17} + 64 \beta_{18} - 8 \beta_{19} ) q^{74} + 625 \beta_{2} q^{75} + ( -7724 + 1328 \beta_{1} + 1280 \beta_{2} + 300 \beta_{3} + 92 \beta_{4} - 824 \beta_{5} + 10 \beta_{6} + 150 \beta_{7} - 34 \beta_{8} + 64 \beta_{9} - 28 \beta_{10} - 12 \beta_{11} + 56 \beta_{12} + 86 \beta_{13} - 42 \beta_{14} + 44 \beta_{15} - 6 \beta_{16} + 10 \beta_{17} + 30 \beta_{19} ) q^{76} + ( -91 - 1903 \beta_{1} + 640 \beta_{2} - 576 \beta_{3} - 10 \beta_{4} + 19 \beta_{5} - 19 \beta_{6} + 150 \beta_{7} - 39 \beta_{8} - 259 \beta_{9} - 38 \beta_{10} + 38 \beta_{11} + 207 \beta_{12} + 61 \beta_{13} - 67 \beta_{14} - 129 \beta_{15} + 18 \beta_{16} - 83 \beta_{17} + 89 \beta_{18} ) q^{77} + ( 4734 + 2492 \beta_{1} + 686 \beta_{2} + 154 \beta_{3} - 8 \beta_{4} + 1258 \beta_{5} + 82 \beta_{6} + 182 \beta_{7} + 72 \beta_{8} + 18 \beta_{9} - 56 \beta_{10} - 2 \beta_{11} + 132 \beta_{12} + 62 \beta_{13} - 14 \beta_{14} - 54 \beta_{15} + 18 \beta_{16} - 22 \beta_{17} + 30 \beta_{18} + 80 \beta_{19} ) q^{78} + ( 14568 + 2194 \beta_{1} - 196 \beta_{2} - 650 \beta_{3} + 50 \beta_{4} - 10 \beta_{5} - 6 \beta_{6} + 214 \beta_{7} - 48 \beta_{8} - 210 \beta_{9} + 20 \beta_{10} + 6 \beta_{11} + 24 \beta_{12} - 18 \beta_{13} + 48 \beta_{14} - 42 \beta_{15} - 4 \beta_{16} - 22 \beta_{17} - 42 \beta_{18} + 6 \beta_{19} ) q^{79} + ( 790 - 462 \beta_{1} - 416 \beta_{2} - 145 \beta_{3} - 30 \beta_{4} + 216 \beta_{5} + 13 \beta_{6} - 15 \beta_{7} + 85 \beta_{8} + 32 \beta_{9} + 44 \beta_{10} - 30 \beta_{11} - 16 \beta_{12} - 23 \beta_{13} - 3 \beta_{14} + 20 \beta_{15} - 17 \beta_{16} - 29 \beta_{17} + 2 \beta_{18} - 9 \beta_{19} ) q^{80} + ( 3358 + 1736 \beta_{1} - 92 \beta_{3} + 75 \beta_{4} + 219 \beta_{5} + 123 \beta_{6} + 49 \beta_{7} + 27 \beta_{8} + 315 \beta_{9} - 44 \beta_{10} + 76 \beta_{11} + 140 \beta_{12} + 40 \beta_{13} + 12 \beta_{14} - 4 \beta_{15} - 20 \beta_{16} - 32 \beta_{17} + 12 \beta_{18} - 80 \beta_{19} ) q^{81} + ( -11049 - 348 \beta_{1} + 1016 \beta_{2} - 237 \beta_{3} - 42 \beta_{4} - 937 \beta_{5} + 23 \beta_{6} - 23 \beta_{7} + 40 \beta_{8} - 73 \beta_{9} - 6 \beta_{10} + 153 \beta_{11} - 54 \beta_{12} - 32 \beta_{13} + 119 \beta_{14} - 120 \beta_{15} - 107 \beta_{16} - 81 \beta_{17} - 89 \beta_{18} - 66 \beta_{19} ) q^{82} + ( 189 + 3828 \beta_{1} + 979 \beta_{2} + 201 \beta_{3} + 65 \beta_{4} + 87 \beta_{5} + 97 \beta_{6} - 62 \beta_{7} - 65 \beta_{8} - 377 \beta_{9} - 89 \beta_{10} + 155 \beta_{11} + 257 \beta_{12} - 50 \beta_{13} + 73 \beta_{14} - 127 \beta_{15} - 73 \beta_{16} - 184 \beta_{17} + 95 \beta_{18} ) q^{83} + ( -14852 + 526 \beta_{1} - 700 \beta_{2} + 22 \beta_{3} + 16 \beta_{4} - 1370 \beta_{5} - 4 \beta_{6} + 94 \beta_{7} - 128 \beta_{8} + 76 \beta_{9} - 56 \beta_{10} + 74 \beta_{11} - 88 \beta_{12} - 48 \beta_{13} + 20 \beta_{14} + 8 \beta_{15} + 92 \beta_{16} + 28 \beta_{17} + 84 \beta_{18} + 82 \beta_{19} ) q^{84} + ( 293 + 861 \beta_{1} + 180 \beta_{2} - 434 \beta_{3} - 40 \beta_{4} + 77 \beta_{5} - 63 \beta_{6} - 144 \beta_{7} + 15 \beta_{8} + 133 \beta_{9} + 46 \beta_{10} - 74 \beta_{11} - 15 \beta_{12} + 23 \beta_{13} - 57 \beta_{14} - 15 \beta_{15} + 32 \beta_{16} + 49 \beta_{17} + 13 \beta_{18} ) q^{85} + ( 1548 + 488 \beta_{1} + 1355 \beta_{2} - 30 \beta_{3} + 84 \beta_{4} + 882 \beta_{5} - 54 \beta_{6} + 86 \beta_{7} + 96 \beta_{8} - 29 \beta_{9} + 108 \beta_{10} - 26 \beta_{11} - 284 \beta_{12} - 52 \beta_{13} - 6 \beta_{14} + 20 \beta_{15} + 70 \beta_{16} + 50 \beta_{17} + 58 \beta_{18} + 20 \beta_{19} ) q^{86} + ( -17187 - 2316 \beta_{1} + 288 \beta_{2} + 1143 \beta_{3} - 7 \beta_{4} - 259 \beta_{5} - 163 \beta_{6} - 200 \beta_{7} + 101 \beta_{8} - 67 \beta_{9} - 25 \beta_{10} - 43 \beta_{11} - 75 \beta_{12} - 46 \beta_{13} - 87 \beta_{14} - 47 \beta_{15} - 43 \beta_{16} + 32 \beta_{17} + 57 \beta_{18} - 60 \beta_{19} ) q^{87} + ( 4060 - 3248 \beta_{1} + 340 \beta_{2} + 636 \beta_{3} - 68 \beta_{4} + 740 \beta_{5} - 26 \beta_{6} - 224 \beta_{7} - 74 \beta_{8} + 66 \beta_{9} + 132 \beta_{10} + 14 \beta_{11} - 200 \beta_{12} - 90 \beta_{13} - 44 \beta_{14} + 76 \beta_{15} + 44 \beta_{16} + 76 \beta_{17} - 86 \beta_{18} - 30 \beta_{19} ) q^{88} + ( 228 - 712 \beta_{1} - 356 \beta_{2} - 790 \beta_{3} + 178 \beta_{4} + 142 \beta_{5} + 26 \beta_{6} + 138 \beta_{7} - 50 \beta_{8} - 98 \beta_{9} - 6 \beta_{10} + 332 \beta_{11} - 104 \beta_{12} + 38 \beta_{13} + 58 \beta_{14} - 108 \beta_{15} - 138 \beta_{16} - 42 \beta_{17} - 36 \beta_{18} + 52 \beta_{19} ) q^{89} + ( 2550 - 312 \beta_{1} - 855 \beta_{2} - 83 \beta_{3} - 30 \beta_{4} - 127 \beta_{5} + 85 \beta_{6} - 33 \beta_{7} + 80 \beta_{8} + 55 \beta_{9} - 30 \beta_{10} + 30 \beta_{11} + 18 \beta_{12} + 30 \beta_{13} + 25 \beta_{14} + 70 \beta_{15} - 25 \beta_{16} - 35 \beta_{17} - 35 \beta_{18} + 2 \beta_{19} ) q^{90} + ( 302 + 772 \beta_{1} - 1482 \beta_{2} - 164 \beta_{3} + 8 \beta_{4} + 82 \beta_{5} - 134 \beta_{6} - 90 \beta_{7} - 24 \beta_{8} + 250 \beta_{9} + 70 \beta_{10} - 22 \beta_{11} - 72 \beta_{12} - 66 \beta_{13} - 56 \beta_{14} + 120 \beta_{15} + 40 \beta_{16} + 68 \beta_{17} + 6 \beta_{18} ) q^{91} + ( 5331 + 667 \beta_{1} - 1608 \beta_{2} - 374 \beta_{3} + 4 \beta_{4} - 884 \beta_{5} + 160 \beta_{6} + 23 \beta_{7} + 69 \beta_{8} - 247 \beta_{9} - 46 \beta_{10} - 60 \beta_{11} + 110 \beta_{12} - 25 \beta_{13} + 86 \beta_{14} - 126 \beta_{15} - 46 \beta_{16} - 252 \beta_{17} - 13 \beta_{18} + 38 \beta_{19} ) q^{92} + ( -616 - 4322 \beta_{1} - 700 \beta_{2} + 316 \beta_{3} + 44 \beta_{4} - 512 \beta_{5} - 60 \beta_{6} + 202 \beta_{7} - 38 \beta_{8} + 400 \beta_{9} + 98 \beta_{10} - 142 \beta_{11} - 42 \beta_{12} - 136 \beta_{13} + 82 \beta_{14} + 118 \beta_{15} - 76 \beta_{16} + 106 \beta_{17} - 8 \beta_{18} ) q^{93} + ( 7283 - 2420 \beta_{1} - 1143 \beta_{2} - 39 \beta_{3} - 14 \beta_{4} + 909 \beta_{5} - 199 \beta_{6} - 217 \beta_{7} - 146 \beta_{8} + 49 \beta_{9} + 10 \beta_{10} - 165 \beta_{11} + 104 \beta_{12} + 25 \beta_{13} - 79 \beta_{14} + 133 \beta_{15} + 121 \beta_{16} + 91 \beta_{17} + 135 \beta_{18} + 64 \beta_{19} ) q^{94} + ( 7010 + 839 \beta_{1} + 160 \beta_{2} + 596 \beta_{3} + 75 \beta_{4} + 55 \beta_{5} + 55 \beta_{6} - 9 \beta_{7} + 30 \beta_{8} + 215 \beta_{9} - 35 \beta_{10} - 25 \beta_{11} - 16 \beta_{12} - 10 \beta_{13} - 45 \beta_{14} - 5 \beta_{15} - 25 \beta_{16} + 35 \beta_{18} - 29 \beta_{19} ) q^{95} + ( 20102 + 3872 \beta_{1} - 2196 \beta_{2} - 318 \beta_{3} + 56 \beta_{4} + 1982 \beta_{5} + 206 \beta_{6} - 486 \beta_{7} + 60 \beta_{8} + 182 \beta_{9} - 116 \beta_{10} - 282 \beta_{11} - 72 \beta_{12} - 76 \beta_{13} + 70 \beta_{14} - 28 \beta_{15} + 98 \beta_{16} - 6 \beta_{17} + 42 \beta_{18} - 28 \beta_{19} ) q^{96} + ( 6661 - 1488 \beta_{1} + 578 \beta_{2} + 1723 \beta_{3} + 13 \beta_{4} - 253 \beta_{5} - 107 \beta_{6} - 131 \beta_{7} - 205 \beta_{8} + 179 \beta_{9} - 13 \beta_{10} + 62 \beta_{11} + 176 \beta_{12} + 53 \beta_{13} - 81 \beta_{14} - 26 \beta_{15} + 81 \beta_{16} + 97 \beta_{17} + 110 \beta_{18} - 134 \beta_{19} ) q^{97} + ( 10329 - 755 \beta_{1} + 912 \beta_{2} + 677 \beta_{3} + 6 \beta_{4} - 1431 \beta_{5} + 137 \beta_{6} + 415 \beta_{7} - 140 \beta_{8} - 55 \beta_{9} - 62 \beta_{10} + 111 \beta_{11} - 38 \beta_{12} + 184 \beta_{13} - 23 \beta_{14} - 84 \beta_{15} + 83 \beta_{16} - 59 \beta_{17} + 65 \beta_{18} + 10 \beta_{19} ) q^{98} + ( -1324 - 6494 \beta_{1} + 41 \beta_{2} + 1802 \beta_{3} - 86 \beta_{4} - 213 \beta_{5} - 44 \beta_{6} + 608 \beta_{7} - 138 \beta_{8} - 175 \beta_{9} + 60 \beta_{10} + 215 \beta_{11} + 266 \beta_{12} + 114 \beta_{13} - 70 \beta_{14} + 10 \beta_{15} - 154 \beta_{16} - 43 \beta_{17} - 276 \beta_{18} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{2} - 32q^{4} + 204q^{6} - 196q^{7} + 248q^{8} - 1620q^{9} + O(q^{10}) \) \( 20q + 2q^{2} - 32q^{4} + 204q^{6} - 196q^{7} + 248q^{8} - 1620q^{9} - 50q^{10} - 1876q^{12} + 2708q^{14} + 900q^{15} + 3080q^{16} - 5294q^{18} - 1900q^{20} + 13836q^{22} - 4676q^{23} + 1032q^{24} - 12500q^{25} - 8084q^{26} + 2108q^{28} + 5800q^{30} + 7160q^{31} + 6792q^{32} + 5672q^{33} + 21132q^{34} + 18344q^{36} - 19580q^{38} - 44904q^{39} + 6200q^{40} + 11608q^{41} - 17116q^{42} + 72296q^{44} - 28516q^{46} + 44180q^{47} - 88856q^{48} + 18756q^{49} - 1250q^{50} - 39680q^{52} - 100584q^{54} - 24200q^{55} - 53624q^{56} + 5032q^{57} + 59496q^{58} - 31300q^{60} + 59824q^{62} + 240620q^{63} - 11264q^{64} - 56688q^{66} + 11576q^{68} + 29800q^{70} - 200312q^{71} + 235912q^{72} - 105136q^{73} + 78876q^{74} - 153872q^{76} + 95864q^{78} + 282080q^{79} + 16000q^{80} + 65172q^{81} - 223032q^{82} - 297128q^{84} + 27452q^{86} - 332592q^{87} + 86896q^{88} - 3160q^{89} + 51750q^{90} + 107916q^{92} + 148820q^{94} + 144400q^{95} + 395168q^{96} + 147376q^{97} + 216942q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} - 109104 x^{12} - 96128 x^{11} + 3580672 x^{10} - 1538048 x^{9} - 27930624 x^{8} + 79364096 x^{7} + 157024256 x^{6} - 926941184 x^{5} + 4244635648 x^{4} + 20937965568 x^{3} - 73014444032 x^{2} - 137438953472 x + 1099511627776\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-67985 \nu^{19} + 62912 \nu^{18} + 1230021 \nu^{17} - 3396972 \nu^{16} - 12508553 \nu^{15} + 17579546 \nu^{14} - 167940308 \nu^{13} - 1059006312 \nu^{12} + 6542957136 \nu^{11} + 17963145184 \nu^{10} - 144586364416 \nu^{9} - 144808361472 \nu^{8} + 906022600704 \nu^{7} - 1373389299712 \nu^{6} + 2583188733952 \nu^{5} + 53728355287040 \nu^{4} - 64235463770112 \nu^{3} - 1192022035660800 \nu^{2} - 297081277251584 \nu + 11464324274978816\)\()/ 1286331967733760 \)
\(\beta_{2}\)\(=\)\((\)\(-79348087 \nu^{19} + 164467438 \nu^{18} + 2750967783 \nu^{17} + 2347892670 \nu^{16} - 17200470427 \nu^{15} + 69605863660 \nu^{14} - 24303499972 \nu^{13} - 719687080656 \nu^{12} + 16148446793808 \nu^{11} + 66989955357824 \nu^{10} - 209376218715392 \nu^{9} - 308378042677248 \nu^{8} + 2858199953264640 \nu^{7} + 4761986870738944 \nu^{6} + 7314399222824960 \nu^{5} + 210707338900799488 \nu^{4} + 19270179452092416 \nu^{3} - 2314359262031118336 \nu^{2} + 3379581800966782976 \nu + 38936417104430104576\)\()/ 713485464769658880 \)
\(\beta_{3}\)\(=\)\((\)\(36529 \nu^{19} - 37138 \nu^{18} - 952929 \nu^{17} - 2345826 \nu^{16} + 21259597 \nu^{15} + 2524124 \nu^{14} - 129135524 \nu^{13} + 437239152 \nu^{12} - 5713941552 \nu^{11} - 49422810752 \nu^{10} + 124686277376 \nu^{9} + 496420435968 \nu^{8} - 2011089383424 \nu^{7} - 6629241389056 \nu^{6} + 4644870553600 \nu^{5} - 112168045379584 \nu^{4} - 115722776739840 \nu^{3} + 1987318143516672 \nu^{2} - 1060268511592448 \nu - 36731983023308800\)\()/ 321582991933440 \)
\(\beta_{4}\)\(=\)\((\)\(-97985891 \nu^{19} - 2347107250 \nu^{18} + 8955951363 \nu^{17} + 45014733006 \nu^{16} - 72698446823 \nu^{15} - 718292536252 \nu^{14} + 1739742865996 \nu^{13} - 2795569027632 \nu^{12} - 13959106014192 \nu^{11} + 414570538595584 \nu^{10} + 83264729637632 \nu^{9} - 6297386944217088 \nu^{8} + 3880844619042816 \nu^{7} + 38744507211677696 \nu^{6} - 11302647475994624 \nu^{5} + 285378387308969984 \nu^{4} + 1657402226575933440 \nu^{3} - 13317643676248178688 \nu^{2} - 26666393143378706432 \nu + 177034869779757793280\)\()/ 535114098577244160 \)
\(\beta_{5}\)\(=\)\((\)\(6673115 \nu^{19} + 30164170 \nu^{18} - 153706635 \nu^{17} - 266710470 \nu^{16} + 3862360175 \nu^{15} + 2106440260 \nu^{14} + 4737874100 \nu^{13} + 236780073360 \nu^{12} - 50299499280 \nu^{11} - 4828965765760 \nu^{10} + 12397823115520 \nu^{9} + 82271702046720 \nu^{8} - 93706914631680 \nu^{7} - 50248724971520 \nu^{6} + 1926810069893120 \nu^{5} - 7838825908797440 \nu^{4} - 6061205571502080 \nu^{3} + 180832148914176000 \nu^{2} + 275660321136312320 \nu - 1550266383906897920\)\()/ 32930098373984256 \)
\(\beta_{6}\)\(=\)\((\)\(-69160775 \nu^{19} + 35235854 \nu^{18} + 3743303607 \nu^{17} - 5689165602 \nu^{16} - 40339659563 \nu^{15} + 104897574572 \nu^{14} + 71837847676 \nu^{13} - 570023436240 \nu^{12} + 25433106255696 \nu^{11} + 31872784133248 \nu^{10} - 381936366044416 \nu^{9} + 58866098325504 \nu^{8} + 5578982574280704 \nu^{7} + 458283032969216 \nu^{6} + 31120845131481088 \nu^{5} + 191585425528193024 \nu^{4} - 282744569160794112 \nu^{3} - 2032038956299714560 \nu^{2} + 15111222022950092800 \nu + 33390654176765149184\)\()/ 164650491869921280 \)
\(\beta_{7}\)\(=\)\((\)\(2121323 \nu^{19} + 2680522 \nu^{18} - 3689307 \nu^{17} + 6591738 \nu^{16} + 264714623 \nu^{15} + 2131216804 \nu^{14} + 7316097524 \nu^{13} + 46845209616 \nu^{12} + 18788644848 \nu^{11} - 227096981632 \nu^{10} + 2708453376256 \nu^{9} + 9296654579712 \nu^{8} + 25859031429120 \nu^{7} + 79128649990144 \nu^{6} + 314856479916032 \nu^{5} + 64149778333696 \nu^{4} + 1241058499362816 \nu^{3} + 38568090132283392 \nu^{2} + 23971157371781120 \nu + 15000465339056128\)\()/ 3430218580623360 \)
\(\beta_{8}\)\(=\)\((\)\(-1652193653 \nu^{19} - 1541606518 \nu^{18} + 22547436165 \nu^{17} - 28897641030 \nu^{16} - 311157762785 \nu^{15} + 1467708071972 \nu^{14} - 5081228940428 \nu^{13} - 23360926939632 \nu^{12} + 137433907436016 \nu^{11} + 419549708390272 \nu^{10} - 1949627409612544 \nu^{9} - 1648804311975936 \nu^{8} + 26730765666349056 \nu^{7} - 43553585263083520 \nu^{6} - 27352748135284736 \nu^{5} + 1798632542207737856 \nu^{4} - 3505658856896200704 \nu^{3} - 1899663293065199616 \nu^{2} + 23419227737505660928 \nu + 183124395323945910272\)\()/ 2140456394308976640 \)
\(\beta_{9}\)\(=\)\((\)\(-191297309 \nu^{19} - 782315974 \nu^{18} + 2082932205 \nu^{17} + 2536434474 \nu^{16} - 58787611817 \nu^{15} - 85483833820 \nu^{14} - 110745422444 \nu^{13} - 5669980042992 \nu^{12} - 6791259700368 \nu^{11} + 108433773083008 \nu^{10} - 238768006227712 \nu^{9} - 1876370651080704 \nu^{8} + 437212383965184 \nu^{7} + 3665853378265088 \nu^{6} - 51318721181646848 \nu^{5} + 93131896266948608 \nu^{4} + 494759853466583040 \nu^{3} - 4201827059883835392 \nu^{2} - 9585649942702063616 \nu + 46519200282337869824\)\()/ 237828488256552960 \)
\(\beta_{10}\)\(=\)\((\)\(242971859 \nu^{19} + 2811987922 \nu^{18} + 3207553677 \nu^{17} - 20184358830 \nu^{16} + 39588349847 \nu^{15} + 377520776188 \nu^{14} + 1752375279092 \nu^{13} + 20864739367728 \nu^{12} + 35738566645488 \nu^{11} - 142981561273600 \nu^{10} - 162276836741888 \nu^{9} + 3892854667689984 \nu^{8} + 11740860134240256 \nu^{7} + 41076144459907072 \nu^{6} + 148465276505292800 \nu^{5} - 150813533844537344 \nu^{4} - 1168261401563627520 \nu^{3} + 9628539359821037568 \nu^{2} + 33898917718208282624 \nu + 77184783368618770432\)\()/ 178371366192414720 \)
\(\beta_{11}\)\(=\)\((\)\(-3066220261 \nu^{19} + 3110541514 \nu^{18} + 59421776949 \nu^{17} - 134270357190 \nu^{16} - 571957216081 \nu^{15} + 940126704580 \nu^{14} - 7059227312716 \nu^{13} - 46213723821168 \nu^{12} + 320632357239024 \nu^{11} + 948236705727872 \nu^{10} - 6642921415851776 \nu^{9} - 6949161965266944 \nu^{8} + 46265140049080320 \nu^{7} - 42847034255802368 \nu^{6} + 129403849000878080 \nu^{5} + 2867221596643262464 \nu^{4} - 2614384754480381952 \nu^{3} - 56531194469582635008 \nu^{2} + 104802983984683286528 \nu + 587303771969482129408\)\()/ 2140456394308976640 \)
\(\beta_{12}\)\(=\)\((\)\(-1100290003 \nu^{19} - 721259066 \nu^{18} + 12218919363 \nu^{17} - 23374232682 \nu^{16} - 231772749511 \nu^{15} - 25085087684 \nu^{14} - 3574170330196 \nu^{13} - 19050811187856 \nu^{12} + 62227502417808 \nu^{11} + 223123539658880 \nu^{10} - 1915645378459904 \nu^{9} - 3933233687967744 \nu^{8} + 2439478277935104 \nu^{7} - 32689349068783616 \nu^{6} - 14829055780323328 \nu^{5} + 733271696324165632 \nu^{4} - 1882563927456350208 \nu^{3} - 10469340056559550464 \nu^{2} + 2627948754501632000 \nu + 93662911967977799680\)\()/ 713485464769658880 \)
\(\beta_{13}\)\(=\)\((\)\(-1149574177 \nu^{19} + 757890370 \nu^{18} + 47847494961 \nu^{17} - 17357641998 \nu^{16} - 524496762781 \nu^{15} + 1537773146356 \nu^{14} + 3652922233892 \nu^{13} - 20243528374704 \nu^{12} + 191782756727856 \nu^{11} + 967750251636608 \nu^{10} - 4205227830653696 \nu^{9} - 6972745424111616 \nu^{8} + 69682632365125632 \nu^{7} + 36167708893315072 \nu^{6} - 74790839713005568 \nu^{5} + 2185607719501692928 \nu^{4} + 122314952508702720 \nu^{3} - 39732701074896715776 \nu^{2} + 62591819346876563456 \nu + 620142114173983129600\)\()/ 713485464769658880 \)
\(\beta_{14}\)\(=\)\((\)\(1378084499 \nu^{19} + 1748073274 \nu^{18} - 38128546179 \nu^{17} - 35922205590 \nu^{16} + 102019001735 \nu^{15} + 371256988996 \nu^{14} + 1610896355156 \nu^{13} + 8925259005072 \nu^{12} - 150554252170128 \nu^{11} - 1001559295189120 \nu^{10} + 1621979202386176 \nu^{9} + 6429888600324096 \nu^{8} - 30601039758594048 \nu^{7} - 84546880885293056 \nu^{6} - 482615504156753920 \nu^{5} - 1710305209625870336 \nu^{4} - 3766338582151692288 \nu^{3} + 24897738341168971776 \nu^{2} - 31709836695618715648 \nu - 606461036125101228032\)\()/ 713485464769658880 \)
\(\beta_{15}\)\(=\)\((\)\(-2613608813 \nu^{19} + 397290362 \nu^{18} + 34407860733 \nu^{17} - 7553046486 \nu^{16} - 442322344697 \nu^{15} - 431598392188 \nu^{14} - 6230330395052 \nu^{13} - 32492244466800 \nu^{12} + 163217146923120 \nu^{11} + 805549508612992 \nu^{10} - 3745521356023552 \nu^{9} - 10263922788108288 \nu^{8} - 1779445647249408 \nu^{7} - 22367264744734720 \nu^{6} - 21435013297602560 \nu^{5} + 1028656954785923072 \nu^{4} - 2464278133868593152 \nu^{3} - 37552806709791031296 \nu^{2} - 30505133227045289984 \nu + 242829437331499384832\)\()/ 1070228197154488320 \)
\(\beta_{16}\)\(=\)\((\)\(-2626005709 \nu^{19} - 1743424454 \nu^{18} + 57278130525 \nu^{17} - 5901549846 \nu^{16} - 828211435225 \nu^{15} - 1037579128316 \nu^{14} - 6720187802668 \nu^{13} - 40791194023536 \nu^{12} + 200684140345968 \nu^{11} + 1091725509280640 \nu^{10} - 6097421383894784 \nu^{9} - 24535050131245056 \nu^{8} + 26278680292651008 \nu^{7} + 50459561438150656 \nu^{6} - 3593514379902976 \nu^{5} + 2093694875525447680 \nu^{4} - 3927365040746790912 \nu^{3} - 74567918626554249216 \nu^{2} - 32398542020156588032 \nu + 794041402225567203328\)\()/ 1070228197154488320 \)
\(\beta_{17}\)\(=\)\((\)\(2753749541 \nu^{19} - 750933770 \nu^{18} - 67876189173 \nu^{17} - 119712276474 \nu^{16} + 704230722833 \nu^{15} + 1504396565116 \nu^{14} - 8296989691828 \nu^{13} + 6658243946352 \nu^{12} - 126022076786928 \nu^{11} - 2028801123217792 \nu^{10} + 3443007790849792 \nu^{9} + 24671853793634304 \nu^{8} - 55675128487735296 \nu^{7} - 314493543457292288 \nu^{6} + 65992036296949760 \nu^{5} - 1426642452441202688 \nu^{4} - 11020745276318023680 \nu^{3} + 64430477724943908864 \nu^{2} + 65785216987031404544 \nu - 1255325012022498689024\)\()/ 1070228197154488320 \)
\(\beta_{18}\)\(=\)\((\)\(6092522125 \nu^{19} - 6606216634 \nu^{18} - 167461819677 \nu^{17} - 161753870442 \nu^{16} + 2039605194649 \nu^{15} + 488403661052 \nu^{14} - 14791748101844 \nu^{13} + 78937971524208 \nu^{12} - 875554655592048 \nu^{11} - 3845777127261056 \nu^{10} + 14777299412748032 \nu^{9} + 39120500097189888 \nu^{8} - 187724555019251712 \nu^{7} - 506020226235301888 \nu^{6} - 350263662661074944 \nu^{5} - 9744803736730992640 \nu^{4} + 2512921342714576896 \nu^{3} + 175906584071195590656 \nu^{2} - 362805856335552315392 \nu - 2297213194579040272384\)\()/ 2140456394308976640 \)
\(\beta_{19}\)\(=\)\((\)\(4725901625 \nu^{19} + 2761897870 \nu^{18} - 58020083145 \nu^{17} + 63571279710 \nu^{16} + 970185399893 \nu^{15} - 733051948244 \nu^{14} + 15987573176444 \nu^{13} + 73027906008624 \nu^{12} - 338152442675376 \nu^{11} - 1060312386677632 \nu^{10} + 8273579040324352 \nu^{9} + 15553269216233472 \nu^{8} - 14803926030987264 \nu^{7} + 129719095658217472 \nu^{6} - 86318799004106752 \nu^{5} - 4209641071912681472 \nu^{4} + 9956219996835151872 \nu^{3} + 83375240244294057984 \nu^{2} - 6938996296388706304 \nu - 539728464311666868224\)\()/ 1070228197154488320 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{2} - 25 \beta_{1} + 3\)\()/50\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{19} + 4 \beta_{12} - \beta_{11} + \beta_{7} + \beta_{5} + \beta_{3} + 6 \beta_{2} - \beta_{1} + 92\)\()/50\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{19} + 7 \beta_{18} - 14 \beta_{17} - 2 \beta_{16} - 10 \beta_{15} + 2 \beta_{14} - 3 \beta_{13} + 16 \beta_{12} + 5 \beta_{11} - 6 \beta_{10} - 13 \beta_{9} - 15 \beta_{8} + 8 \beta_{7} + 18 \beta_{6} + 3 \beta_{5} - 10 \beta_{4} - 2 \beta_{3} + 54 \beta_{2} - 130 \beta_{1} - 655\)\()/100\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{19} - \beta_{17} - \beta_{16} - 4 \beta_{15} + \beta_{14} - \beta_{13} + 4 \beta_{12} + 2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 16 \beta_{2} + 18 \beta_{1} - 154\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-98 \beta_{19} - 59 \beta_{18} + 23 \beta_{17} - 41 \beta_{16} - 50 \beta_{15} - 159 \beta_{14} + 96 \beta_{13} + 248 \beta_{12} - 99 \beta_{11} - 18 \beta_{10} - 209 \beta_{9} - 400 \beta_{8} + 459 \beta_{7} - 241 \beta_{6} - 229 \beta_{5} - 100 \beta_{4} - 221 \beta_{3} - 2454 \beta_{2} + 680 \beta_{1} - 13307\)\()/100\)
\(\nu^{6}\)\(=\)\((\)\(203 \beta_{19} - 321 \beta_{18} + 502 \beta_{17} + 446 \beta_{16} + 350 \beta_{15} - 246 \beta_{14} - 431 \beta_{13} - 1748 \beta_{12} - 683 \beta_{11} + 298 \beta_{10} + 2679 \beta_{9} + 585 \beta_{8} - 384 \beta_{7} + 46 \beta_{6} + 83 \beta_{5} - 710 \beta_{4} + 1046 \beta_{3} + 4126 \beta_{2} + 10750 \beta_{1} - 154599\)\()/100\)
\(\nu^{7}\)\(=\)\((\)\(815 \beta_{19} + 188 \beta_{18} - 181 \beta_{17} + 947 \beta_{16} - 1820 \beta_{15} - 947 \beta_{14} + 1463 \beta_{13} - 660 \beta_{12} - 3122 \beta_{11} - 264 \beta_{10} - 5302 \beta_{9} + 5695 \beta_{8} - 7219 \beta_{7} - 1213 \beta_{6} - 310 \beta_{5} - 470 \beta_{4} - 9409 \beta_{3} - 18952 \beta_{2} + 70186 \beta_{1} - 605926\)\()/100\)
\(\nu^{8}\)\(=\)\((\)\(-58 \beta_{19} - 11 \beta_{18} + 263 \beta_{17} - 281 \beta_{16} + 334 \beta_{15} - 159 \beta_{14} - 96 \beta_{13} - 1032 \beta_{12} - 459 \beta_{11} + 222 \beta_{10} + 607 \beta_{9} - 144 \beta_{8} - 1141 \beta_{7} + 143 \beta_{6} - 2781 \beta_{5} - 100 \beta_{4} + 179 \beta_{3} + 2330 \beta_{2} + 13912 \beta_{1} + 54821\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-6317 \beta_{19} - 8969 \beta_{18} - 13322 \beta_{17} - 3986 \beta_{16} - 8290 \beta_{15} + 11386 \beta_{14} - 25319 \beta_{13} - 86148 \beta_{12} + 97357 \beta_{11} - 8918 \beta_{10} - 87249 \beta_{9} + 105665 \beta_{8} + 37200 \beta_{7} - 16106 \beta_{6} - 159245 \beta_{5} + 27610 \beta_{4} - 39930 \beta_{3} - 414770 \beta_{2} + 170334 \beta_{1} + 7594441\)\()/100\)
\(\nu^{10}\)\(=\)\((\)\(58639 \beta_{19} + 17716 \beta_{18} + 148603 \beta_{17} - 72781 \beta_{16} + 284420 \beta_{15} - 59939 \beta_{14} + 46031 \beta_{13} - 54084 \beta_{12} + 143502 \beta_{11} + 50312 \beta_{10} + 129346 \beta_{9} - 63705 \beta_{8} + 88941 \beta_{7} - 116101 \beta_{6} - 256622 \beta_{5} + 100010 \beta_{4} - 941809 \beta_{3} - 732360 \beta_{2} - 2171862 \beta_{1} - 22761814\)\()/100\)
\(\nu^{11}\)\(=\)\((\)\(383022 \beta_{19} + 19549 \beta_{18} - 168673 \beta_{17} + 201311 \beta_{16} + 834430 \beta_{15} + 376889 \beta_{14} - 429696 \beta_{13} - 509192 \beta_{12} - 806123 \beta_{11} - 323042 \beta_{10} + 118359 \beta_{9} + 1057520 \beta_{8} + 726435 \beta_{7} + 1695351 \beta_{6} + 3489891 \beta_{5} + 9980 \beta_{4} - 1487685 \beta_{3} + 1298586 \beta_{2} + 17390776 \beta_{1} + 103852861\)\()/100\)
\(\nu^{12}\)\(=\)\((\)\(-57389 \beta_{19} + 90535 \beta_{18} - 86266 \beta_{17} - 34098 \beta_{16} - 40242 \beta_{15} + 77370 \beta_{14} - 22071 \beta_{13} + 69772 \beta_{12} + 121389 \beta_{11} + 8378 \beta_{10} - 101313 \beta_{9} + 29825 \beta_{8} - 26912 \beta_{7} + 139214 \beta_{6} + 272363 \beta_{5} + 135402 \beta_{4} - 245466 \beta_{3} + 334894 \beta_{2} - 3258194 \beta_{1} - 19775167\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-5174193 \beta_{19} - 5852340 \beta_{18} - 4240725 \beta_{17} - 9784845 \beta_{16} - 4087580 \beta_{15} + 6880045 \beta_{14} + 2672775 \beta_{13} + 23306188 \beta_{12} - 4223186 \beta_{11} - 2201640 \beta_{10} + 647450 \beta_{9} - 16156945 \beta_{8} + 28895437 \beta_{7} - 108685 \beta_{6} + 59986618 \beta_{5} + 12902570 \beta_{4} + 36744287 \beta_{3} - 53370824 \beta_{2} + 178496298 \beta_{1} + 2125002202\)\()/100\)
\(\nu^{14}\)\(=\)\((\)\(-22835066 \beta_{19} - 872211 \beta_{18} - 820753 \beta_{17} + 14290831 \beta_{16} + 18184990 \beta_{15} - 7229671 \beta_{14} + 23437664 \beta_{13} - 21828424 \beta_{12} + 8482013 \beta_{11} - 6741042 \beta_{10} + 73504199 \beta_{9} - 29041680 \beta_{8} + 140936467 \beta_{7} - 22346409 \beta_{6} + 172841467 \beta_{5} - 50040260 \beta_{4} + 79590107 \beta_{3} - 104076950 \beta_{2} - 1885638056 \beta_{1} - 2914135811\)\()/100\)
\(\nu^{15}\)\(=\)\((\)\(79597763 \beta_{19} - 24695161 \beta_{18} + 59342582 \beta_{17} + 113252366 \beta_{16} - 71557410 \beta_{15} - 894566 \beta_{14} + 91308489 \beta_{13} + 129913052 \beta_{12} - 181894627 \beta_{11} - 4670742 \beta_{10} + 371243519 \beta_{9} + 130339985 \beta_{8} - 266034064 \beta_{7} - 97732314 \beta_{6} + 2028925715 \beta_{5} + 69633690 \beta_{4} + 851372966 \beta_{3} + 1896488430 \beta_{2} - 51530530 \beta_{1} - 25887410871\)\()/100\)
\(\nu^{16}\)\(=\)\((\)\(-3041913 \beta_{19} + 257348 \beta_{18} - 3990861 \beta_{17} + 5918283 \beta_{16} + 4133092 \beta_{15} - 15273371 \beta_{14} + 22439799 \beta_{13} + 10740028 \beta_{12} - 77937858 \beta_{11} - 3694616 \beta_{10} - 36180926 \beta_{9} - 21104081 \beta_{8} - 64766699 \beta_{7} - 6471037 \beta_{6} - 53272398 \beta_{5} - 39508486 \beta_{4} + 67189287 \beta_{3} + 165989112 \beta_{2} - 558532230 \beta_{1} + 2454452570\)\()/4\)
\(\nu^{17}\)\(=\)\((\)\(134431070 \beta_{19} - 52573091 \beta_{18} - 65313953 \beta_{17} + 658056031 \beta_{16} - 1776630530 \beta_{15} + 266743769 \beta_{14} - 985457856 \beta_{13} - 3600529800 \beta_{12} + 3513174725 \beta_{11} + 1065700158 \beta_{10} + 1511536599 \beta_{9} - 551823120 \beta_{8} - 3489450077 \beta_{7} - 939732009 \beta_{6} - 20112198413 \beta_{5} + 727457020 \beta_{4} + 33455272603 \beta_{3} + 40736511098 \beta_{2} - 7901449992 \beta_{1} + 419029465245\)\()/100\)
\(\nu^{18}\)\(=\)\((\)\(1589925323 \beta_{19} - 2663673185 \beta_{18} - 2473747690 \beta_{17} - 772959170 \beta_{16} - 5842034210 \beta_{15} - 2664210870 \beta_{14} + 5739559825 \beta_{13} + 17589050892 \beta_{12} + 19607853429 \beta_{11} - 1793185430 \beta_{10} - 44976049225 \beta_{9} - 13332125015 \beta_{8} + 7778563328 \beta_{7} - 32844813010 \beta_{6} - 68495214509 \beta_{5} - 6857381350 \beta_{4} - 24835889962 \beta_{3} - 76527737314 \beta_{2} - 88408366338 \beta_{1} + 510369921177\)\()/100\)
\(\nu^{19}\)\(=\)\((\)\(40320191407 \beta_{19} + 6137817148 \beta_{18} + 53603331819 \beta_{17} + 31875087827 \beta_{16} + 53419435300 \beta_{15} - 18528567827 \beta_{14} - 4600296937 \beta_{13} - 42116813332 \beta_{12} - 14844838610 \beta_{11} + 8324488696 \beta_{10} + 58328557898 \beta_{9} - 87966573025 \beta_{8} + 30641079853 \beta_{7} + 15053706627 \beta_{6} - 280673706102 \beta_{5} - 26950544150 \beta_{4} - 78322136737 \beta_{3} + 408214603576 \beta_{2} + 403468045674 \beta_{1} + 2088153966970\)\()/100\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
−2.63430 3.01006i
−2.63430 + 3.01006i
−3.90102 + 0.884346i
−3.90102 0.884346i
0.593959 3.95566i
0.593959 + 3.95566i
−3.80026 1.24819i
−3.80026 + 1.24819i
−2.80358 2.85306i
−2.80358 + 2.85306i
3.18502 2.41984i
3.18502 + 2.41984i
3.46430 1.99965i
3.46430 + 1.99965i
0.236693 3.99299i
0.236693 + 3.99299i
3.72553 + 1.45618i
3.72553 1.45618i
2.93366 + 2.71913i
2.93366 2.71913i
−5.64436 0.375761i 6.67450i 31.7176 + 4.24186i 25.0000i 2.50801 37.6733i −38.2812 −177.432 35.8608i 198.451 −9.39401 + 141.109i
21.2 −5.64436 + 0.375761i 6.67450i 31.7176 4.24186i 25.0000i 2.50801 + 37.6733i −38.2812 −177.432 + 35.8608i 198.451 −9.39401 141.109i
21.3 −4.78536 3.01667i 25.4343i 13.7994 + 28.8717i 25.0000i 76.7270 121.712i −56.4938 21.0614 179.790i −403.904 75.4168 119.634i
21.4 −4.78536 + 3.01667i 25.4343i 13.7994 28.8717i 25.0000i 76.7270 + 121.712i −56.4938 21.0614 + 179.790i −403.904 75.4168 + 119.634i
21.5 −3.36170 4.54962i 6.93089i −9.39799 + 30.5888i 25.0000i 31.5329 23.2996i 47.1406 170.761 60.0732i 194.963 −113.740 + 84.0424i
21.6 −3.36170 + 4.54962i 6.93089i −9.39799 30.5888i 25.0000i 31.5329 + 23.2996i 47.1406 170.761 + 60.0732i 194.963 −113.740 84.0424i
21.7 −2.55207 5.04846i 11.5927i −18.9739 + 25.7680i 25.0000i −58.5253 + 29.5854i −231.529 178.512 + 30.0270i 108.609 126.211 63.8017i
21.8 −2.55207 + 5.04846i 11.5927i −18.9739 25.7680i 25.0000i −58.5253 29.5854i −231.529 178.512 30.0270i 108.609 126.211 + 63.8017i
21.9 0.0494789 5.65664i 10.7455i −31.9951 0.559768i 25.0000i 60.7833 + 0.531674i 198.733 −4.74949 + 180.957i 127.535 141.416 + 1.23697i
21.10 0.0494789 + 5.65664i 10.7455i −31.9951 + 0.559768i 25.0000i 60.7833 0.531674i 198.733 −4.74949 180.957i 127.535 141.416 1.23697i
21.11 0.765181 5.60486i 17.3148i −30.8290 8.57748i 25.0000i −97.0471 13.2490i −9.19080 −71.6654 + 166.229i −56.8021 −140.122 19.1295i
21.12 0.765181 + 5.60486i 17.3148i −30.8290 + 8.57748i 25.0000i −97.0471 + 13.2490i −9.19080 −71.6654 166.229i −56.8021 −140.122 + 19.1295i
21.13 1.46465 5.46395i 29.2080i −27.7096 16.0056i 25.0000i 159.591 + 42.7797i −168.173 −128.039 + 127.961i −610.110 −136.599 36.6164i
21.14 1.46465 + 5.46395i 29.2080i −27.7096 + 16.0056i 25.0000i 159.591 42.7797i −168.173 −128.039 127.961i −610.110 −136.599 + 36.6164i
21.15 4.22968 3.75630i 25.0521i 3.78045 31.7759i 25.0000i −94.1031 105.962i 103.624 −103.370 148.603i −384.607 93.9075 + 105.742i
21.16 4.22968 + 3.75630i 25.0521i 3.78045 + 31.7759i 25.0000i −94.1031 + 105.962i 103.624 −103.370 + 148.603i −384.607 93.9075 105.742i
21.17 5.18171 2.26935i 10.8240i 21.7001 23.5182i 25.0000i 24.5634 + 56.0868i 163.706 59.0729 171.109i 125.841 −56.7336 129.543i
21.18 5.18171 + 2.26935i 10.8240i 21.7001 + 23.5182i 25.0000i 24.5634 56.0868i 163.706 59.0729 + 171.109i 125.841 −56.7336 + 129.543i
21.19 5.65278 0.214529i 18.7876i 31.9080 2.42537i 25.0000i −4.03048 106.202i −107.536 179.848 20.5552i −109.975 −5.36321 141.320i
21.20 5.65278 + 0.214529i 18.7876i 31.9080 + 2.42537i 25.0000i −4.03048 + 106.202i −107.536 179.848 + 20.5552i −109.975 −5.36321 + 141.320i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.6.d.a 20
3.b odd 2 1 360.6.k.b 20
4.b odd 2 1 160.6.d.a 20
5.b even 2 1 200.6.d.b 20
5.c odd 4 1 200.6.f.b 20
5.c odd 4 1 200.6.f.c 20
8.b even 2 1 inner 40.6.d.a 20
8.d odd 2 1 160.6.d.a 20
20.d odd 2 1 800.6.d.c 20
20.e even 4 1 800.6.f.b 20
20.e even 4 1 800.6.f.c 20
24.h odd 2 1 360.6.k.b 20
40.e odd 2 1 800.6.d.c 20
40.f even 2 1 200.6.d.b 20
40.i odd 4 1 200.6.f.b 20
40.i odd 4 1 200.6.f.c 20
40.k even 4 1 800.6.f.b 20
40.k even 4 1 800.6.f.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.d.a 20 1.a even 1 1 trivial
40.6.d.a 20 8.b even 2 1 inner
160.6.d.a 20 4.b odd 2 1
160.6.d.a 20 8.d odd 2 1
200.6.d.b 20 5.b even 2 1
200.6.d.b 20 40.f even 2 1
200.6.f.b 20 5.c odd 4 1
200.6.f.b 20 40.i odd 4 1
200.6.f.c 20 5.c odd 4 1
200.6.f.c 20 40.i odd 4 1
360.6.k.b 20 3.b odd 2 1
360.6.k.b 20 24.h odd 2 1
800.6.d.c 20 20.d odd 2 1
800.6.d.c 20 40.e odd 2 1
800.6.f.b 20 20.e even 4 1
800.6.f.b 20 40.k even 4 1
800.6.f.c 20 20.e even 4 1
800.6.f.c 20 40.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 18 T^{2} - 116 T^{3} - 444 T^{4} - 1584 T^{5} - 2144 T^{6} + 186496 T^{7} - 497408 T^{8} + 567296 T^{9} - 25550848 T^{10} + 18153472 T^{11} - 509345792 T^{12} + 6111100928 T^{13} - 2248146944 T^{14} - 53150220288 T^{15} - 476741369856 T^{16} - 3985729650688 T^{17} + 19791209299968 T^{18} - 70368744177664 T^{19} + 1125899906842624 T^{20} \)
$3$ \( 1 - 1620 T^{2} + 1394322 T^{4} - 834927892 T^{6} + 392823148221 T^{8} - 155295909553872 T^{10} + 53814329151823576 T^{12} - 16772396183204761872 T^{14} + \)\(47\!\cdots\!30\)\( T^{16} - \)\(12\!\cdots\!00\)\( T^{18} + \)\(31\!\cdots\!00\)\( T^{20} - \)\(75\!\cdots\!00\)\( T^{22} + \)\(16\!\cdots\!30\)\( T^{24} - \)\(34\!\cdots\!28\)\( T^{26} + \)\(65\!\cdots\!76\)\( T^{28} - \)\(11\!\cdots\!28\)\( T^{30} + \)\(16\!\cdots\!21\)\( T^{32} - \)\(20\!\cdots\!08\)\( T^{34} + \)\(20\!\cdots\!22\)\( T^{36} - \)\(14\!\cdots\!80\)\( T^{38} + \)\(51\!\cdots\!01\)\( T^{40} \)
$5$ \( ( 1 + 625 T^{2} )^{10} \)
$7$ \( ( 1 + 98 T + 84148 T^{2} + 8017214 T^{3} + 3556570901 T^{4} + 345044190776 T^{5} + 103920201897616 T^{6} + 10159390994080936 T^{7} + 2363994840482709802 T^{8} + \)\(22\!\cdots\!76\)\( T^{9} + \)\(43\!\cdots\!72\)\( T^{10} + \)\(37\!\cdots\!32\)\( T^{11} + \)\(66\!\cdots\!98\)\( T^{12} + \)\(48\!\cdots\!48\)\( T^{13} + \)\(82\!\cdots\!16\)\( T^{14} + \)\(46\!\cdots\!32\)\( T^{15} + \)\(80\!\cdots\!49\)\( T^{16} + \)\(30\!\cdots\!02\)\( T^{17} + \)\(53\!\cdots\!48\)\( T^{18} + \)\(10\!\cdots\!86\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} )^{2} \)
$11$ \( 1 - 1510004 T^{2} + 1155249727726 T^{4} - 592766540112799924 T^{6} + \)\(22\!\cdots\!17\)\( T^{8} - \)\(70\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(40\!\cdots\!64\)\( T^{14} + \)\(79\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!76\)\( T^{20} - \)\(36\!\cdots\!04\)\( T^{22} + \)\(53\!\cdots\!82\)\( T^{24} - \)\(70\!\cdots\!64\)\( T^{26} + \)\(82\!\cdots\!36\)\( T^{28} - \)\(82\!\cdots\!04\)\( T^{30} + \)\(69\!\cdots\!17\)\( T^{32} - \)\(46\!\cdots\!24\)\( T^{34} + \)\(23\!\cdots\!26\)\( T^{36} - \)\(80\!\cdots\!04\)\( T^{38} + \)\(13\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 - 3520332 T^{2} + 6227853839054 T^{4} - 7441447313525468748 T^{6} + \)\(67\!\cdots\!57\)\( T^{8} - \)\(50\!\cdots\!12\)\( T^{10} + \)\(32\!\cdots\!64\)\( T^{12} - \)\(17\!\cdots\!28\)\( T^{14} + \)\(87\!\cdots\!42\)\( T^{16} - \)\(38\!\cdots\!32\)\( T^{18} + \)\(14\!\cdots\!64\)\( T^{20} - \)\(52\!\cdots\!68\)\( T^{22} + \)\(16\!\cdots\!42\)\( T^{24} - \)\(46\!\cdots\!72\)\( T^{26} + \)\(11\!\cdots\!64\)\( T^{28} - \)\(25\!\cdots\!88\)\( T^{30} + \)\(46\!\cdots\!57\)\( T^{32} - \)\(70\!\cdots\!52\)\( T^{34} + \)\(81\!\cdots\!54\)\( T^{36} - \)\(63\!\cdots\!68\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \)
$17$ \( ( 1 + 5447662 T^{2} + 1859072000 T^{3} + 15317572824845 T^{4} + 7413542230528000 T^{5} + 32518332783929091752 T^{6} + \)\(14\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!10\)\( T^{8} + \)\(21\!\cdots\!00\)\( T^{9} + \)\(84\!\cdots\!72\)\( T^{10} + \)\(31\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!90\)\( T^{12} + \)\(41\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!52\)\( T^{14} + \)\(42\!\cdots\!00\)\( T^{15} + \)\(12\!\cdots\!05\)\( T^{16} + \)\(21\!\cdots\!00\)\( T^{17} + \)\(89\!\cdots\!62\)\( T^{18} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \)
$19$ \( 1 - 23638692 T^{2} + 296336418074734 T^{4} - \)\(25\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!97\)\( T^{8} - \)\(93\!\cdots\!80\)\( T^{10} + \)\(42\!\cdots\!52\)\( T^{12} - \)\(16\!\cdots\!48\)\( T^{14} + \)\(56\!\cdots\!66\)\( T^{16} - \)\(16\!\cdots\!12\)\( T^{18} + \)\(44\!\cdots\!28\)\( T^{20} - \)\(10\!\cdots\!12\)\( T^{22} + \)\(21\!\cdots\!66\)\( T^{24} - \)\(38\!\cdots\!48\)\( T^{26} + \)\(60\!\cdots\!52\)\( T^{28} - \)\(80\!\cdots\!80\)\( T^{30} + \)\(91\!\cdots\!97\)\( T^{32} - \)\(83\!\cdots\!32\)\( T^{34} + \)\(59\!\cdots\!34\)\( T^{36} - \)\(28\!\cdots\!92\)\( T^{38} + \)\(75\!\cdots\!01\)\( T^{40} \)
$23$ \( ( 1 + 2338 T + 35997660 T^{2} + 45007042654 T^{3} + 520972471777845 T^{4} + 50426407997609208 T^{5} + \)\(39\!\cdots\!60\)\( T^{6} - \)\(66\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} - \)\(91\!\cdots\!52\)\( T^{9} + \)\(76\!\cdots\!60\)\( T^{10} - \)\(58\!\cdots\!36\)\( T^{11} + \)\(74\!\cdots\!90\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{13} + \)\(68\!\cdots\!60\)\( T^{14} + \)\(55\!\cdots\!44\)\( T^{15} + \)\(37\!\cdots\!05\)\( T^{16} + \)\(20\!\cdots\!78\)\( T^{17} + \)\(10\!\cdots\!60\)\( T^{18} + \)\(44\!\cdots\!34\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} )^{2} \)
$29$ \( 1 - 215092900 T^{2} + 23243889276296494 T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(90\!\cdots\!13\)\( T^{8} - \)\(38\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!86\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{18} + \)\(58\!\cdots\!28\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{22} + \)\(20\!\cdots\!86\)\( T^{24} - \)\(31\!\cdots\!00\)\( T^{26} + \)\(43\!\cdots\!92\)\( T^{28} - \)\(51\!\cdots\!00\)\( T^{30} + \)\(50\!\cdots\!13\)\( T^{32} - \)\(39\!\cdots\!00\)\( T^{34} + \)\(22\!\cdots\!94\)\( T^{36} - \)\(88\!\cdots\!00\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( ( 1 - 3580 T + 132421614 T^{2} - 484936307876 T^{3} + 8983138546835629 T^{4} - 33755686429634218608 T^{5} + \)\(43\!\cdots\!88\)\( T^{6} - \)\(16\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!38\)\( T^{8} - \)\(60\!\cdots\!32\)\( T^{9} + \)\(54\!\cdots\!44\)\( T^{10} - \)\(17\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!38\)\( T^{12} - \)\(38\!\cdots\!96\)\( T^{13} + \)\(29\!\cdots\!88\)\( T^{14} - \)\(64\!\cdots\!08\)\( T^{15} + \)\(49\!\cdots\!29\)\( T^{16} - \)\(76\!\cdots\!76\)\( T^{17} + \)\(59\!\cdots\!14\)\( T^{18} - \)\(46\!\cdots\!80\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( 1 - 565372668 T^{2} + 171401952644913934 T^{4} - \)\(36\!\cdots\!00\)\( T^{6} + \)\(59\!\cdots\!09\)\( T^{8} - \)\(80\!\cdots\!44\)\( T^{10} + \)\(93\!\cdots\!16\)\( T^{12} - \)\(93\!\cdots\!40\)\( T^{14} + \)\(83\!\cdots\!86\)\( T^{16} - \)\(67\!\cdots\!48\)\( T^{18} + \)\(48\!\cdots\!08\)\( T^{20} - \)\(32\!\cdots\!52\)\( T^{22} + \)\(19\!\cdots\!86\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(49\!\cdots\!16\)\( T^{28} - \)\(20\!\cdots\!56\)\( T^{30} + \)\(73\!\cdots\!09\)\( T^{32} - \)\(21\!\cdots\!00\)\( T^{34} + \)\(48\!\cdots\!34\)\( T^{36} - \)\(77\!\cdots\!32\)\( T^{38} + \)\(66\!\cdots\!01\)\( T^{40} \)
$41$ \( ( 1 - 5804 T + 580737234 T^{2} - 4176614475628 T^{3} + 181136735899530493 T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(39\!\cdots\!40\)\( T^{6} - \)\(28\!\cdots\!68\)\( T^{7} + \)\(63\!\cdots\!42\)\( T^{8} - \)\(43\!\cdots\!24\)\( T^{9} + \)\(82\!\cdots\!24\)\( T^{10} - \)\(49\!\cdots\!24\)\( T^{11} + \)\(85\!\cdots\!42\)\( T^{12} - \)\(44\!\cdots\!68\)\( T^{13} + \)\(70\!\cdots\!40\)\( T^{14} - \)\(28\!\cdots\!48\)\( T^{15} + \)\(43\!\cdots\!93\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{17} + \)\(18\!\cdots\!34\)\( T^{18} - \)\(21\!\cdots\!04\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} )^{2} \)
$43$ \( 1 - 1208126740 T^{2} + 787740581508660018 T^{4} - \)\(36\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!73\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{10} + \)\(95\!\cdots\!40\)\( T^{12} - \)\(20\!\cdots\!56\)\( T^{14} + \)\(40\!\cdots\!70\)\( T^{16} - \)\(70\!\cdots\!00\)\( T^{18} + \)\(10\!\cdots\!96\)\( T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(18\!\cdots\!70\)\( T^{24} - \)\(21\!\cdots\!44\)\( T^{26} + \)\(20\!\cdots\!40\)\( T^{28} - \)\(17\!\cdots\!20\)\( T^{30} + \)\(13\!\cdots\!73\)\( T^{32} - \)\(79\!\cdots\!04\)\( T^{34} + \)\(37\!\cdots\!18\)\( T^{36} - \)\(12\!\cdots\!60\)\( T^{38} + \)\(22\!\cdots\!01\)\( T^{40} \)
$47$ \( ( 1 - 22090 T + 1548510076 T^{2} - 25779450599270 T^{3} + 1065218725316011845 T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!96\)\( T^{6} - \)\(51\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!10\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(36\!\cdots\!56\)\( T^{10} - \)\(32\!\cdots\!00\)\( T^{11} + \)\(76\!\cdots\!90\)\( T^{12} - \)\(62\!\cdots\!80\)\( T^{13} + \)\(12\!\cdots\!96\)\( T^{14} - \)\(90\!\cdots\!40\)\( T^{15} + \)\(15\!\cdots\!05\)\( T^{16} - \)\(86\!\cdots\!10\)\( T^{17} + \)\(11\!\cdots\!76\)\( T^{18} - \)\(38\!\cdots\!30\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} )^{2} \)
$53$ \( 1 - 4003669356 T^{2} + 7954725909905649454 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!77\)\( T^{8} - \)\(91\!\cdots\!76\)\( T^{10} + \)\(65\!\cdots\!16\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(22\!\cdots\!54\)\( T^{16} - \)\(11\!\cdots\!56\)\( T^{18} + \)\(48\!\cdots\!96\)\( T^{20} - \)\(19\!\cdots\!44\)\( T^{22} + \)\(68\!\cdots\!54\)\( T^{24} - \)\(21\!\cdots\!20\)\( T^{26} + \)\(61\!\cdots\!16\)\( T^{28} - \)\(14\!\cdots\!24\)\( T^{30} + \)\(31\!\cdots\!77\)\( T^{32} - \)\(53\!\cdots\!40\)\( T^{34} + \)\(69\!\cdots\!54\)\( T^{36} - \)\(61\!\cdots\!44\)\( T^{38} + \)\(26\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 9996949828 T^{2} + 49650800837889885966 T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(39\!\cdots\!17\)\( T^{8} - \)\(73\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!54\)\( T^{16} - \)\(13\!\cdots\!88\)\( T^{18} + \)\(10\!\cdots\!24\)\( T^{20} - \)\(70\!\cdots\!88\)\( T^{22} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(19\!\cdots\!80\)\( T^{26} + \)\(77\!\cdots\!44\)\( T^{28} - \)\(25\!\cdots\!28\)\( T^{30} + \)\(69\!\cdots\!17\)\( T^{32} - \)\(14\!\cdots\!20\)\( T^{34} + \)\(23\!\cdots\!66\)\( T^{36} - \)\(23\!\cdots\!28\)\( T^{38} + \)\(12\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 7152852348 T^{2} + 26523669582205677166 T^{4} - \)\(67\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!17\)\( T^{8} - \)\(22\!\cdots\!48\)\( T^{10} + \)\(31\!\cdots\!84\)\( T^{12} - \)\(38\!\cdots\!60\)\( T^{14} + \)\(42\!\cdots\!14\)\( T^{16} - \)\(41\!\cdots\!08\)\( T^{18} + \)\(36\!\cdots\!64\)\( T^{20} - \)\(29\!\cdots\!08\)\( T^{22} + \)\(21\!\cdots\!14\)\( T^{24} - \)\(13\!\cdots\!60\)\( T^{26} + \)\(81\!\cdots\!84\)\( T^{28} - \)\(41\!\cdots\!48\)\( T^{30} + \)\(17\!\cdots\!17\)\( T^{32} - \)\(63\!\cdots\!00\)\( T^{34} + \)\(17\!\cdots\!66\)\( T^{36} - \)\(34\!\cdots\!48\)\( T^{38} + \)\(34\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 - 11828518964 T^{2} + 68669252417166303634 T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(76\!\cdots\!57\)\( T^{8} - \)\(18\!\cdots\!04\)\( T^{10} + \)\(38\!\cdots\!16\)\( T^{12} - \)\(70\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!54\)\( T^{16} - \)\(18\!\cdots\!64\)\( T^{18} + \)\(25\!\cdots\!76\)\( T^{20} - \)\(33\!\cdots\!36\)\( T^{22} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(42\!\cdots\!60\)\( T^{26} + \)\(42\!\cdots\!16\)\( T^{28} - \)\(36\!\cdots\!96\)\( T^{30} + \)\(28\!\cdots\!57\)\( T^{32} - \)\(17\!\cdots\!40\)\( T^{34} + \)\(83\!\cdots\!34\)\( T^{36} - \)\(26\!\cdots\!36\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \)
$71$ \( ( 1 + 100156 T + 13448448446 T^{2} + 957445823975748 T^{3} + 76873855601451932317 T^{4} + \)\(43\!\cdots\!20\)\( T^{5} + \)\(26\!\cdots\!28\)\( T^{6} + \)\(12\!\cdots\!92\)\( T^{7} + \)\(67\!\cdots\!74\)\( T^{8} + \)\(28\!\cdots\!84\)\( T^{9} + \)\(13\!\cdots\!68\)\( T^{10} + \)\(51\!\cdots\!84\)\( T^{11} + \)\(21\!\cdots\!74\)\( T^{12} + \)\(74\!\cdots\!92\)\( T^{13} + \)\(28\!\cdots\!28\)\( T^{14} + \)\(82\!\cdots\!20\)\( T^{15} + \)\(26\!\cdots\!17\)\( T^{16} + \)\(59\!\cdots\!48\)\( T^{17} + \)\(15\!\cdots\!46\)\( T^{18} + \)\(20\!\cdots\!56\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$73$ \( ( 1 + 52568 T + 9379342894 T^{2} + 374528688123736 T^{3} + 37721970904921608509 T^{4} + \)\(11\!\cdots\!56\)\( T^{5} + \)\(82\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!58\)\( T^{8} + \)\(24\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!04\)\( T^{10} + \)\(51\!\cdots\!28\)\( T^{11} + \)\(49\!\cdots\!42\)\( T^{12} + \)\(17\!\cdots\!28\)\( T^{13} + \)\(15\!\cdots\!04\)\( T^{14} + \)\(44\!\cdots\!08\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} + \)\(61\!\cdots\!52\)\( T^{17} + \)\(31\!\cdots\!94\)\( T^{18} + \)\(37\!\cdots\!24\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \)
$79$ \( ( 1 - 141040 T + 30508439526 T^{2} - 3027236690742416 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} - \)\(29\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!48\)\( T^{6} - \)\(17\!\cdots\!16\)\( T^{7} + \)\(13\!\cdots\!30\)\( T^{8} - \)\(73\!\cdots\!40\)\( T^{9} + \)\(47\!\cdots\!04\)\( T^{10} - \)\(22\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!30\)\( T^{12} - \)\(51\!\cdots\!84\)\( T^{13} + \)\(24\!\cdots\!48\)\( T^{14} - \)\(81\!\cdots\!12\)\( T^{15} + \)\(32\!\cdots\!93\)\( T^{16} - \)\(79\!\cdots\!84\)\( T^{17} + \)\(24\!\cdots\!26\)\( T^{18} - \)\(34\!\cdots\!60\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \)
$83$ \( 1 - 34768653380 T^{2} + \)\(55\!\cdots\!58\)\( T^{4} - \)\(53\!\cdots\!56\)\( T^{6} + \)\(34\!\cdots\!33\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!40\)\( T^{12} - \)\(24\!\cdots\!36\)\( T^{14} - \)\(64\!\cdots\!10\)\( T^{16} + \)\(49\!\cdots\!20\)\( T^{18} - \)\(23\!\cdots\!44\)\( T^{20} + \)\(76\!\cdots\!80\)\( T^{22} - \)\(15\!\cdots\!10\)\( T^{24} - \)\(90\!\cdots\!64\)\( T^{26} + \)\(25\!\cdots\!40\)\( T^{28} - \)\(14\!\cdots\!80\)\( T^{30} + \)\(48\!\cdots\!33\)\( T^{32} - \)\(11\!\cdots\!44\)\( T^{34} + \)\(18\!\cdots\!58\)\( T^{36} - \)\(18\!\cdots\!20\)\( T^{38} + \)\(80\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 + 1580 T + 27398194046 T^{2} + 460040940498284 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} + \)\(90\!\cdots\!72\)\( T^{5} + \)\(38\!\cdots\!08\)\( T^{6} + \)\(95\!\cdots\!24\)\( T^{7} + \)\(29\!\cdots\!70\)\( T^{8} + \)\(73\!\cdots\!60\)\( T^{9} + \)\(18\!\cdots\!64\)\( T^{10} + \)\(40\!\cdots\!40\)\( T^{11} + \)\(91\!\cdots\!70\)\( T^{12} + \)\(16\!\cdots\!76\)\( T^{13} + \)\(37\!\cdots\!08\)\( T^{14} + \)\(49\!\cdots\!28\)\( T^{15} + \)\(11\!\cdots\!93\)\( T^{16} + \)\(77\!\cdots\!16\)\( T^{17} + \)\(25\!\cdots\!46\)\( T^{18} + \)\(83\!\cdots\!20\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$97$ \( ( 1 - 73688 T + 43672916862 T^{2} - 2817316775448856 T^{3} + \)\(98\!\cdots\!25\)\( T^{4} - \)\(59\!\cdots\!28\)\( T^{5} + \)\(15\!\cdots\!52\)\( T^{6} - \)\(87\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(96\!\cdots\!88\)\( T^{9} + \)\(17\!\cdots\!72\)\( T^{10} - \)\(82\!\cdots\!16\)\( T^{11} + \)\(13\!\cdots\!30\)\( T^{12} - \)\(55\!\cdots\!88\)\( T^{13} + \)\(84\!\cdots\!52\)\( T^{14} - \)\(27\!\cdots\!96\)\( T^{15} + \)\(39\!\cdots\!25\)\( T^{16} - \)\(97\!\cdots\!08\)\( T^{17} + \)\(12\!\cdots\!62\)\( T^{18} - \)\(18\!\cdots\!16\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \)
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