Properties

Label 33.7.b.a
Level $33$
Weight $7$
Character orbit 33.b
Analytic conductor $7.592$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 33.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.59178475946\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 1026 x^{18} + 441321 x^{16} + 103808124 x^{14} + 14594358456 x^{12} + 1256133373152 x^{10} + 64843235559312 x^{8} + 1864534894961472 x^{6} + 25120735435348224 x^{4} + 103653954147713024 x^{2} + 32367107232497664\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{17}\cdot 11^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( -39 + \beta_{2} ) q^{4} + \beta_{6} q^{5} + ( 14 - 2 \beta_{1} + \beta_{8} ) q^{6} + ( 8 + \beta_{3} - \beta_{4} ) q^{7} + ( 1 - 37 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{8} + ( -53 - 8 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( -39 + \beta_{2} ) q^{4} + \beta_{6} q^{5} + ( 14 - 2 \beta_{1} + \beta_{8} ) q^{6} + ( 8 + \beta_{3} - \beta_{4} ) q^{7} + ( 1 - 37 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{8} + ( -53 - 8 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{11} ) q^{9} + ( 51 + \beta_{1} + \beta_{2} - 7 \beta_{3} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{14} ) q^{10} + ( -\beta_{3} - \beta_{12} ) q^{11} + ( 294 + 25 \beta_{1} + 3 \beta_{2} + 40 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{10} + \beta_{11} - \beta_{14} - \beta_{16} ) q^{12} + ( 41 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{15} ) q^{13} + ( -3 + 19 \beta_{1} + \beta_{2} + 11 \beta_{3} + \beta_{6} - \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{14} + ( -154 - 57 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} - 6 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{15} + ( 1427 + 4 \beta_{1} - 49 \beta_{2} - 45 \beta_{3} - 4 \beta_{4} - \beta_{6} + 4 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{13} - 5 \beta_{14} - 2 \beta_{15} - 3 \beta_{17} ) q^{16} + ( -28 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 11 \beta_{6} - \beta_{10} + 2 \beta_{12} + \beta_{15} + 2 \beta_{18} ) q^{17} + ( 948 - 15 \beta_{1} - 22 \beta_{2} - 22 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 8 \beta_{6} - \beta_{7} - 3 \beta_{8} + 3 \beta_{10} + \beta_{11} - 4 \beta_{12} - 3 \beta_{14} + \beta_{16} - \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{18} + ( 271 + 2 \beta_{1} + 42 \beta_{2} + 38 \beta_{3} + 4 \beta_{4} + 3 \beta_{6} - \beta_{7} + 6 \beta_{8} - \beta_{10} + \beta_{11} - 3 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} - \beta_{17} ) q^{19} + ( -19 - 7 \beta_{1} + 11 \beta_{2} + 60 \beta_{3} - 2 \beta_{5} - 38 \beta_{6} + 11 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + 7 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} - 3 \beta_{16} + \beta_{17} - 5 \beta_{18} - 3 \beta_{19} ) q^{20} + ( -1039 - 75 \beta_{1} - 8 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - 6 \beta_{12} + 3 \beta_{13} + \beta_{14} + 4 \beta_{15} - \beta_{16} + 2 \beta_{17} + 3 \beta_{19} ) q^{21} + ( 8 + 5 \beta_{1} - 3 \beta_{2} - 34 \beta_{3} - 2 \beta_{4} - \beta_{6} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{22} + ( -3 - 114 \beta_{1} - \beta_{2} + 19 \beta_{3} + 2 \beta_{4} + 7 \beta_{6} - \beta_{7} + 6 \beta_{8} - 2 \beta_{9} + \beta_{10} + 3 \beta_{11} + 10 \beta_{12} - 3 \beta_{13} + \beta_{14} - 2 \beta_{18} + 6 \beta_{19} ) q^{23} + ( -2441 + 41 \beta_{1} + 71 \beta_{2} + 52 \beta_{3} + 17 \beta_{4} + 2 \beta_{5} + 29 \beta_{6} + 4 \beta_{7} - 25 \beta_{8} + 8 \beta_{9} - 6 \beta_{10} + 3 \beta_{11} - 14 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} + 5 \beta_{17} + 4 \beta_{18} + \beta_{19} ) q^{24} + ( -5006 + 35 \beta_{1} + 26 \beta_{2} - 167 \beta_{3} - 18 \beta_{4} + 3 \beta_{6} + 12 \beta_{8} + 3 \beta_{9} - 10 \beta_{10} - 12 \beta_{11} + 6 \beta_{13} + 13 \beta_{14} + 8 \beta_{15} + 3 \beta_{17} ) q^{25} + ( 40 + 123 \beta_{1} - 22 \beta_{2} - 113 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 42 \beta_{6} + \beta_{7} - 11 \beta_{8} - \beta_{9} - 5 \beta_{10} + 10 \beta_{11} + 15 \beta_{12} - 7 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} + 7 \beta_{16} - 3 \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{26} + ( -935 + 383 \beta_{1} + 81 \beta_{2} + 92 \beta_{3} - 3 \beta_{4} - 11 \beta_{5} + 20 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 6 \beta_{11} - 17 \beta_{12} - 5 \beta_{13} + 8 \beta_{14} + 3 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 5 \beta_{18} + \beta_{19} ) q^{27} + ( -1698 - 31 \beta_{1} + 72 \beta_{2} + 65 \beta_{3} + 7 \beta_{4} - 15 \beta_{6} + 4 \beta_{7} - 29 \beta_{8} - 3 \beta_{9} - 11 \beta_{10} - 13 \beta_{11} - 2 \beta_{13} + 7 \beta_{14} + 7 \beta_{15} + 8 \beta_{17} ) q^{28} + ( 68 - 245 \beta_{1} - 16 \beta_{2} - 225 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} + 6 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 18 \beta_{12} - 9 \beta_{13} - 3 \beta_{14} - 15 \beta_{15} + 14 \beta_{16} + \beta_{17} - 6 \beta_{18} + 2 \beta_{19} ) q^{29} + ( 5572 - 569 \beta_{1} - 172 \beta_{2} - 125 \beta_{3} - 8 \beta_{4} + 15 \beta_{5} - 43 \beta_{6} + \beta_{7} - 7 \beta_{8} + 16 \beta_{9} + 7 \beta_{10} + 25 \beta_{11} - 30 \beta_{12} - 3 \beta_{13} - 29 \beta_{14} - 13 \beta_{15} + 6 \beta_{16} - 11 \beta_{17} - 6 \beta_{18} ) q^{30} + ( -4493 + 31 \beta_{1} - 44 \beta_{2} - 194 \beta_{3} + 6 \beta_{6} - 14 \beta_{7} - 14 \beta_{8} + 12 \beta_{9} + 6 \beta_{10} + 20 \beta_{11} + 7 \beta_{13} - 16 \beta_{14} - 6 \beta_{15} - 4 \beta_{17} ) q^{31} + ( -159 + 1982 \beta_{1} + 37 \beta_{2} + 581 \beta_{3} - \beta_{4} - 51 \beta_{5} - 26 \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{9} + 10 \beta_{10} - 23 \beta_{11} + 42 \beta_{12} + 17 \beta_{13} + \beta_{14} + 8 \beta_{15} - 20 \beta_{16} + 3 \beta_{17} + 6 \beta_{18} + 4 \beta_{19} ) q^{32} + ( -1077 - 222 \beta_{1} + 11 \beta_{2} + 16 \beta_{3} + 11 \beta_{4} + 5 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - 2 \beta_{12} + \beta_{13} + 6 \beta_{15} - 3 \beta_{16} + 2 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{33} + ( 3459 - 14 \beta_{1} - 29 \beta_{2} - 24 \beta_{3} + 44 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} + 18 \beta_{8} - 27 \beta_{9} + 7 \beta_{10} - 33 \beta_{11} - 6 \beta_{13} + 20 \beta_{14} + 16 \beta_{15} + 3 \beta_{17} ) q^{34} + ( 72 + 107 \beta_{1} - 15 \beta_{2} - 192 \beta_{3} - 13 \beta_{4} + 18 \beta_{5} - 58 \beta_{6} + 2 \beta_{7} - 31 \beta_{8} + 16 \beta_{9} - 2 \beta_{10} - 25 \beta_{11} + 32 \beta_{12} + \beta_{13} - 5 \beta_{14} - 20 \beta_{15} + 14 \beta_{16} + 3 \beta_{17} + 8 \beta_{18} + 2 \beta_{19} ) q^{35} + ( -1385 + 1554 \beta_{1} - 82 \beta_{2} - 344 \beta_{3} + 13 \beta_{4} - 35 \beta_{5} - 74 \beta_{6} - 4 \beta_{7} + 43 \beta_{8} - 21 \beta_{9} + 5 \beta_{10} - 50 \beta_{11} - 47 \beta_{12} + 22 \beta_{13} + 30 \beta_{14} + 17 \beta_{15} - 16 \beta_{16} - 9 \beta_{17} + \beta_{18} - 5 \beta_{19} ) q^{36} + ( 11874 - 61 \beta_{1} - 27 \beta_{2} + 471 \beta_{3} + 14 \beta_{4} + 6 \beta_{6} - 3 \beta_{7} + 38 \beta_{8} + 14 \beta_{9} + 15 \beta_{10} - \beta_{11} - 20 \beta_{13} - 37 \beta_{14} + 16 \beta_{15} - 14 \beta_{17} ) q^{37} + ( 73 - 2138 \beta_{1} - 11 \beta_{2} - 84 \beta_{3} - 8 \beta_{4} + 56 \beta_{5} - 119 \beta_{6} + \beta_{7} - 46 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 60 \beta_{12} + 6 \beta_{13} + 6 \beta_{14} + 12 \beta_{15} - 2 \beta_{16} - 7 \beta_{17} - 8 \beta_{19} ) q^{38} + ( -869 + 704 \beta_{1} + 100 \beta_{2} + 59 \beta_{3} + 76 \beta_{4} - 35 \beta_{5} - 61 \beta_{6} + \beta_{7} + 25 \beta_{8} - 21 \beta_{9} - 20 \beta_{10} + 4 \beta_{11} - 54 \beta_{12} - 14 \beta_{13} - 8 \beta_{14} - \beta_{15} + 7 \beta_{16} + 5 \beta_{17} + 9 \beta_{19} ) q^{39} + ( 1474 - 204 \beta_{1} + 73 \beta_{2} + 1219 \beta_{3} + 64 \beta_{4} - 19 \beta_{6} + 20 \beta_{7} - 150 \beta_{8} + 45 \beta_{9} + 5 \beta_{10} + 110 \beta_{11} - 23 \beta_{13} - 58 \beta_{14} - 49 \beta_{15} + 11 \beta_{17} ) q^{40} + ( 130 + 966 \beta_{1} - 53 \beta_{2} - 259 \beta_{3} - 21 \beta_{4} - 11 \beta_{5} + 199 \beta_{6} + 6 \beta_{7} - 94 \beta_{8} + 6 \beta_{9} - 3 \beta_{10} + 8 \beta_{11} + 58 \beta_{12} - 2 \beta_{13} + 6 \beta_{14} + \beta_{15} + 20 \beta_{16} - 12 \beta_{17} - 2 \beta_{18} - 16 \beta_{19} ) q^{41} + ( 7627 - 505 \beta_{1} - 245 \beta_{2} - 103 \beta_{3} - 66 \beta_{4} + 2 \beta_{5} - 93 \beta_{6} + 4 \beta_{7} + 29 \beta_{8} - 14 \beta_{9} - 20 \beta_{10} - 23 \beta_{11} - 77 \beta_{12} + 17 \beta_{13} + 8 \beta_{14} - 39 \beta_{15} - 9 \beta_{16} + 12 \beta_{17} - 5 \beta_{18} - 5 \beta_{19} ) q^{42} + ( -6003 + 125 \beta_{1} - 273 \beta_{2} - 904 \beta_{3} + 35 \beta_{4} + 5 \beta_{6} + 9 \beta_{7} + 3 \beta_{8} - 6 \beta_{9} - 17 \beta_{10} - 20 \beta_{11} + 40 \beta_{13} + 46 \beta_{14} + 15 \beta_{15} + 18 \beta_{17} ) q^{43} + ( -65 + 275 \beta_{1} + 17 \beta_{2} + 147 \beta_{3} + 15 \beta_{4} - 25 \beta_{5} - 71 \beta_{6} - \beta_{7} + 56 \beta_{8} - 8 \beta_{9} - 3 \beta_{10} + 17 \beta_{11} + 15 \beta_{12} - 5 \beta_{13} - 2 \beta_{14} + 5 \beta_{15} - 7 \beta_{16} + 3 \beta_{17} - 5 \beta_{18} - \beta_{19} ) q^{44} + ( -12209 + 1964 \beta_{1} + 88 \beta_{2} + 167 \beta_{3} - 138 \beta_{4} - 62 \beta_{5} + 27 \beta_{6} + 9 \beta_{7} - 40 \beta_{8} + 27 \beta_{9} - 29 \beta_{10} + 2 \beta_{11} - 68 \beta_{12} - 10 \beta_{13} - 24 \beta_{14} + 28 \beta_{15} - 4 \beta_{16} - \beta_{17} + 16 \beta_{18} - 8 \beta_{19} ) q^{45} + ( 12309 - 40 \beta_{1} - 352 \beta_{2} + 523 \beta_{3} - 189 \beta_{4} + 28 \beta_{6} - 10 \beta_{7} - 64 \beta_{8} - \beta_{9} + 58 \beta_{10} + 92 \beta_{11} + 9 \beta_{13} - 20 \beta_{14} - 26 \beta_{15} + 9 \beta_{17} ) q^{46} + ( -2 + 19 \beta_{1} - \beta_{2} + 299 \beta_{3} - 38 \beta_{4} + 8 \beta_{5} + 121 \beta_{6} - 11 \beta_{7} - 146 \beta_{8} - 18 \beta_{9} + 51 \beta_{10} - 43 \beta_{11} + 82 \beta_{12} + 22 \beta_{13} + 27 \beta_{14} + 40 \beta_{15} - 48 \beta_{16} - 16 \beta_{17} - 14 \beta_{18} + 18 \beta_{19} ) q^{47} + ( 15497 - 5407 \beta_{1} + 127 \beta_{2} - 1124 \beta_{3} + 28 \beta_{4} + 130 \beta_{5} + 401 \beta_{6} + 7 \beta_{7} + 91 \beta_{8} - 78 \beta_{9} - 23 \beta_{10} - 68 \beta_{11} - 105 \beta_{12} - 56 \beta_{13} + 73 \beta_{14} + 23 \beta_{15} + 19 \beta_{16} - 40 \beta_{17} + 15 \beta_{18} + 15 \beta_{19} ) q^{48} + ( -9550 + 114 \beta_{1} + 579 \beta_{2} - 210 \beta_{3} + 118 \beta_{4} + 51 \beta_{6} + 15 \beta_{7} + 138 \beta_{8} + 13 \beta_{9} - 9 \beta_{10} - 11 \beta_{11} + 4 \beta_{13} + 42 \beta_{14} + 10 \beta_{15} - 5 \beta_{17} ) q^{49} + ( -459 - 6073 \beta_{1} + 183 \beta_{2} + 1362 \beta_{3} + 113 \beta_{4} + 76 \beta_{5} + 199 \beta_{6} - 2 \beta_{7} + 414 \beta_{8} - 34 \beta_{9} - 47 \beta_{10} + 108 \beta_{11} + 124 \beta_{12} - 28 \beta_{14} + 7 \beta_{15} - 18 \beta_{16} + 30 \beta_{17} - 8 \beta_{18} - 6 \beta_{19} ) q^{50} + ( -1153 - 381 \beta_{1} + 193 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} + 41 \beta_{5} - 367 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 7 \beta_{9} - 2 \beta_{10} + 13 \beta_{11} - 68 \beta_{12} - 39 \beta_{13} + 15 \beta_{14} - 59 \beta_{15} + 9 \beta_{16} + 24 \beta_{17} - 14 \beta_{18} + \beta_{19} ) q^{51} + ( -5982 + 207 \beta_{1} + 344 \beta_{2} - 985 \beta_{3} - 115 \beta_{4} + 63 \beta_{6} + 4 \beta_{7} + 237 \beta_{8} - 75 \beta_{9} + 69 \beta_{10} - 67 \beta_{11} + 36 \beta_{13} + 49 \beta_{14} + 41 \beta_{15} - 20 \beta_{17} ) q^{52} + ( -347 + 1587 \beta_{1} + 130 \beta_{2} + 901 \beta_{3} + 48 \beta_{4} - 2 \beta_{5} - 64 \beta_{6} - 4 \beta_{7} + 232 \beta_{8} + 27 \beta_{9} - 28 \beta_{10} - 54 \beta_{11} + 44 \beta_{12} + 18 \beta_{13} - 39 \beta_{14} - 50 \beta_{15} - 4 \beta_{16} + 43 \beta_{17} + 44 \beta_{18} + 20 \beta_{19} ) q^{53} + ( -39111 - 5601 \beta_{1} + 1030 \beta_{2} - 538 \beta_{3} - 197 \beta_{4} + 191 \beta_{5} - 107 \beta_{6} - 9 \beta_{7} + 36 \beta_{8} + 23 \beta_{9} - 9 \beta_{10} - 25 \beta_{11} - 100 \beta_{12} + 56 \beta_{13} + 29 \beta_{14} - 25 \beta_{15} - 20 \beta_{16} - 4 \beta_{17} - 4 \beta_{18} - 22 \beta_{19} ) q^{54} + ( -9695 - 96 \beta_{1} + 238 \beta_{2} + 259 \beta_{3} - 32 \beta_{4} - 38 \beta_{6} - 73 \beta_{8} - 27 \beta_{9} - 10 \beta_{10} - 37 \beta_{11} - 11 \beta_{13} + 16 \beta_{14} + 17 \beta_{15} + 17 \beta_{17} ) q^{55} + ( 608 - 4833 \beta_{1} - 140 \beta_{2} - 2393 \beta_{3} + 24 \beta_{4} + 142 \beta_{5} + 174 \beta_{6} + 9 \beta_{7} + 33 \beta_{8} - 7 \beta_{9} - 31 \beta_{10} + 50 \beta_{11} + 75 \beta_{12} - 65 \beta_{13} - 4 \beta_{14} + 25 \beta_{15} + 7 \beta_{16} - 5 \beta_{17} + 15 \beta_{18} - 47 \beta_{19} ) q^{56} + ( -12810 + 443 \beta_{1} - 128 \beta_{2} - 239 \beta_{3} + 251 \beta_{4} - 48 \beta_{5} + 295 \beta_{6} - 16 \beta_{7} + 16 \beta_{8} + 47 \beta_{9} + 86 \beta_{10} + 98 \beta_{11} - 18 \beta_{12} + 42 \beta_{13} - 142 \beta_{14} - 101 \beta_{15} - 9 \beta_{16} - 19 \beta_{17} - 48 \beta_{18} - 9 \beta_{19} ) q^{57} + ( 30187 + 324 \beta_{1} - 609 \beta_{2} + 232 \beta_{3} - 444 \beta_{4} + 201 \beta_{6} - 25 \beta_{7} + 524 \beta_{8} + 27 \beta_{9} + 121 \beta_{10} + 31 \beta_{11} - 4 \beta_{13} - 36 \beta_{14} + 28 \beta_{15} - 89 \beta_{17} ) q^{58} + ( 82 + 7087 \beta_{1} - 125 \beta_{2} - 70 \beta_{3} - 12 \beta_{4} - 190 \beta_{5} - 301 \beta_{6} - \beta_{7} - 156 \beta_{8} - 53 \beta_{9} + 21 \beta_{10} + 99 \beta_{11} + 42 \beta_{12} - 33 \beta_{13} + 46 \beta_{14} + 82 \beta_{15} - 12 \beta_{16} - 45 \beta_{17} - 42 \beta_{18} - 6 \beta_{19} ) q^{59} + ( 49272 + 12179 \beta_{1} - 1657 \beta_{2} - 675 \beta_{3} - 351 \beta_{4} - 221 \beta_{5} + 1286 \beta_{6} + 31 \beta_{7} - 26 \beta_{8} - 18 \beta_{9} + 20 \beta_{10} - 94 \beta_{11} - 64 \beta_{12} + 95 \beta_{13} - 40 \beta_{14} + 52 \beta_{15} - 3 \beta_{16} - 78 \beta_{17} + 44 \beta_{18} - 4 \beta_{19} ) q^{60} + ( 55885 - 360 \beta_{1} - 718 \beta_{2} - 385 \beta_{3} - 13 \beta_{4} - 243 \beta_{6} - 37 \beta_{7} - 556 \beta_{8} - 24 \beta_{9} - 41 \beta_{10} + 19 \beta_{11} - \beta_{13} - 171 \beta_{14} - 77 \beta_{15} + 7 \beta_{17} ) q^{61} + ( 175 - 2701 \beta_{1} - 14 \beta_{2} - 1352 \beta_{3} + 38 \beta_{4} - 8 \beta_{5} + 541 \beta_{6} - 2 \beta_{7} + 195 \beta_{8} + 34 \beta_{9} - 27 \beta_{10} - 79 \beta_{11} + 34 \beta_{12} - 41 \beta_{13} - 41 \beta_{14} - 43 \beta_{15} - 14 \beta_{16} + 43 \beta_{17} + 58 \beta_{18} - 2 \beta_{19} ) q^{62} + ( 28782 - 2464 \beta_{1} - 390 \beta_{2} + 809 \beta_{3} + 58 \beta_{4} + 97 \beta_{5} - 672 \beta_{6} + 2 \beta_{7} - 179 \beta_{8} + 63 \beta_{9} - 7 \beta_{10} - 39 \beta_{11} + 58 \beta_{12} + 55 \beta_{13} + 123 \beta_{14} + 152 \beta_{15} - 13 \beta_{16} + 30 \beta_{17} + 52 \beta_{18} + \beta_{19} ) q^{63} + ( -128069 - 760 \beta_{1} + 3005 \beta_{2} + 1835 \beta_{3} + 540 \beta_{4} - 285 \beta_{6} + 100 \beta_{7} - 948 \beta_{8} + 39 \beta_{9} - 148 \beta_{10} + 140 \beta_{11} - 3 \beta_{13} - 7 \beta_{14} - 80 \beta_{15} + 169 \beta_{17} ) q^{64} + ( -50 + 6911 \beta_{1} - 37 \beta_{2} + 548 \beta_{3} - 83 \beta_{4} - 178 \beta_{5} - 154 \beta_{6} + 22 \beta_{7} - 233 \beta_{8} + 48 \beta_{9} - 6 \beta_{10} - 7 \beta_{11} - 152 \beta_{12} + 37 \beta_{13} - 3 \beta_{14} - 68 \beta_{15} + 90 \beta_{16} - 19 \beta_{17} - 32 \beta_{18} - 42 \beta_{19} ) q^{65} + ( 23560 - 1585 \beta_{1} - 620 \beta_{2} - 148 \beta_{3} + 133 \beta_{4} + 43 \beta_{5} + 242 \beta_{6} + 7 \beta_{7} + 10 \beta_{8} - 12 \beta_{9} - 5 \beta_{10} + 40 \beta_{11} - 3 \beta_{12} - 53 \beta_{13} - 50 \beta_{14} - 37 \beta_{15} + 16 \beta_{16} - \beta_{17} - 9 \beta_{18} + 18 \beta_{19} ) q^{66} + ( 30879 - 221 \beta_{1} - 1174 \beta_{2} + 1956 \beta_{3} + 502 \beta_{4} - 26 \beta_{6} + 18 \beta_{7} + 122 \beta_{8} + 18 \beta_{9} - 138 \beta_{10} - 100 \beta_{11} - 155 \beta_{13} + 66 \beta_{14} - 48 \beta_{15} - 28 \beta_{17} ) q^{67} + ( -438 + 3144 \beta_{1} + 216 \beta_{2} + 1332 \beta_{3} + 2 \beta_{4} - 42 \beta_{5} - 1858 \beta_{6} - 10 \beta_{7} + 212 \beta_{8} - 20 \beta_{9} + 52 \beta_{10} - 10 \beta_{11} + 32 \beta_{12} + 34 \beta_{13} - 14 \beta_{14} - 26 \beta_{15} - 64 \beta_{16} + 24 \beta_{17} - 88 \beta_{18} - 4 \beta_{19} ) q^{68} + ( 16424 + 5257 \beta_{1} - 304 \beta_{2} - 248 \beta_{3} + 396 \beta_{4} - 98 \beta_{5} - 414 \beta_{6} + 5 \beta_{7} - 152 \beta_{8} + 14 \beta_{9} + 125 \beta_{10} + 38 \beta_{11} + 176 \beta_{12} - 134 \beta_{13} + 103 \beta_{14} + 84 \beta_{15} + 66 \beta_{16} - 48 \beta_{17} - 4 \beta_{18} + 50 \beta_{19} ) q^{69} + ( -11772 + 412 \beta_{1} - 30 \beta_{2} - 1204 \beta_{3} - 70 \beta_{4} + 100 \beta_{6} + 30 \beta_{7} + 540 \beta_{8} - 22 \beta_{9} - 82 \beta_{10} - 266 \beta_{11} + 4 \beta_{13} + 182 \beta_{14} + 140 \beta_{15} - 22 \beta_{17} ) q^{70} + ( 164 - 8310 \beta_{1} - 106 \beta_{2} + 791 \beta_{3} - 158 \beta_{4} + 182 \beta_{5} - 1484 \beta_{6} - 4 \beta_{7} - 690 \beta_{8} + 23 \beta_{9} + 72 \beta_{10} - 104 \beta_{11} - 280 \beta_{12} + 95 \beta_{13} + 71 \beta_{14} + 22 \beta_{15} + 44 \beta_{16} - 67 \beta_{17} + 8 \beta_{18} + 68 \beta_{19} ) q^{71} + ( -103789 + 3769 \beta_{1} + 3094 \beta_{2} + 3597 \beta_{3} + 517 \beta_{4} - 158 \beta_{5} - 1756 \beta_{6} - 22 \beta_{7} + 31 \beta_{8} + 27 \beta_{9} - 133 \beta_{10} + 125 \beta_{11} + 130 \beta_{12} + 157 \beta_{13} - 69 \beta_{14} - 40 \beta_{15} - 61 \beta_{16} + 162 \beta_{17} - 20 \beta_{18} - 71 \beta_{19} ) q^{72} + ( -68124 + 49 \beta_{1} - 1413 \beta_{2} - 1138 \beta_{3} - 221 \beta_{4} - 80 \beta_{6} - 138 \beta_{7} - 61 \beta_{8} - 112 \beta_{9} - 36 \beta_{10} - 251 \beta_{11} - 33 \beta_{13} + 75 \beta_{14} + 110 \beta_{15} + 7 \beta_{17} ) q^{73} + ( -1133 + 14400 \beta_{1} + 153 \beta_{2} + 5246 \beta_{3} - 125 \beta_{4} - 282 \beta_{5} + 2007 \beta_{6} - 44 \beta_{7} - 574 \beta_{8} - 32 \beta_{9} + 151 \beta_{10} - 194 \beta_{11} - 138 \beta_{12} + 176 \beta_{13} + 94 \beta_{14} + 101 \beta_{15} - 108 \beta_{16} - 50 \beta_{17} + 18 \beta_{18} + 156 \beta_{19} ) q^{74} + ( 124628 - 8576 \beta_{1} - 3097 \beta_{2} + 3854 \beta_{3} + 15 \beta_{4} - 41 \beta_{5} + 276 \beta_{6} - 9 \beta_{7} - 313 \beta_{8} + 56 \beta_{9} - 18 \beta_{10} + 204 \beta_{11} + 351 \beta_{12} - 56 \beta_{13} - 126 \beta_{14} - 9 \beta_{15} + 52 \beta_{16} + 178 \beta_{17} + 39 \beta_{18} + 51 \beta_{19} ) q^{75} + ( 235546 + 390 \beta_{1} - 3852 \beta_{2} - 1608 \beta_{3} + 310 \beta_{4} + 156 \beta_{6} - 96 \beta_{7} + 254 \beta_{8} + 134 \beta_{9} + 154 \beta_{10} + 302 \beta_{11} + 28 \beta_{13} - 302 \beta_{14} - 184 \beta_{15} - 170 \beta_{17} ) q^{76} + ( 250 - 1441 \beta_{1} - 84 \beta_{2} - 1436 \beta_{3} + 43 \beta_{4} - 13 \beta_{5} - 216 \beta_{6} + 3 \beta_{7} + 140 \beta_{8} + 24 \beta_{9} - 46 \beta_{10} - 7 \beta_{11} - 50 \beta_{12} - 62 \beta_{13} - 27 \beta_{14} - 37 \beta_{15} + 32 \beta_{16} + 24 \beta_{17} + 48 \beta_{18} + 14 \beta_{19} ) q^{77} + ( -78780 - 8131 \beta_{1} + 2508 \beta_{2} + 665 \beta_{3} - 310 \beta_{4} - 32 \beta_{5} - 710 \beta_{6} - 65 \beta_{7} + 217 \beta_{8} - 29 \beta_{9} + 109 \beta_{10} + 214 \beta_{11} + 179 \beta_{12} - 437 \beta_{13} - 6 \beta_{14} - 149 \beta_{15} + 81 \beta_{16} - \beta_{17} - 145 \beta_{18} + 65 \beta_{19} ) q^{78} + ( 18434 + 231 \beta_{1} - 1737 \beta_{2} - 1005 \beta_{3} - 516 \beta_{4} - 116 \beta_{6} + 20 \beta_{7} - 189 \beta_{8} + 94 \beta_{9} - 314 \beta_{10} - 101 \beta_{11} + 9 \beta_{13} + 143 \beta_{14} - 80 \beta_{15} + 39 \beta_{17} ) q^{79} + ( 3962 - 4691 \beta_{1} - 1274 \beta_{2} - 12651 \beta_{3} - 374 \beta_{4} + 392 \beta_{5} + 2464 \beta_{6} + 71 \beta_{7} - 1667 \beta_{8} + 137 \beta_{9} - 9 \beta_{10} - 64 \beta_{11} - 347 \beta_{12} - 215 \beta_{13} + 104 \beta_{14} - 21 \beta_{15} + 309 \beta_{16} - 175 \beta_{17} + 49 \beta_{18} - 117 \beta_{19} ) q^{80} + ( -54225 - 4581 \beta_{1} - 430 \beta_{2} + 412 \beta_{3} - 256 \beta_{4} - 68 \beta_{5} + 569 \beta_{6} - 87 \beta_{7} + 24 \beta_{8} + 166 \beta_{9} + 255 \beta_{10} + 64 \beta_{11} + 592 \beta_{12} + 220 \beta_{13} - 137 \beta_{14} + 244 \beta_{15} - 130 \beta_{16} - 14 \beta_{17} - 80 \beta_{18} - 98 \beta_{19} ) q^{81} + ( -87627 + 1380 \beta_{1} + 2353 \beta_{2} - 7814 \beta_{3} - 20 \beta_{4} + 293 \beta_{6} - 107 \beta_{7} + 884 \beta_{8} - 133 \beta_{9} + 159 \beta_{10} - 331 \beta_{11} + 266 \beta_{13} + 230 \beta_{14} + 320 \beta_{15} - 37 \beta_{17} ) q^{82} + ( -2544 - 12330 \beta_{1} + 914 \beta_{2} + 8086 \beta_{3} + 262 \beta_{4} + 152 \beta_{5} + 1276 \beta_{6} + 20 \beta_{7} + 1306 \beta_{8} + 70 \beta_{9} - 196 \beta_{10} + 178 \beta_{11} - 752 \beta_{12} + 164 \beta_{13} - 176 \beta_{14} - 308 \beta_{15} + 172 \beta_{16} + 156 \beta_{17} - 32 \beta_{18} + 52 \beta_{19} ) q^{83} + ( -16854 + 20857 \beta_{1} - 1284 \beta_{2} + 327 \beta_{3} - 63 \beta_{4} - 186 \beta_{5} - 1657 \beta_{6} + 80 \beta_{7} - 141 \beta_{8} + 31 \beta_{9} - 195 \beta_{10} - 15 \beta_{11} + 282 \beta_{12} - 218 \beta_{13} - 71 \beta_{14} + 31 \beta_{15} - 42 \beta_{16} + 114 \beta_{17} + 30 \beta_{18} + 36 \beta_{19} ) q^{84} + ( -205702 + 1431 \beta_{1} + 5091 \beta_{2} - 7304 \beta_{3} - 73 \beta_{4} + 168 \beta_{6} + 50 \beta_{7} + 817 \beta_{8} + 162 \beta_{9} - 196 \beta_{10} - 185 \beta_{11} + 223 \beta_{13} + 79 \beta_{14} + 28 \beta_{15} - 123 \beta_{17} ) q^{85} + ( -1005 + 9672 \beta_{1} + 413 \beta_{2} + 1838 \beta_{3} + 304 \beta_{4} - 52 \beta_{5} - 1123 \beta_{6} + 21 \beta_{7} + 1352 \beta_{8} - 29 \beta_{9} - 183 \beta_{10} + 369 \beta_{11} - 168 \beta_{12} - 102 \beta_{13} - 134 \beta_{14} - 170 \beta_{15} + 106 \beta_{16} + 113 \beta_{17} - 108 \beta_{18} - 20 \beta_{19} ) q^{86} + ( -150733 + 5355 \beta_{1} + 4091 \beta_{2} + 1338 \beta_{3} - 267 \beta_{4} + 112 \beta_{5} + 1013 \beta_{6} + 91 \beta_{7} - 469 \beta_{8} + 66 \beta_{9} + 125 \beta_{10} + 326 \beta_{11} + 686 \beta_{12} + 86 \beta_{13} - 310 \beta_{14} - 41 \beta_{15} + 162 \beta_{16} - 90 \beta_{17} + 38 \beta_{18} + 44 \beta_{19} ) q^{87} + ( -32597 - 668 \beta_{1} + 1426 \beta_{2} + 4692 \beta_{3} + 56 \beta_{4} - 38 \beta_{6} - 326 \beta_{8} + 116 \beta_{9} + \beta_{10} + 238 \beta_{11} - 154 \beta_{13} - 149 \beta_{14} - 115 \beta_{15} + 6 \beta_{17} ) q^{88} + ( 2789 + 13633 \beta_{1} - 778 \beta_{2} - 12945 \beta_{3} - 56 \beta_{4} - 88 \beta_{5} + 67 \beta_{6} - 36 \beta_{7} - 74 \beta_{8} - 13 \beta_{9} + 118 \beta_{10} - 168 \beta_{11} - 704 \beta_{12} - 408 \beta_{13} + 23 \beta_{14} + 4 \beta_{15} - 112 \beta_{16} + 13 \beta_{17} - 32 \beta_{18} + 104 \beta_{19} ) q^{89} + ( -207137 - 16835 \beta_{1} + 6647 \beta_{2} - 1057 \beta_{3} + 762 \beta_{4} + 108 \beta_{5} + 4096 \beta_{6} - 39 \beta_{7} + 102 \beta_{8} - 201 \beta_{9} - 309 \beta_{10} - 446 \beta_{11} + 252 \beta_{12} - 90 \beta_{13} + 546 \beta_{14} + 426 \beta_{15} - 27 \beta_{16} + 87 \beta_{17} + 234 \beta_{18} + 27 \beta_{19} ) q^{90} + ( 46216 - 876 \beta_{1} + 2322 \beta_{2} + 9176 \beta_{3} - 180 \beta_{4} + 144 \beta_{6} + 200 \beta_{7} + 688 \beta_{8} - 94 \beta_{9} + 32 \beta_{10} - 168 \beta_{11} - 342 \beta_{13} + 190 \beta_{14} + 90 \beta_{15} - 16 \beta_{17} ) q^{91} + ( 661 + 25801 \beta_{1} - 253 \beta_{2} - 2064 \beta_{3} - 264 \beta_{4} + 14 \beta_{5} - 1070 \beta_{6} - 20 \beta_{7} - 1097 \beta_{8} + 115 \beta_{9} + 134 \beta_{10} - 567 \beta_{11} - 469 \beta_{12} + 150 \beta_{13} + 55 \beta_{14} + 83 \beta_{15} - 127 \beta_{16} - 35 \beta_{17} + 239 \beta_{18} - 7 \beta_{19} ) q^{92} + ( 117314 + 12136 \beta_{1} + 711 \beta_{2} + 4821 \beta_{3} + 8 \beta_{4} + 334 \beta_{5} + 3339 \beta_{6} - 38 \beta_{7} + 102 \beta_{8} + 14 \beta_{9} + 140 \beta_{10} + 41 \beta_{11} + 852 \beta_{12} + 260 \beta_{13} + 408 \beta_{14} - 46 \beta_{15} - 62 \beta_{16} + 42 \beta_{17} - 120 \beta_{18} - 18 \beta_{19} ) q^{93} + ( -6934 + 1037 \beta_{1} - 2234 \beta_{2} - 13329 \beta_{3} + 47 \beta_{4} - 423 \beta_{6} + 122 \beta_{7} - 1693 \beta_{8} + 57 \beta_{9} - 327 \beta_{10} + 265 \beta_{11} + 704 \beta_{13} + 199 \beta_{14} - 183 \beta_{15} + 352 \beta_{17} ) q^{94} + ( 618 - 28183 \beta_{1} - 43 \beta_{2} - 7396 \beta_{3} + 331 \beta_{4} - 550 \beta_{5} + 342 \beta_{6} - 54 \beta_{7} + 1409 \beta_{8} - 284 \beta_{9} + 78 \beta_{10} + 327 \beta_{11} - 540 \beta_{12} - 393 \beta_{13} - 17 \beta_{14} + 268 \beta_{15} - 434 \beta_{16} + 71 \beta_{17} - 216 \beta_{18} - 110 \beta_{19} ) q^{95} + ( 445743 + 9887 \beta_{1} - 10603 \beta_{2} - 5724 \beta_{3} - 410 \beta_{4} - 303 \beta_{5} - 4059 \beta_{6} - 34 \beta_{7} + 1024 \beta_{8} - 591 \beta_{9} + 313 \beta_{10} - 698 \beta_{11} + 294 \beta_{12} + 605 \beta_{13} + 204 \beta_{14} - 113 \beta_{15} - 150 \beta_{16} - 293 \beta_{17} - 138 \beta_{18} + 6 \beta_{19} ) q^{96} + ( -159474 - 199 \beta_{1} + 7321 \beta_{2} - 2117 \beta_{3} - 212 \beta_{4} - 218 \beta_{6} - 189 \beta_{7} - 18 \beta_{8} - 344 \beta_{9} + 273 \beta_{10} - 123 \beta_{11} - 78 \beta_{13} - 389 \beta_{14} - 190 \beta_{15} - 242 \beta_{17} ) q^{97} + ( -2296 - 44011 \beta_{1} + 732 \beta_{2} + 10366 \beta_{3} + 148 \beta_{4} + 618 \beta_{5} + 574 \beta_{6} - 20 \beta_{7} + 342 \beta_{8} - 120 \beta_{9} - 30 \beta_{10} + 298 \beta_{11} - 76 \beta_{12} + 266 \beta_{13} + 42 \beta_{14} + 74 \beta_{15} + 20 \beta_{16} - 22 \beta_{17} - 76 \beta_{18} + 140 \beta_{19} ) q^{98} + ( 11006 + 8770 \beta_{1} - 978 \beta_{2} + 584 \beta_{3} + 8 \beta_{4} - 30 \beta_{5} + 451 \beta_{6} + 34 \beta_{7} - 140 \beta_{8} + 36 \beta_{9} - 106 \beta_{10} - 54 \beta_{11} + 72 \beta_{12} - 83 \beta_{13} + 117 \beta_{14} - 43 \beta_{15} - 15 \beta_{16} + 113 \beta_{17} + 27 \beta_{18} - 54 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 16q^{3} - 772q^{4} + 286q^{6} + 160q^{7} - 1072q^{9} + O(q^{10}) \) \( 20q + 16q^{3} - 772q^{4} + 286q^{6} + 160q^{7} - 1072q^{9} + 996q^{10} + 6092q^{12} + 808q^{13} - 3032q^{15} + 28004q^{16} + 18686q^{18} + 5920q^{19} - 20888q^{21} - 48096q^{24} - 100612q^{25} - 17624q^{27} - 33296q^{28} + 109582q^{30} - 90896q^{31} - 21296q^{33} + 68928q^{34} - 28988q^{36} + 239656q^{37} - 15416q^{39} + 34632q^{40} + 150364q^{42} - 125840q^{43} - 242428q^{45} + 244380q^{46} + 305492q^{48} - 186204q^{49} - 21992q^{51} - 120368q^{52} - 777728q^{54} - 191664q^{55} - 255840q^{57} + 601176q^{58} + 970736q^{60} + 1108360q^{61} + 574088q^{63} - 2533132q^{64} + 465850q^{66} + 617728q^{67} + 323804q^{69} - 238680q^{70} - 2031648q^{72} - 1379960q^{73} + 2481512q^{75} + 4678408q^{76} - 1556840q^{78} + 347152q^{79} - 1086136q^{81} - 1760328q^{82} - 345760q^{84} - 4097232q^{85} - 2983056q^{87} - 622908q^{88} - 4093630q^{90} + 979616q^{91} + 2363236q^{93} - 217752q^{94} + 8811824q^{96} - 3139256q^{97} + 212960q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 1026 x^{18} + 441321 x^{16} + 103808124 x^{14} + 14594358456 x^{12} + 1256133373152 x^{10} + 64843235559312 x^{8} + 1864534894961472 x^{6} + 25120735435348224 x^{4} + 103653954147713024 x^{2} + 32367107232497664\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 103 \)
\(\beta_{3}\)\(=\)\((\)\(\)\(18\!\cdots\!53\)\( \nu^{19} + \)\(26\!\cdots\!04\)\( \nu^{18} + \)\(16\!\cdots\!34\)\( \nu^{17} + \)\(24\!\cdots\!36\)\( \nu^{16} + \)\(59\!\cdots\!65\)\( \nu^{15} + \)\(93\!\cdots\!16\)\( \nu^{14} + \)\(11\!\cdots\!88\)\( \nu^{13} + \)\(18\!\cdots\!00\)\( \nu^{12} + \)\(10\!\cdots\!00\)\( \nu^{11} + \)\(21\!\cdots\!40\)\( \nu^{10} + \)\(50\!\cdots\!16\)\( \nu^{9} + \)\(13\!\cdots\!88\)\( \nu^{8} + \)\(34\!\cdots\!28\)\( \nu^{7} + \)\(47\!\cdots\!72\)\( \nu^{6} - \)\(48\!\cdots\!40\)\( \nu^{5} + \)\(73\!\cdots\!72\)\( \nu^{4} - \)\(11\!\cdots\!56\)\( \nu^{3} + \)\(33\!\cdots\!04\)\( \nu^{2} - \)\(46\!\cdots\!68\)\( \nu - \)\(91\!\cdots\!40\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(18\!\cdots\!53\)\( \nu^{19} + \)\(10\!\cdots\!68\)\( \nu^{18} + \)\(16\!\cdots\!34\)\( \nu^{17} + \)\(10\!\cdots\!92\)\( \nu^{16} + \)\(59\!\cdots\!65\)\( \nu^{15} + \)\(38\!\cdots\!92\)\( \nu^{14} + \)\(11\!\cdots\!88\)\( \nu^{13} + \)\(77\!\cdots\!40\)\( \nu^{12} + \)\(10\!\cdots\!00\)\( \nu^{11} + \)\(89\!\cdots\!60\)\( \nu^{10} + \)\(50\!\cdots\!16\)\( \nu^{9} + \)\(57\!\cdots\!36\)\( \nu^{8} + \)\(34\!\cdots\!28\)\( \nu^{7} + \)\(19\!\cdots\!64\)\( \nu^{6} - \)\(48\!\cdots\!40\)\( \nu^{5} + \)\(28\!\cdots\!84\)\( \nu^{4} - \)\(11\!\cdots\!56\)\( \nu^{3} + \)\(10\!\cdots\!08\)\( \nu^{2} - \)\(46\!\cdots\!68\)\( \nu - \)\(15\!\cdots\!60\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(18\!\cdots\!53\)\( \nu^{19} + \)\(26\!\cdots\!04\)\( \nu^{18} + \)\(16\!\cdots\!34\)\( \nu^{17} + \)\(24\!\cdots\!36\)\( \nu^{16} + \)\(59\!\cdots\!65\)\( \nu^{15} + \)\(93\!\cdots\!16\)\( \nu^{14} + \)\(11\!\cdots\!88\)\( \nu^{13} + \)\(18\!\cdots\!00\)\( \nu^{12} + \)\(10\!\cdots\!00\)\( \nu^{11} + \)\(21\!\cdots\!40\)\( \nu^{10} + \)\(50\!\cdots\!16\)\( \nu^{9} + \)\(13\!\cdots\!88\)\( \nu^{8} + \)\(34\!\cdots\!28\)\( \nu^{7} + \)\(47\!\cdots\!72\)\( \nu^{6} - \)\(48\!\cdots\!40\)\( \nu^{5} + \)\(73\!\cdots\!72\)\( \nu^{4} - \)\(11\!\cdots\!16\)\( \nu^{3} + \)\(33\!\cdots\!04\)\( \nu^{2} + \)\(10\!\cdots\!32\)\( \nu - \)\(12\!\cdots\!80\)\(\)\()/ \)\(34\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(776808349103473966772762515 \nu^{19} + 728982860376555934532369124502 \nu^{17} + 278212437841038470328727294688203 \nu^{15} + 55541963969824196351424040663447588 \nu^{13} + 6187673425345664900841838814918085640 \nu^{11} + 374970230451303684536094489891020644960 \nu^{9} + 10544408957130840409868539298963582480944 \nu^{7} + 45063507190360336612183055231196584161216 \nu^{5} - 2586118030747151275679279125281260735270144 \nu^{3} - 22922228766703639602501551971866113382187008 \nu\)\()/ \)\(86\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(14\!\cdots\!85\)\( \nu^{19} + \)\(30\!\cdots\!88\)\( \nu^{18} + \)\(14\!\cdots\!58\)\( \nu^{17} + \)\(28\!\cdots\!36\)\( \nu^{16} + \)\(56\!\cdots\!37\)\( \nu^{15} + \)\(11\!\cdots\!08\)\( \nu^{14} + \)\(12\!\cdots\!12\)\( \nu^{13} + \)\(22\!\cdots\!56\)\( \nu^{12} + \)\(15\!\cdots\!80\)\( \nu^{11} + \)\(25\!\cdots\!80\)\( \nu^{10} + \)\(12\!\cdots\!20\)\( \nu^{9} + \)\(16\!\cdots\!56\)\( \nu^{8} + \)\(55\!\cdots\!56\)\( \nu^{7} + \)\(56\!\cdots\!52\)\( \nu^{6} + \)\(13\!\cdots\!24\)\( \nu^{5} + \)\(87\!\cdots\!36\)\( \nu^{4} + \)\(15\!\cdots\!44\)\( \nu^{3} + \)\(46\!\cdots\!80\)\( \nu^{2} + \)\(46\!\cdots\!08\)\( \nu + \)\(56\!\cdots\!64\)\(\)\()/ \)\(13\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(14\!\cdots\!79\)\( \nu^{19} + \)\(12\!\cdots\!19\)\( \nu^{18} - \)\(13\!\cdots\!36\)\( \nu^{17} + \)\(11\!\cdots\!48\)\( \nu^{16} - \)\(52\!\cdots\!91\)\( \nu^{15} + \)\(43\!\cdots\!59\)\( \nu^{14} - \)\(10\!\cdots\!50\)\( \nu^{13} + \)\(88\!\cdots\!18\)\( \nu^{12} - \)\(12\!\cdots\!40\)\( \nu^{11} + \)\(10\!\cdots\!40\)\( \nu^{10} - \)\(78\!\cdots\!88\)\( \nu^{9} + \)\(64\!\cdots\!08\)\( \nu^{8} - \)\(26\!\cdots\!72\)\( \nu^{7} + \)\(21\!\cdots\!56\)\( \nu^{6} - \)\(41\!\cdots\!72\)\( \nu^{5} + \)\(32\!\cdots\!28\)\( \nu^{4} - \)\(19\!\cdots\!04\)\( \nu^{3} + \)\(13\!\cdots\!40\)\( \nu^{2} + \)\(22\!\cdots\!00\)\( \nu + \)\(25\!\cdots\!12\)\(\)\()/ \)\(59\!\cdots\!20\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(28\!\cdots\!17\)\( \nu^{19} + \)\(12\!\cdots\!40\)\( \nu^{18} - \)\(27\!\cdots\!86\)\( \nu^{17} + \)\(11\!\cdots\!44\)\( \nu^{16} - \)\(11\!\cdots\!25\)\( \nu^{15} + \)\(44\!\cdots\!96\)\( \nu^{14} - \)\(23\!\cdots\!32\)\( \nu^{13} + \)\(89\!\cdots\!96\)\( \nu^{12} - \)\(30\!\cdots\!20\)\( \nu^{11} + \)\(10\!\cdots\!60\)\( \nu^{10} - \)\(22\!\cdots\!04\)\( \nu^{9} + \)\(65\!\cdots\!80\)\( \nu^{8} - \)\(10\!\cdots\!32\)\( \nu^{7} + \)\(22\!\cdots\!88\)\( \nu^{6} - \)\(24\!\cdots\!60\)\( \nu^{5} + \)\(32\!\cdots\!92\)\( \nu^{4} - \)\(29\!\cdots\!16\)\( \nu^{3} + \)\(12\!\cdots\!32\)\( \nu^{2} - \)\(86\!\cdots\!68\)\( \nu - \)\(75\!\cdots\!36\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(10\!\cdots\!41\)\( \nu^{19} + \)\(15\!\cdots\!88\)\( \nu^{18} - \)\(10\!\cdots\!18\)\( \nu^{17} + \)\(14\!\cdots\!56\)\( \nu^{16} - \)\(40\!\cdots\!85\)\( \nu^{15} + \)\(56\!\cdots\!88\)\( \nu^{14} - \)\(86\!\cdots\!36\)\( \nu^{13} + \)\(11\!\cdots\!76\)\( \nu^{12} - \)\(10\!\cdots\!40\)\( \nu^{11} + \)\(13\!\cdots\!00\)\( \nu^{10} - \)\(79\!\cdots\!12\)\( \nu^{9} + \)\(82\!\cdots\!16\)\( \nu^{8} - \)\(33\!\cdots\!96\)\( \nu^{7} + \)\(27\!\cdots\!92\)\( \nu^{6} - \)\(80\!\cdots\!60\)\( \nu^{5} + \)\(37\!\cdots\!16\)\( \nu^{4} - \)\(89\!\cdots\!68\)\( \nu^{3} + \)\(12\!\cdots\!00\)\( \nu^{2} - \)\(25\!\cdots\!04\)\( \nu + \)\(60\!\cdots\!64\)\(\)\()/ \)\(34\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(34\!\cdots\!23\)\( \nu^{19} - \)\(15\!\cdots\!52\)\( \nu^{18} + \)\(33\!\cdots\!94\)\( \nu^{17} - \)\(14\!\cdots\!16\)\( \nu^{16} + \)\(13\!\cdots\!15\)\( \nu^{15} - \)\(56\!\cdots\!40\)\( \nu^{14} + \)\(29\!\cdots\!48\)\( \nu^{13} - \)\(11\!\cdots\!32\)\( \nu^{12} + \)\(37\!\cdots\!80\)\( \nu^{11} - \)\(13\!\cdots\!80\)\( \nu^{10} + \)\(28\!\cdots\!76\)\( \nu^{9} - \)\(86\!\cdots\!64\)\( \nu^{8} + \)\(13\!\cdots\!08\)\( \nu^{7} - \)\(29\!\cdots\!12\)\( \nu^{6} + \)\(34\!\cdots\!80\)\( \nu^{5} - \)\(46\!\cdots\!40\)\( \nu^{4} + \)\(42\!\cdots\!84\)\( \nu^{3} - \)\(21\!\cdots\!56\)\( \nu^{2} + \)\(14\!\cdots\!52\)\( \nu - \)\(97\!\cdots\!08\)\(\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(76\!\cdots\!51\)\( \nu^{19} - \)\(52\!\cdots\!08\)\( \nu^{18} - \)\(73\!\cdots\!38\)\( \nu^{17} - \)\(48\!\cdots\!72\)\( \nu^{16} - \)\(29\!\cdots\!95\)\( \nu^{15} - \)\(18\!\cdots\!32\)\( \nu^{14} - \)\(63\!\cdots\!16\)\( \nu^{13} - \)\(37\!\cdots\!00\)\( \nu^{12} - \)\(78\!\cdots\!20\)\( \nu^{11} - \)\(42\!\cdots\!80\)\( \nu^{10} - \)\(57\!\cdots\!72\)\( \nu^{9} - \)\(27\!\cdots\!76\)\( \nu^{8} - \)\(24\!\cdots\!16\)\( \nu^{7} - \)\(94\!\cdots\!44\)\( \nu^{6} - \)\(53\!\cdots\!40\)\( \nu^{5} - \)\(14\!\cdots\!44\)\( \nu^{4} - \)\(55\!\cdots\!08\)\( \nu^{3} - \)\(67\!\cdots\!08\)\( \nu^{2} - \)\(15\!\cdots\!44\)\( \nu + \)\(18\!\cdots\!80\)\(\)\()/ \)\(20\!\cdots\!40\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(18\!\cdots\!53\)\( \nu^{19} - \)\(32\!\cdots\!80\)\( \nu^{18} + \)\(16\!\cdots\!34\)\( \nu^{17} - \)\(30\!\cdots\!20\)\( \nu^{16} + \)\(59\!\cdots\!65\)\( \nu^{15} - \)\(11\!\cdots\!20\)\( \nu^{14} + \)\(11\!\cdots\!88\)\( \nu^{13} - \)\(23\!\cdots\!00\)\( \nu^{12} + \)\(10\!\cdots\!00\)\( \nu^{11} - \)\(26\!\cdots\!00\)\( \nu^{10} + \)\(50\!\cdots\!16\)\( \nu^{9} - \)\(17\!\cdots\!60\)\( \nu^{8} + \)\(34\!\cdots\!28\)\( \nu^{7} - \)\(59\!\cdots\!40\)\( \nu^{6} - \)\(48\!\cdots\!40\)\( \nu^{5} - \)\(91\!\cdots\!40\)\( \nu^{4} - \)\(11\!\cdots\!56\)\( \nu^{3} - \)\(42\!\cdots\!40\)\( \nu^{2} - \)\(47\!\cdots\!08\)\( \nu - \)\(74\!\cdots\!20\)\(\)\()/ \)\(43\!\cdots\!80\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(14\!\cdots\!27\)\( \nu^{19} - \)\(39\!\cdots\!28\)\( \nu^{18} + \)\(13\!\cdots\!94\)\( \nu^{17} - \)\(37\!\cdots\!04\)\( \nu^{16} + \)\(52\!\cdots\!87\)\( \nu^{15} - \)\(14\!\cdots\!60\)\( \nu^{14} + \)\(10\!\cdots\!44\)\( \nu^{13} - \)\(29\!\cdots\!68\)\( \nu^{12} + \)\(12\!\cdots\!00\)\( \nu^{11} - \)\(34\!\cdots\!40\)\( \nu^{10} + \)\(78\!\cdots\!24\)\( \nu^{9} - \)\(23\!\cdots\!36\)\( \nu^{8} + \)\(27\!\cdots\!48\)\( \nu^{7} - \)\(82\!\cdots\!48\)\( \nu^{6} + \)\(45\!\cdots\!04\)\( \nu^{5} - \)\(13\!\cdots\!40\)\( \nu^{4} + \)\(29\!\cdots\!60\)\( \nu^{3} - \)\(72\!\cdots\!24\)\( \nu^{2} + \)\(72\!\cdots\!96\)\( \nu - \)\(75\!\cdots\!12\)\(\)\()/ \)\(34\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(96\!\cdots\!95\)\( \nu^{19} + \)\(36\!\cdots\!70\)\( \nu^{18} + \)\(92\!\cdots\!01\)\( \nu^{17} + \)\(34\!\cdots\!30\)\( \nu^{16} + \)\(36\!\cdots\!39\)\( \nu^{15} + \)\(13\!\cdots\!70\)\( \nu^{14} + \)\(76\!\cdots\!99\)\( \nu^{13} + \)\(27\!\cdots\!50\)\( \nu^{12} + \)\(93\!\cdots\!30\)\( \nu^{11} + \)\(31\!\cdots\!60\)\( \nu^{10} + \)\(67\!\cdots\!20\)\( \nu^{9} + \)\(21\!\cdots\!20\)\( \nu^{8} + \)\(28\!\cdots\!72\)\( \nu^{7} + \)\(73\!\cdots\!40\)\( \nu^{6} + \)\(64\!\cdots\!08\)\( \nu^{5} + \)\(11\!\cdots\!80\)\( \nu^{4} + \)\(69\!\cdots\!88\)\( \nu^{3} + \)\(57\!\cdots\!80\)\( \nu^{2} + \)\(21\!\cdots\!56\)\( \nu + \)\(31\!\cdots\!60\)\(\)\()/ \)\(16\!\cdots\!80\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(29\!\cdots\!93\)\( \nu^{19} + \)\(49\!\cdots\!00\)\( \nu^{18} - \)\(29\!\cdots\!90\)\( \nu^{17} + \)\(46\!\cdots\!08\)\( \nu^{16} - \)\(12\!\cdots\!09\)\( \nu^{15} + \)\(17\!\cdots\!72\)\( \nu^{14} - \)\(27\!\cdots\!32\)\( \nu^{13} + \)\(36\!\cdots\!52\)\( \nu^{12} - \)\(37\!\cdots\!80\)\( \nu^{11} + \)\(42\!\cdots\!40\)\( \nu^{10} - \)\(29\!\cdots\!16\)\( \nu^{9} + \)\(27\!\cdots\!80\)\( \nu^{8} - \)\(14\!\cdots\!60\)\( \nu^{7} + \)\(97\!\cdots\!76\)\( \nu^{6} - \)\(36\!\cdots\!68\)\( \nu^{5} + \)\(15\!\cdots\!64\)\( \nu^{4} - \)\(42\!\cdots\!72\)\( \nu^{3} + \)\(73\!\cdots\!44\)\( \nu^{2} - \)\(13\!\cdots\!28\)\( \nu + \)\(35\!\cdots\!88\)\(\)\()/ \)\(34\!\cdots\!40\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(60\!\cdots\!35\)\( \nu^{19} - \)\(49\!\cdots\!72\)\( \nu^{18} - \)\(56\!\cdots\!42\)\( \nu^{17} - \)\(45\!\cdots\!96\)\( \nu^{16} - \)\(21\!\cdots\!23\)\( \nu^{15} - \)\(17\!\cdots\!40\)\( \nu^{14} - \)\(43\!\cdots\!08\)\( \nu^{13} - \)\(34\!\cdots\!12\)\( \nu^{12} - \)\(50\!\cdots\!00\)\( \nu^{11} - \)\(39\!\cdots\!80\)\( \nu^{10} - \)\(33\!\cdots\!40\)\( \nu^{9} - \)\(24\!\cdots\!24\)\( \nu^{8} - \)\(12\!\cdots\!44\)\( \nu^{7} - \)\(82\!\cdots\!52\)\( \nu^{6} - \)\(21\!\cdots\!96\)\( \nu^{5} - \)\(11\!\cdots\!20\)\( \nu^{4} - \)\(15\!\cdots\!16\)\( \nu^{3} - \)\(33\!\cdots\!16\)\( \nu^{2} - \)\(33\!\cdots\!72\)\( \nu + \)\(48\!\cdots\!12\)\(\)\()/ \)\(34\!\cdots\!40\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(10\!\cdots\!99\)\( \nu^{19} - \)\(30\!\cdots\!92\)\( \nu^{18} - \)\(98\!\cdots\!34\)\( \nu^{17} - \)\(28\!\cdots\!16\)\( \nu^{16} - \)\(37\!\cdots\!23\)\( \nu^{15} - \)\(10\!\cdots\!40\)\( \nu^{14} - \)\(75\!\cdots\!72\)\( \nu^{13} - \)\(22\!\cdots\!12\)\( \nu^{12} - \)\(85\!\cdots\!00\)\( \nu^{11} - \)\(25\!\cdots\!80\)\( \nu^{10} - \)\(54\!\cdots\!28\)\( \nu^{9} - \)\(16\!\cdots\!44\)\( \nu^{8} - \)\(18\!\cdots\!48\)\( \nu^{7} - \)\(59\!\cdots\!92\)\( \nu^{6} - \)\(23\!\cdots\!36\)\( \nu^{5} - \)\(93\!\cdots\!00\)\( \nu^{4} - \)\(25\!\cdots\!28\)\( \nu^{3} - \)\(44\!\cdots\!16\)\( \nu^{2} + \)\(60\!\cdots\!12\)\( \nu - \)\(16\!\cdots\!28\)\(\)\()/ \)\(20\!\cdots\!40\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(34\!\cdots\!07\)\( \nu^{19} + \)\(10\!\cdots\!56\)\( \nu^{18} + \)\(32\!\cdots\!30\)\( \nu^{17} + \)\(10\!\cdots\!68\)\( \nu^{16} + \)\(12\!\cdots\!51\)\( \nu^{15} + \)\(40\!\cdots\!40\)\( \nu^{14} + \)\(25\!\cdots\!08\)\( \nu^{13} + \)\(82\!\cdots\!76\)\( \nu^{12} + \)\(30\!\cdots\!80\)\( \nu^{11} + \)\(96\!\cdots\!60\)\( \nu^{10} + \)\(19\!\cdots\!84\)\( \nu^{9} + \)\(64\!\cdots\!32\)\( \nu^{8} + \)\(70\!\cdots\!60\)\( \nu^{7} + \)\(23\!\cdots\!76\)\( \nu^{6} + \)\(11\!\cdots\!12\)\( \nu^{5} + \)\(39\!\cdots\!00\)\( \nu^{4} + \)\(76\!\cdots\!08\)\( \nu^{3} + \)\(25\!\cdots\!48\)\( \nu^{2} + \)\(21\!\cdots\!32\)\( \nu + \)\(46\!\cdots\!64\)\(\)\()/ \)\(52\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 103\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 3 \beta_{3} - 165 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-3 \beta_{17} - 2 \beta_{15} - 5 \beta_{14} + \beta_{13} + 2 \beta_{10} - \beta_{9} + 4 \beta_{8} - \beta_{6} - 4 \beta_{4} - 45 \beta_{3} - 241 \beta_{2} + 4 \beta_{1} + 17107\)
\(\nu^{5}\)\(=\)\(4 \beta_{19} + 6 \beta_{18} + 3 \beta_{17} - 20 \beta_{16} + 8 \beta_{15} + \beta_{14} + 17 \beta_{13} + 42 \beta_{12} - 23 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} + \beta_{8} - 4 \beta_{7} - 26 \beta_{6} - 307 \beta_{5} - \beta_{4} + 1349 \beta_{3} + 37 \beta_{2} + 31934 \beta_{1} - 415\)
\(\nu^{6}\)\(=\)\(1129 \beta_{17} + 560 \beta_{15} + 1593 \beta_{14} - 323 \beta_{13} + 140 \beta_{11} - 788 \beta_{10} + 359 \beta_{9} - 2228 \beta_{8} + 100 \beta_{7} + 35 \beta_{6} + 1820 \beta_{4} + 16235 \beta_{3} + 55549 \beta_{2} - 2040 \beta_{1} - 3333125\)
\(\nu^{7}\)\(=\)\(-938 \beta_{19} - 1952 \beta_{18} - 1409 \beta_{17} + 9250 \beta_{16} - 3750 \beta_{15} - 289 \beta_{14} - 8963 \beta_{13} - 15644 \beta_{12} + 9167 \beta_{11} - 4380 \beta_{10} + 1256 \beta_{9} - 2919 \beta_{8} + 1698 \beta_{7} + 16714 \beta_{6} + 79269 \beta_{5} + 17 \beta_{4} - 469017 \beta_{3} - 20473 \beta_{2} - 6751566 \beta_{1} + 138641\)
\(\nu^{8}\)\(=\)\(-328131 \beta_{17} - 120258 \beta_{15} - 394481 \beta_{14} + 93045 \beta_{13} - 83594 \beta_{11} + 238758 \beta_{10} - 114479 \beta_{9} + 810634 \beta_{8} - 45080 \beta_{7} + 57929 \beta_{6} - 595134 \beta_{4} - 4854491 \beta_{3} - 12870171 \beta_{2} + 733298 \beta_{1} + 709315871\)
\(\nu^{9}\)\(=\)\(135292 \beta_{19} + 442738 \beta_{18} + 497969 \beta_{17} - 3080732 \beta_{16} + 1210824 \beta_{15} + 38035 \beta_{14} + 3177507 \beta_{13} + 4502398 \beta_{12} - 2741589 \beta_{11} + 1409974 \beta_{10} - 483062 \beta_{9} + 1672283 \beta_{8} - 536004 \beta_{7} - 7498702 \beta_{6} - 19636139 \beta_{5} + 107061 \beta_{4} + 142313725 \beta_{3} + 7551279 \beta_{2} + 1503012028 \beta_{1} - 41316039\)
\(\nu^{10}\)\(=\)\(87691193 \beta_{17} + 23376692 \beta_{15} + 90193309 \beta_{14} - 26065059 \beta_{13} + 33780156 \beta_{11} - 65590160 \beta_{10} + 35055815 \beta_{9} - 250298604 \beta_{8} + 14928396 \beta_{7} - 28032229 \beta_{6} + 171250612 \beta_{4} + 1373458283 \beta_{3} + 3017776929 \beta_{2} - 228445912 \beta_{1} - 158844273061\)
\(\nu^{11}\)\(=\)\(-5986582 \beta_{19} - 79733948 \beta_{18} - 156760521 \beta_{17} + 906047582 \beta_{16} - 339525106 \beta_{15} + 4754827 \beta_{14} - 961636139 \beta_{13} - 1209969272 \beta_{12} + 739102475 \beta_{11} - 405007128 \beta_{10} + 155400556 \beta_{9} - 673777111 \beta_{8} + 152005694 \beta_{7} + 2750087506 \beta_{6} + 4816830437 \beta_{5} - 59159643 \beta_{4} - 40070171285 \beta_{3} - 2368224709 \beta_{2} - 345357799414 \beta_{1} + 11541186337\)
\(\nu^{12}\)\(=\)\(-22630072363 \beta_{17} - 4261775030 \beta_{15} - 19962492901 \beta_{14} + 7177428821 \beta_{13} - 11569111526 \beta_{11} + 17159758154 \beta_{10} - 10360572347 \beta_{9} + 71364642646 \beta_{8} - 4398144144 \beta_{7} + 9466765269 \beta_{6} - 46373895330 \beta_{4} - 378551663527 \beta_{3} - 715908863291 \beta_{2} + 66165161910 \beta_{1} + 36686162740695\)
\(\nu^{13}\)\(=\)\(-4820394472 \beta_{19} + 10249514342 \beta_{18} + 46295830421 \beta_{17} - 250810612648 \beta_{16} + 89182521780 \beta_{15} - 5297460725 \beta_{14} + 269201865647 \beta_{13} + 318641653250 \beta_{12} - 189669916109 \beta_{11} + 110034799722 \beta_{10} - 45946658402 \beta_{9} + 231591674367 \beta_{8} - 40998369696 \beta_{7} - 895706537182 \beta_{6} - 1181229977979 \beta_{5} + 23206058797 \beta_{4} + 10815996964041 \beta_{3} + 684950306287 \beta_{2} + 81015087328640 \beta_{1} - 3105586855463\)
\(\nu^{14}\)\(=\)\(5745892287009 \beta_{17} + 724396369776 \beta_{15} + 4352303058137 \beta_{14} - 1949142583491 \beta_{13} + 3617821759256 \beta_{11} - 4369511551692 \beta_{10} + 2965837393715 \beta_{9} - 19481789756776 \beta_{8} + 1222343350196 \beta_{7} - 2797263386849 \beta_{6} + 12164061706824 \beta_{4} + 102598831637495 \beta_{3} + 171534328263305 \beta_{2} - 18382896738716 \beta_{1} - 8643276108707293\)
\(\nu^{15}\)\(=\)\(2544515036342 \beta_{19} - 56507963560 \beta_{18} - 13132279560317 \beta_{17} + 67148844261794 \beta_{16} - 22670007531894 \beta_{15} + 2364891356075 \beta_{14} - 72229156748679 \beta_{13} - 83347814538100 \beta_{12} + 47417731762875 \beta_{11} - 28981665813508 \beta_{10} + 12966999183776 \beta_{9} - 72807114882971 \beta_{8} + 10767388204242 \beta_{7} + 270920860652722 \beta_{6} + 290387362763029 \beta_{5} - 7885001134827 \beta_{4} - 2848395207863545 \beta_{3} - 189318753993501 \beta_{2} - 19280434751141098 \beta_{1} + 817247420494337\)
\(\nu^{16}\)\(=\)\(-1447387788088643 \beta_{17} - 109733308050138 \beta_{15} - 941841102783785 \beta_{14} + 523129194167573 \beta_{13} - 1068779363783610 \beta_{11} + 1095651166350222 \beta_{10} - 826776050095327 \beta_{9} + 5183769428987114 \beta_{8} - 328629114093288 \beta_{7} + 774421801559753 \beta_{6} - 3134708440269054 \beta_{4} - 27456726724676267 \beta_{3} - 41437698194696579 \beta_{2} + 4977109593287682 \beta_{1} + 2064454373448184479\)
\(\nu^{17}\)\(=\)\(-913399405568316 \beta_{19} - 583315569631942 \beta_{18} + 3624054288566193 \beta_{17} - 17623653014331204 \beta_{16} + 5666771940421712 \beta_{15} - 839012020439045 \beta_{14} + 18910596888616323 \beta_{13} + 21723225322353542 \beta_{12} - 11687626205850189 \beta_{11} + 7493814764904254 \beta_{10} - 3555699774765422 \beta_{9} + 21631165384083051 \beta_{8} - 2785042268127148 \beta_{7} - 78057064825437470 \beta_{6} - 71603942554716715 \beta_{5} + 2470981566554509 \beta_{4} + 738993575174681637 \beta_{3} + 50908641758678631 \beta_{2} + 4637103900059923196 \beta_{1} - 212117161161165351\)
\(\nu^{18}\)\(=\)\(363187125755177385 \beta_{17} + 12611933842231500 \beta_{15} + 203060610689319925 \beta_{14} - 138995964778172067 \beta_{13} + 303832926097560804 \beta_{11} - 272401047673614408 \beta_{10} + 225658781858229327 \beta_{9} - 1357372367726663348 \beta_{8} + 86572371643737308 \beta_{7} - 206868780881242541 \beta_{6} + 799631903691827980 \beta_{4} + 7272541749935592755 \beta_{3} + 10077078058656130753 \beta_{2} - 1324528765762093872 \beta_{1} - 498029618404848791989\)
\(\nu^{19}\)\(=\)\(285390727067283234 \beta_{19} + 282355977768047340 \beta_{18} - 980537935363288929 \beta_{17} + 4567964633076022854 \beta_{16} - 1404780758033713434 \beta_{15} + 266775617695165659 \beta_{14} - 4878794425724096307 \beta_{13} - 5643283738213063728 \beta_{12} + 2859883600483690107 \beta_{11} - 1915870321983854832 \beta_{10} + 956276683411137876 \beta_{9} - 6183210931576773807 \beta_{8} + 713762317668123270 \beta_{7} + 21753339101322100722 \beta_{6} + 17707164676850656965 \beta_{5} - 734706614458695243 \beta_{4} - 189933215291644412541 \beta_{3} - 13449489632069636325 \beta_{2} - 1124247227423037066062 \beta_{1} + 54569912547203197665\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
15.8546i
14.5373i
12.6955i
11.1144i
9.36068i
9.18066i
8.58590i
5.13283i
2.50689i
0.582665i
0.582665i
2.50689i
5.13283i
8.58590i
9.18066i
9.36068i
11.1144i
12.6955i
14.5373i
15.8546i
15.8546i 10.1117 + 25.0350i −187.368 128.653i 396.920 160.317i 185.193 1955.95i −524.506 + 506.295i 2039.74
23.2 14.5373i −26.2741 6.21862i −147.333 220.756i −90.4019 + 381.955i 126.134 1211.44i 651.658 + 326.777i −3209.19
23.3 12.6955i 16.0155 21.7372i −97.1752 9.46491i −275.964 203.324i −297.499 421.175i −216.009 696.262i −120.162
23.4 11.1144i −19.8885 18.2606i −59.5295 223.873i −202.955 + 221.048i −152.100 49.6872i 62.1023 + 726.350i 2488.21
23.5 9.36068i 26.7168 + 3.90029i −23.6224 36.7722i 36.5094 250.088i 305.058 377.962i 698.575 + 208.407i −344.213
23.6 9.18066i −4.20063 + 26.6712i −20.2844 92.7007i 244.859 + 38.5645i −527.539 401.338i −693.709 224.072i −851.053
23.7 8.58590i −21.1258 + 16.8137i −9.71763 61.7994i 144.360 + 181.384i 427.595 466.063i 163.602 710.405i 530.603
23.8 5.13283i −8.59633 25.5950i 37.6541 122.121i −131.375 + 44.1235i 168.062 521.773i −581.206 + 440.046i −626.825
23.9 2.50689i 23.0874 + 13.9990i 57.7155 201.357i 35.0940 57.8777i −481.867 305.128i 337.056 + 646.401i 504.781
23.10 0.582665i 12.1539 24.1098i 63.6605 147.782i −14.0479 7.08168i 326.963 74.3834i −433.563 586.058i 86.1076
23.11 0.582665i 12.1539 + 24.1098i 63.6605 147.782i −14.0479 + 7.08168i 326.963 74.3834i −433.563 + 586.058i 86.1076
23.12 2.50689i 23.0874 13.9990i 57.7155 201.357i 35.0940 + 57.8777i −481.867 305.128i 337.056 646.401i 504.781
23.13 5.13283i −8.59633 + 25.5950i 37.6541 122.121i −131.375 44.1235i 168.062 521.773i −581.206 440.046i −626.825
23.14 8.58590i −21.1258 16.8137i −9.71763 61.7994i 144.360 181.384i 427.595 466.063i 163.602 + 710.405i 530.603
23.15 9.18066i −4.20063 26.6712i −20.2844 92.7007i 244.859 38.5645i −527.539 401.338i −693.709 + 224.072i −851.053
23.16 9.36068i 26.7168 3.90029i −23.6224 36.7722i 36.5094 + 250.088i 305.058 377.962i 698.575 208.407i −344.213
23.17 11.1144i −19.8885 + 18.2606i −59.5295 223.873i −202.955 221.048i −152.100 49.6872i 62.1023 726.350i 2488.21
23.18 12.6955i 16.0155 + 21.7372i −97.1752 9.46491i −275.964 + 203.324i −297.499 421.175i −216.009 + 696.262i −120.162
23.19 14.5373i −26.2741 + 6.21862i −147.333 220.756i −90.4019 381.955i 126.134 1211.44i 651.658 326.777i −3209.19
23.20 15.8546i 10.1117 25.0350i −187.368 128.653i 396.920 + 160.317i 185.193 1955.95i −524.506 506.295i 2039.74
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.7.b.a 20
3.b odd 2 1 inner 33.7.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.7.b.a 20 1.a even 1 1 trivial
33.7.b.a 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 254 T^{2} + 37609 T^{4} - 3966852 T^{6} + 314837688 T^{8} - 19731233568 T^{10} + 966330701712 T^{12} - 34234869955776 T^{14} + 580345385056512 T^{16} + 30989693969301504 T^{18} - 3496424513357217792 T^{20} + \)\(12\!\cdots\!84\)\( T^{22} + \)\(97\!\cdots\!92\)\( T^{24} - \)\(23\!\cdots\!36\)\( T^{26} + \)\(27\!\cdots\!72\)\( T^{28} - \)\(22\!\cdots\!68\)\( T^{30} + \)\(14\!\cdots\!48\)\( T^{32} - \)\(76\!\cdots\!32\)\( T^{34} + \)\(29\!\cdots\!24\)\( T^{36} - \)\(82\!\cdots\!24\)\( T^{38} + \)\(13\!\cdots\!76\)\( T^{40} \)
$3$ \( 1 - 16 T + 664 T^{2} - 3384 T^{3} + 392526 T^{4} - 9884592 T^{5} + 152787222 T^{6} - 7557922080 T^{7} + 126155672721 T^{8} + 167981492952 T^{9} - 133177734979164 T^{10} + 122458508362008 T^{11} + 67044296866520961 T^{12} - 2928093868057497120 T^{13} + 43151624289679645782 T^{14} - \)\(20\!\cdots\!08\)\( T^{15} + \)\(58\!\cdots\!46\)\( T^{16} - \)\(37\!\cdots\!56\)\( T^{17} + \)\(52\!\cdots\!04\)\( T^{18} - \)\(93\!\cdots\!04\)\( T^{19} + \)\(42\!\cdots\!01\)\( T^{20} \)
$5$ \( 1 - 105944 T^{2} + 5828372572 T^{4} - 223491085826124 T^{6} + 6796594245046654950 T^{8} - \)\(17\!\cdots\!00\)\( T^{10} + \)\(40\!\cdots\!50\)\( T^{12} - \)\(86\!\cdots\!00\)\( T^{14} + \)\(16\!\cdots\!25\)\( T^{16} - \)\(29\!\cdots\!00\)\( T^{18} + \)\(47\!\cdots\!00\)\( T^{20} - \)\(71\!\cdots\!00\)\( T^{22} + \)\(98\!\cdots\!25\)\( T^{24} - \)\(12\!\cdots\!00\)\( T^{26} + \)\(14\!\cdots\!50\)\( T^{28} - \)\(15\!\cdots\!00\)\( T^{30} + \)\(14\!\cdots\!50\)\( T^{32} - \)\(11\!\cdots\!00\)\( T^{34} + \)\(73\!\cdots\!00\)\( T^{36} - \)\(32\!\cdots\!00\)\( T^{38} + \)\(75\!\cdots\!25\)\( T^{40} \)
$7$ \( ( 1 - 80 T + 637996 T^{2} + 3183312 T^{3} + 208575242877 T^{4} + 10337171228448 T^{5} + 47323593286298448 T^{6} + 3191350635114380256 T^{7} + \)\(81\!\cdots\!82\)\( T^{8} + \)\(55\!\cdots\!40\)\( T^{9} + \)\(10\!\cdots\!24\)\( T^{10} + \)\(65\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!82\)\( T^{12} + \)\(51\!\cdots\!44\)\( T^{13} + \)\(90\!\cdots\!48\)\( T^{14} + \)\(23\!\cdots\!52\)\( T^{15} + \)\(55\!\cdots\!77\)\( T^{16} + \)\(99\!\cdots\!88\)\( T^{17} + \)\(23\!\cdots\!96\)\( T^{18} - \)\(34\!\cdots\!20\)\( T^{19} + \)\(50\!\cdots\!01\)\( T^{20} )^{2} \)
$11$ \( ( 1 + 161051 T^{2} )^{10} \)
$13$ \( ( 1 - 404 T + 27747598 T^{2} - 11751652500 T^{3} + 406880014516797 T^{4} - 155026051136922576 T^{5} + \)\(39\!\cdots\!72\)\( T^{6} - \)\(13\!\cdots\!12\)\( T^{7} + \)\(28\!\cdots\!78\)\( T^{8} - \)\(83\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!64\)\( T^{10} - \)\(40\!\cdots\!20\)\( T^{11} + \)\(66\!\cdots\!18\)\( T^{12} - \)\(14\!\cdots\!48\)\( T^{13} + \)\(21\!\cdots\!92\)\( T^{14} - \)\(40\!\cdots\!24\)\( T^{15} + \)\(51\!\cdots\!77\)\( T^{16} - \)\(71\!\cdots\!00\)\( T^{17} + \)\(81\!\cdots\!58\)\( T^{18} - \)\(57\!\cdots\!56\)\( T^{19} + \)\(68\!\cdots\!01\)\( T^{20} )^{2} \)
$17$ \( 1 - 326153840 T^{2} + 51772798381932250 T^{4} - \)\(53\!\cdots\!92\)\( T^{6} + \)\(39\!\cdots\!57\)\( T^{8} - \)\(23\!\cdots\!72\)\( T^{10} + \)\(10\!\cdots\!64\)\( T^{12} - \)\(42\!\cdots\!76\)\( T^{14} + \)\(14\!\cdots\!78\)\( T^{16} - \)\(41\!\cdots\!96\)\( T^{18} + \)\(10\!\cdots\!92\)\( T^{20} - \)\(23\!\cdots\!56\)\( T^{22} + \)\(47\!\cdots\!38\)\( T^{24} - \)\(83\!\cdots\!56\)\( T^{26} + \)\(12\!\cdots\!24\)\( T^{28} - \)\(15\!\cdots\!72\)\( T^{30} + \)\(15\!\cdots\!77\)\( T^{32} - \)\(12\!\cdots\!32\)\( T^{34} + \)\(68\!\cdots\!50\)\( T^{36} - \)\(25\!\cdots\!40\)\( T^{38} + \)\(45\!\cdots\!01\)\( T^{40} \)
$19$ \( ( 1 - 2960 T + 227191252 T^{2} - 1434889516464 T^{3} + 26321072412781149 T^{4} - \)\(22\!\cdots\!64\)\( T^{5} + \)\(23\!\cdots\!60\)\( T^{6} - \)\(19\!\cdots\!36\)\( T^{7} + \)\(17\!\cdots\!78\)\( T^{8} - \)\(11\!\cdots\!36\)\( T^{9} + \)\(96\!\cdots\!36\)\( T^{10} - \)\(53\!\cdots\!16\)\( T^{11} + \)\(38\!\cdots\!58\)\( T^{12} - \)\(20\!\cdots\!76\)\( T^{13} + \)\(11\!\cdots\!60\)\( T^{14} - \)\(52\!\cdots\!64\)\( T^{15} + \)\(28\!\cdots\!69\)\( T^{16} - \)\(73\!\cdots\!04\)\( T^{17} + \)\(54\!\cdots\!32\)\( T^{18} - \)\(33\!\cdots\!60\)\( T^{19} + \)\(53\!\cdots\!01\)\( T^{20} )^{2} \)
$23$ \( 1 - 1376795048 T^{2} + 989675322643423132 T^{4} - \)\(48\!\cdots\!92\)\( T^{6} + \)\(18\!\cdots\!14\)\( T^{8} - \)\(56\!\cdots\!00\)\( T^{10} + \)\(14\!\cdots\!98\)\( T^{12} - \)\(32\!\cdots\!84\)\( T^{14} + \)\(63\!\cdots\!73\)\( T^{16} - \)\(10\!\cdots\!44\)\( T^{18} + \)\(17\!\cdots\!64\)\( T^{20} - \)\(24\!\cdots\!24\)\( T^{22} + \)\(30\!\cdots\!93\)\( T^{24} - \)\(33\!\cdots\!24\)\( T^{26} + \)\(33\!\cdots\!38\)\( T^{28} - \)\(28\!\cdots\!00\)\( T^{30} + \)\(20\!\cdots\!94\)\( T^{32} - \)\(11\!\cdots\!72\)\( T^{34} + \)\(52\!\cdots\!52\)\( T^{36} - \)\(16\!\cdots\!88\)\( T^{38} + \)\(25\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 - 5446349648 T^{2} + 15007440397895201242 T^{4} - \)\(28\!\cdots\!16\)\( T^{6} + \)\(39\!\cdots\!41\)\( T^{8} - \)\(46\!\cdots\!00\)\( T^{10} + \)\(46\!\cdots\!88\)\( T^{12} - \)\(40\!\cdots\!96\)\( T^{14} + \)\(31\!\cdots\!50\)\( T^{16} - \)\(21\!\cdots\!64\)\( T^{18} + \)\(13\!\cdots\!68\)\( T^{20} - \)\(77\!\cdots\!24\)\( T^{22} + \)\(39\!\cdots\!50\)\( T^{24} - \)\(18\!\cdots\!16\)\( T^{26} + \)\(73\!\cdots\!68\)\( T^{28} - \)\(25\!\cdots\!00\)\( T^{30} + \)\(78\!\cdots\!81\)\( T^{32} - \)\(19\!\cdots\!96\)\( T^{34} + \)\(36\!\cdots\!82\)\( T^{36} - \)\(47\!\cdots\!28\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \)
$31$ \( ( 1 + 45448 T + 5844063544 T^{2} + 231425522151072 T^{3} + 16632393739020278238 T^{4} + \)\(57\!\cdots\!04\)\( T^{5} + \)\(30\!\cdots\!38\)\( T^{6} + \)\(94\!\cdots\!88\)\( T^{7} + \)\(40\!\cdots\!93\)\( T^{8} + \)\(11\!\cdots\!92\)\( T^{9} + \)\(40\!\cdots\!16\)\( T^{10} + \)\(98\!\cdots\!52\)\( T^{11} + \)\(31\!\cdots\!73\)\( T^{12} + \)\(65\!\cdots\!08\)\( T^{13} + \)\(18\!\cdots\!98\)\( T^{14} + \)\(31\!\cdots\!04\)\( T^{15} + \)\(81\!\cdots\!78\)\( T^{16} + \)\(10\!\cdots\!92\)\( T^{17} + \)\(22\!\cdots\!04\)\( T^{18} + \)\(15\!\cdots\!08\)\( T^{19} + \)\(30\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( ( 1 - 119828 T + 17863924552 T^{2} - 1743024437706648 T^{3} + \)\(16\!\cdots\!58\)\( T^{4} - \)\(13\!\cdots\!44\)\( T^{5} + \)\(96\!\cdots\!30\)\( T^{6} - \)\(63\!\cdots\!92\)\( T^{7} + \)\(39\!\cdots\!21\)\( T^{8} - \)\(22\!\cdots\!52\)\( T^{9} + \)\(11\!\cdots\!36\)\( T^{10} - \)\(57\!\cdots\!68\)\( T^{11} + \)\(26\!\cdots\!01\)\( T^{12} - \)\(10\!\cdots\!68\)\( T^{13} + \)\(41\!\cdots\!30\)\( T^{14} - \)\(14\!\cdots\!56\)\( T^{15} + \)\(46\!\cdots\!78\)\( T^{16} - \)\(12\!\cdots\!12\)\( T^{17} + \)\(33\!\cdots\!92\)\( T^{18} - \)\(57\!\cdots\!92\)\( T^{19} + \)\(12\!\cdots\!01\)\( T^{20} )^{2} \)
$41$ \( 1 - 53919366848 T^{2} + \)\(14\!\cdots\!02\)\( T^{4} - \)\(26\!\cdots\!84\)\( T^{6} + \)\(35\!\cdots\!33\)\( T^{8} - \)\(38\!\cdots\!40\)\( T^{10} + \)\(34\!\cdots\!08\)\( T^{12} - \)\(26\!\cdots\!44\)\( T^{14} + \)\(17\!\cdots\!58\)\( T^{16} - \)\(99\!\cdots\!04\)\( T^{18} + \)\(50\!\cdots\!52\)\( T^{20} - \)\(22\!\cdots\!24\)\( T^{22} + \)\(88\!\cdots\!38\)\( T^{24} - \)\(30\!\cdots\!04\)\( T^{26} + \)\(89\!\cdots\!68\)\( T^{28} - \)\(22\!\cdots\!40\)\( T^{30} + \)\(47\!\cdots\!73\)\( T^{32} - \)\(78\!\cdots\!24\)\( T^{34} + \)\(98\!\cdots\!82\)\( T^{36} - \)\(81\!\cdots\!08\)\( T^{38} + \)\(34\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 + 62920 T + 47666590516 T^{2} + 2789010979726200 T^{3} + \)\(10\!\cdots\!57\)\( T^{4} + \)\(58\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!40\)\( T^{6} + \)\(75\!\cdots\!56\)\( T^{7} + \)\(15\!\cdots\!06\)\( T^{8} + \)\(67\!\cdots\!64\)\( T^{9} + \)\(11\!\cdots\!92\)\( T^{10} + \)\(42\!\cdots\!36\)\( T^{11} + \)\(61\!\cdots\!06\)\( T^{12} + \)\(19\!\cdots\!44\)\( T^{13} + \)\(24\!\cdots\!40\)\( T^{14} + \)\(58\!\cdots\!48\)\( T^{15} + \)\(68\!\cdots\!57\)\( T^{16} + \)\(11\!\cdots\!00\)\( T^{17} + \)\(12\!\cdots\!16\)\( T^{18} + \)\(10\!\cdots\!80\)\( T^{19} + \)\(10\!\cdots\!01\)\( T^{20} )^{2} \)
$47$ \( 1 - 107446465724 T^{2} + \)\(55\!\cdots\!98\)\( T^{4} - \)\(18\!\cdots\!88\)\( T^{6} + \)\(47\!\cdots\!85\)\( T^{8} - \)\(95\!\cdots\!04\)\( T^{10} + \)\(16\!\cdots\!16\)\( T^{12} - \)\(23\!\cdots\!96\)\( T^{14} + \)\(31\!\cdots\!30\)\( T^{16} - \)\(37\!\cdots\!72\)\( T^{18} + \)\(42\!\cdots\!88\)\( T^{20} - \)\(44\!\cdots\!52\)\( T^{22} + \)\(42\!\cdots\!30\)\( T^{24} - \)\(37\!\cdots\!16\)\( T^{26} + \)\(29\!\cdots\!76\)\( T^{28} - \)\(20\!\cdots\!04\)\( T^{30} + \)\(11\!\cdots\!85\)\( T^{32} - \)\(54\!\cdots\!28\)\( T^{34} + \)\(18\!\cdots\!58\)\( T^{36} - \)\(41\!\cdots\!64\)\( T^{38} + \)\(44\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 - 212494098644 T^{2} + \)\(22\!\cdots\!70\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(81\!\cdots\!73\)\( T^{8} - \)\(33\!\cdots\!60\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(35\!\cdots\!84\)\( T^{14} + \)\(94\!\cdots\!86\)\( T^{16} - \)\(23\!\cdots\!76\)\( T^{18} + \)\(52\!\cdots\!00\)\( T^{20} - \)\(11\!\cdots\!16\)\( T^{22} + \)\(22\!\cdots\!66\)\( T^{24} - \)\(42\!\cdots\!64\)\( T^{26} + \)\(68\!\cdots\!84\)\( T^{28} - \)\(97\!\cdots\!60\)\( T^{30} + \)\(11\!\cdots\!93\)\( T^{32} - \)\(10\!\cdots\!00\)\( T^{34} + \)\(75\!\cdots\!70\)\( T^{36} - \)\(35\!\cdots\!84\)\( T^{38} + \)\(81\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 415514367272 T^{2} + \)\(89\!\cdots\!72\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{6} + \)\(15\!\cdots\!42\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{10} + \)\(10\!\cdots\!50\)\( T^{12} - \)\(70\!\cdots\!36\)\( T^{14} + \)\(40\!\cdots\!53\)\( T^{16} - \)\(20\!\cdots\!68\)\( T^{18} + \)\(91\!\cdots\!44\)\( T^{20} - \)\(36\!\cdots\!08\)\( T^{22} + \)\(12\!\cdots\!33\)\( T^{24} - \)\(39\!\cdots\!76\)\( T^{26} + \)\(10\!\cdots\!50\)\( T^{28} - \)\(24\!\cdots\!28\)\( T^{30} + \)\(47\!\cdots\!02\)\( T^{32} - \)\(75\!\cdots\!16\)\( T^{34} + \)\(90\!\cdots\!52\)\( T^{36} - \)\(74\!\cdots\!12\)\( T^{38} + \)\(31\!\cdots\!01\)\( T^{40} \)
$61$ \( ( 1 - 554180 T + 340353776758 T^{2} - 128121008267951940 T^{3} + \)\(46\!\cdots\!17\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{5} + \)\(39\!\cdots\!96\)\( T^{6} - \)\(10\!\cdots\!84\)\( T^{7} + \)\(26\!\cdots\!02\)\( T^{8} - \)\(61\!\cdots\!56\)\( T^{9} + \)\(14\!\cdots\!84\)\( T^{10} - \)\(31\!\cdots\!16\)\( T^{11} + \)\(69\!\cdots\!42\)\( T^{12} - \)\(13\!\cdots\!04\)\( T^{13} + \)\(27\!\cdots\!36\)\( T^{14} - \)\(49\!\cdots\!76\)\( T^{15} + \)\(86\!\cdots\!37\)\( T^{16} - \)\(12\!\cdots\!40\)\( T^{17} + \)\(16\!\cdots\!98\)\( T^{18} - \)\(14\!\cdots\!80\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \)
$67$ \( ( 1 - 308864 T + 565227244024 T^{2} - 123808515862783800 T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(22\!\cdots\!96\)\( T^{5} + \)\(22\!\cdots\!06\)\( T^{6} - \)\(29\!\cdots\!56\)\( T^{7} + \)\(28\!\cdots\!09\)\( T^{8} - \)\(32\!\cdots\!56\)\( T^{9} + \)\(28\!\cdots\!84\)\( T^{10} - \)\(29\!\cdots\!64\)\( T^{11} + \)\(22\!\cdots\!49\)\( T^{12} - \)\(21\!\cdots\!04\)\( T^{13} + \)\(15\!\cdots\!26\)\( T^{14} - \)\(13\!\cdots\!04\)\( T^{15} + \)\(77\!\cdots\!66\)\( T^{16} - \)\(61\!\cdots\!00\)\( T^{17} + \)\(25\!\cdots\!84\)\( T^{18} - \)\(12\!\cdots\!56\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$71$ \( 1 - 864341724752 T^{2} + \)\(41\!\cdots\!12\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!26\)\( T^{8} - \)\(90\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!42\)\( T^{12} - \)\(33\!\cdots\!64\)\( T^{14} + \)\(55\!\cdots\!69\)\( T^{16} - \)\(81\!\cdots\!24\)\( T^{18} + \)\(10\!\cdots\!92\)\( T^{20} - \)\(13\!\cdots\!84\)\( T^{22} + \)\(14\!\cdots\!89\)\( T^{24} - \)\(14\!\cdots\!44\)\( T^{26} + \)\(13\!\cdots\!62\)\( T^{28} - \)\(10\!\cdots\!04\)\( T^{30} + \)\(75\!\cdots\!66\)\( T^{32} - \)\(45\!\cdots\!00\)\( T^{34} + \)\(21\!\cdots\!52\)\( T^{36} - \)\(74\!\cdots\!72\)\( T^{38} + \)\(14\!\cdots\!01\)\( T^{40} \)
$73$ \( ( 1 + 689980 T + 960708843166 T^{2} + 478433795122687356 T^{3} + \)\(41\!\cdots\!73\)\( T^{4} + \)\(17\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(43\!\cdots\!80\)\( T^{7} + \)\(26\!\cdots\!30\)\( T^{8} + \)\(84\!\cdots\!28\)\( T^{9} + \)\(44\!\cdots\!08\)\( T^{10} + \)\(12\!\cdots\!92\)\( T^{11} + \)\(60\!\cdots\!30\)\( T^{12} + \)\(14\!\cdots\!20\)\( T^{13} + \)\(62\!\cdots\!16\)\( T^{14} + \)\(13\!\cdots\!36\)\( T^{15} + \)\(50\!\cdots\!53\)\( T^{16} + \)\(86\!\cdots\!24\)\( T^{17} + \)\(26\!\cdots\!46\)\( T^{18} + \)\(28\!\cdots\!20\)\( T^{19} + \)\(63\!\cdots\!01\)\( T^{20} )^{2} \)
$79$ \( ( 1 - 173576 T + 1428771684292 T^{2} - 373338458307928056 T^{3} + \)\(10\!\cdots\!09\)\( T^{4} - \)\(33\!\cdots\!40\)\( T^{5} + \)\(47\!\cdots\!32\)\( T^{6} - \)\(17\!\cdots\!16\)\( T^{7} + \)\(16\!\cdots\!82\)\( T^{8} - \)\(58\!\cdots\!44\)\( T^{9} + \)\(46\!\cdots\!96\)\( T^{10} - \)\(14\!\cdots\!24\)\( T^{11} + \)\(10\!\cdots\!62\)\( T^{12} - \)\(24\!\cdots\!76\)\( T^{13} + \)\(16\!\cdots\!92\)\( T^{14} - \)\(28\!\cdots\!40\)\( T^{15} + \)\(20\!\cdots\!89\)\( T^{16} - \)\(18\!\cdots\!96\)\( T^{17} + \)\(17\!\cdots\!12\)\( T^{18} - \)\(51\!\cdots\!56\)\( T^{19} + \)\(72\!\cdots\!01\)\( T^{20} )^{2} \)
$83$ \( 1 - 1928289612260 T^{2} + \)\(21\!\cdots\!66\)\( T^{4} - \)\(17\!\cdots\!76\)\( T^{6} + \)\(12\!\cdots\!97\)\( T^{8} - \)\(69\!\cdots\!76\)\( T^{10} + \)\(35\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!52\)\( T^{14} + \)\(64\!\cdots\!62\)\( T^{16} - \)\(24\!\cdots\!48\)\( T^{18} + \)\(82\!\cdots\!12\)\( T^{20} - \)\(25\!\cdots\!28\)\( T^{22} + \)\(73\!\cdots\!02\)\( T^{24} - \)\(19\!\cdots\!12\)\( T^{26} + \)\(45\!\cdots\!88\)\( T^{28} - \)\(97\!\cdots\!76\)\( T^{30} + \)\(18\!\cdots\!17\)\( T^{32} - \)\(28\!\cdots\!96\)\( T^{34} + \)\(36\!\cdots\!46\)\( T^{36} - \)\(35\!\cdots\!60\)\( T^{38} + \)\(19\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 - 5772784866920 T^{2} + \)\(16\!\cdots\!44\)\( T^{4} - \)\(32\!\cdots\!80\)\( T^{6} + \)\(47\!\cdots\!14\)\( T^{8} - \)\(54\!\cdots\!04\)\( T^{10} + \)\(52\!\cdots\!42\)\( T^{12} - \)\(43\!\cdots\!56\)\( T^{14} + \)\(30\!\cdots\!89\)\( T^{16} - \)\(18\!\cdots\!84\)\( T^{18} + \)\(97\!\cdots\!04\)\( T^{20} - \)\(45\!\cdots\!64\)\( T^{22} + \)\(18\!\cdots\!49\)\( T^{24} - \)\(64\!\cdots\!16\)\( T^{26} + \)\(19\!\cdots\!02\)\( T^{28} - \)\(50\!\cdots\!04\)\( T^{30} + \)\(10\!\cdots\!94\)\( T^{32} - \)\(18\!\cdots\!80\)\( T^{34} + \)\(23\!\cdots\!84\)\( T^{36} - \)\(19\!\cdots\!20\)\( T^{38} + \)\(84\!\cdots\!01\)\( T^{40} \)
$97$ \( ( 1 + 1569628 T + 4780520894296 T^{2} + 5703832984718951904 T^{3} + \)\(11\!\cdots\!94\)\( T^{4} + \)\(11\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!94\)\( T^{6} + \)\(16\!\cdots\!68\)\( T^{7} + \)\(21\!\cdots\!29\)\( T^{8} + \)\(17\!\cdots\!84\)\( T^{9} + \)\(20\!\cdots\!56\)\( T^{10} + \)\(14\!\cdots\!36\)\( T^{11} + \)\(14\!\cdots\!89\)\( T^{12} + \)\(94\!\cdots\!52\)\( T^{13} + \)\(86\!\cdots\!14\)\( T^{14} + \)\(46\!\cdots\!64\)\( T^{15} + \)\(37\!\cdots\!74\)\( T^{16} + \)\(15\!\cdots\!36\)\( T^{17} + \)\(11\!\cdots\!56\)\( T^{18} + \)\(30\!\cdots\!32\)\( T^{19} + \)\(16\!\cdots\!01\)\( T^{20} )^{2} \)
show more
show less