Properties

Label 43.4.c.a
Level $43$
Weight $4$
Character orbit 43.c
Analytic conductor $2.537$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 43.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.53708213025\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - x^{19} + 60 x^{18} - 25 x^{17} + 2336 x^{16} - 645 x^{15} + 52478 x^{14} - 2415 x^{13} + 850704 x^{12} - 4147 x^{11} + 8670544 x^{10} + 873865 x^{9} + 62344097 x^{8} + 3655316 x^{7} + 215661012 x^{6} + 43840208 x^{5} + 507687824 x^{4} - 3907840 x^{3} + 17517568 x^{2} + 245760 x + 589824\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( -\beta_{4} - \beta_{6} ) q^{3} + ( 4 + \beta_{2} ) q^{4} + ( -2 \beta_{4} + \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{6} + ( -5 - \beta_{2} + 5 \beta_{4} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{7} + ( -4 + 3 \beta_{5} - \beta_{8} + \beta_{11} + \beta_{17} + \beta_{18} ) q^{8} + ( -12 + \beta_{1} + 12 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{13} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} + ( -\beta_{4} - \beta_{6} ) q^{3} + ( 4 + \beta_{2} ) q^{4} + ( -2 \beta_{4} + \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{6} + ( -5 - \beta_{2} + 5 \beta_{4} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{7} + ( -4 + 3 \beta_{5} - \beta_{8} + \beta_{11} + \beta_{17} + \beta_{18} ) q^{8} + ( -12 + \beta_{1} + 12 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{13} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{9} + ( 5 \beta_{1} - \beta_{3} + 3 \beta_{4} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{13} ) q^{10} + ( 4 - \beta_{2} + 2 \beta_{5} + \beta_{8} - \beta_{11} - \beta_{14} - \beta_{16} - 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{11} + ( -\beta_{1} - 10 \beta_{4} - 4 \beta_{6} - \beta_{8} - 4 \beta_{9} - \beta_{12} - \beta_{13} ) q^{12} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{13} + ( 9 - 10 \beta_{1} - \beta_{2} - 9 \beta_{4} - 10 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} - 2 \beta_{18} - \beta_{19} ) q^{14} + ( 5 + 5 \beta_{1} - 2 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{15} + ( 10 + 2 \beta_{2} - \beta_{3} - 3 \beta_{5} + 2 \beta_{8} + \beta_{11} - 5 \beta_{14} - \beta_{15} + \beta_{16} - 3 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{16} + ( -10 - 2 \beta_{1} + \beta_{2} + 10 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{9} - 2 \beta_{13} + 4 \beta_{14} - 3 \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{17} + ( 23 - 14 \beta_{1} + 8 \beta_{2} - 23 \beta_{4} - 14 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 8 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{18} + ( 8 \beta_{1} + 6 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{19} + ( -9 \beta_{1} - 2 \beta_{3} - 51 \beta_{4} - \beta_{6} + \beta_{7} - 3 \beta_{8} - 6 \beta_{9} - \beta_{12} + 2 \beta_{13} ) q^{20} + ( 3 + 7 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} - 3 \beta_{8} + 4 \beta_{11} - 10 \beta_{14} + 3 \beta_{15} + 2 \beta_{16} + 5 \beta_{17} + 3 \beta_{18} ) q^{21} + ( 18 - 4 \beta_{2} - \beta_{3} + 8 \beta_{5} + 4 \beta_{11} + \beta_{14} - \beta_{15} - 4 \beta_{16} - 4 \beta_{17} ) q^{22} + ( \beta_{1} - \beta_{3} - 11 \beta_{4} - 9 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{13} ) q^{23} + ( 23 \beta_{1} + 27 \beta_{4} + 9 \beta_{6} - 8 \beta_{7} + 6 \beta_{8} - 7 \beta_{10} + \beta_{12} ) q^{24} + ( -16 + 3 \beta_{1} + 3 \beta_{2} + 16 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 3 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - \beta_{12} + 4 \beta_{14} - 5 \beta_{15} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{25} + ( -4 - 7 \beta_{1} - 8 \beta_{2} + 4 \beta_{4} - 7 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} + 8 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + 3 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} - 3 \beta_{16} - 5 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{26} + ( -6 - 4 \beta_{2} + \beta_{3} - \beta_{5} + 3 \beta_{8} - 5 \beta_{11} - 4 \beta_{14} + \beta_{15} + \beta_{17} - 3 \beta_{18} + \beta_{19} ) q^{27} + ( -86 - 12 \beta_{1} - 15 \beta_{2} + 86 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + 15 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + 6 \beta_{14} + 4 \beta_{15} + \beta_{16} + 6 \beta_{17} + 3 \beta_{18} - \beta_{19} ) q^{28} + ( -8 + 11 \beta_{1} + \beta_{2} + 8 \beta_{4} + 11 \beta_{5} + \beta_{6} - \beta_{9} - 5 \beta_{10} + 5 \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + 6 \beta_{18} - \beta_{19} ) q^{29} + ( 70 + 5 \beta_{2} - 70 \beta_{4} - 5 \beta_{6} + 14 \beta_{7} - 5 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 5 \beta_{14} + 7 \beta_{15} - 2 \beta_{16} - 14 \beta_{17} - 4 \beta_{18} + 2 \beta_{19} ) q^{30} + ( -16 \beta_{1} - 4 \beta_{3} + 24 \beta_{4} - 4 \beta_{6} + 5 \beta_{7} + 3 \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{12} - 5 \beta_{13} ) q^{31} + ( -47 - 8 \beta_{2} - 7 \beta_{3} - 5 \beta_{5} - \beta_{8} - 2 \beta_{11} + 10 \beta_{14} - 7 \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{32} + ( -7 \beta_{1} + 10 \beta_{3} - 48 \beta_{4} - 3 \beta_{6} + 9 \beta_{7} - 6 \beta_{8} - 7 \beta_{9} + 9 \beta_{10} + \beta_{12} ) q^{33} + ( -16 - 16 \beta_{1} + 5 \beta_{2} + 16 \beta_{4} - 16 \beta_{5} - 19 \beta_{6} - 11 \beta_{7} - 5 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - 5 \beta_{13} - 19 \beta_{14} - \beta_{15} + 5 \beta_{16} + 11 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{34} + ( 35 + 8 \beta_{2} - 3 \beta_{3} + \beta_{5} + 2 \beta_{8} - 5 \beta_{11} + 8 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} - 8 \beta_{17} - 2 \beta_{18} ) q^{35} + ( -106 + 39 \beta_{1} - 13 \beta_{2} + 106 \beta_{4} + 39 \beta_{5} - 6 \beta_{6} - 13 \beta_{7} + 13 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{13} - 6 \beta_{14} - \beta_{16} + 13 \beta_{17} + 4 \beta_{18} + \beta_{19} ) q^{36} + ( 18 \beta_{1} - \beta_{3} - 20 \beta_{4} - 14 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} + 19 \beta_{9} - \beta_{12} + 5 \beta_{13} ) q^{37} + ( 10 \beta_{1} + 7 \beta_{3} - 82 \beta_{4} + 5 \beta_{6} + 4 \beta_{7} + \beta_{8} - 15 \beta_{9} - 12 \beta_{10} - \beta_{12} + 3 \beta_{13} ) q^{38} + ( 70 + 8 \beta_{2} + 7 \beta_{3} + 28 \beta_{5} - 5 \beta_{8} + 7 \beta_{11} + 7 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} + 5 \beta_{18} ) q^{39} + ( 50 \beta_{1} + 7 \beta_{3} + 144 \beta_{4} + 23 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 11 \beta_{9} + 2 \beta_{10} - 2 \beta_{13} ) q^{40} + ( 51 + 3 \beta_{3} + 28 \beta_{5} - 4 \beta_{8} - \beta_{11} - 23 \beta_{14} + 3 \beta_{15} - 3 \beta_{16} + 10 \beta_{17} + 4 \beta_{18} - 3 \beta_{19} ) q^{41} + ( -17 + 21 \beta_{2} - 14 \beta_{3} + 34 \beta_{5} - 5 \beta_{8} + 8 \beta_{11} + 41 \beta_{14} - 14 \beta_{15} + 4 \beta_{16} + 5 \beta_{17} + 5 \beta_{18} - \beta_{19} ) q^{42} + ( 51 - 38 \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - 9 \beta_{4} - 7 \beta_{5} + 5 \beta_{6} - 3 \beta_{8} - 5 \beta_{10} + 4 \beta_{11} + \beta_{12} + \beta_{13} + 15 \beta_{14} + 6 \beta_{15} - \beta_{16} + 11 \beta_{17} - 3 \beta_{18} ) q^{43} + ( 54 + 11 \beta_{2} + 8 \beta_{3} - \beta_{5} - 7 \beta_{8} + 8 \beta_{11} - 36 \beta_{14} + 8 \beta_{15} + \beta_{16} - 8 \beta_{17} + 7 \beta_{18} + \beta_{19} ) q^{44} + ( 92 + 20 \beta_{2} + 4 \beta_{3} + 6 \beta_{8} - 8 \beta_{11} + 18 \beta_{14} + 4 \beta_{15} - 4 \beta_{16} - 6 \beta_{17} - 6 \beta_{18} ) q^{45} + ( -24 \beta_{1} + 9 \beta_{3} + 8 \beta_{4} + 21 \beta_{6} + 3 \beta_{7} - \beta_{8} + 7 \beta_{9} + 9 \beta_{10} ) q^{46} + ( -29 - 22 \beta_{2} - 11 \beta_{3} + 41 \beta_{5} + 7 \beta_{8} - 3 \beta_{11} + 5 \beta_{14} - 11 \beta_{15} + 2 \beta_{16} - 25 \beta_{17} - 7 \beta_{18} + \beta_{19} ) q^{47} + ( -25 \beta_{1} - 4 \beta_{3} - 250 \beta_{4} - 10 \beta_{6} + 17 \beta_{7} - 4 \beta_{8} - 34 \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{48} + ( 9 \beta_{1} + 5 \beta_{3} - 19 \beta_{4} - 16 \beta_{6} - 7 \beta_{7} + 2 \beta_{8} - 13 \beta_{9} - 2 \beta_{10} + 3 \beta_{13} ) q^{49} + ( 71 + 10 \beta_{1} + 18 \beta_{2} - 71 \beta_{4} + 10 \beta_{5} - 49 \beta_{6} - 7 \beta_{7} - 18 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} - \beta_{12} - 6 \beta_{13} - 49 \beta_{14} - 6 \beta_{15} + 6 \beta_{16} + 7 \beta_{17} - 7 \beta_{18} + \beta_{19} ) q^{50} + ( 75 + 5 \beta_{2} - 13 \beta_{3} - 72 \beta_{5} + 8 \beta_{8} - 3 \beta_{14} - 13 \beta_{15} + 3 \beta_{16} + 15 \beta_{17} - 8 \beta_{18} - \beta_{19} ) q^{51} + ( -71 - 40 \beta_{1} - 29 \beta_{2} + 71 \beta_{4} - 40 \beta_{5} + 18 \beta_{6} + 5 \beta_{7} + 29 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 18 \beta_{14} + 11 \beta_{15} - 3 \beta_{16} - 5 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{52} + ( 31 \beta_{1} - 3 \beta_{3} + 74 \beta_{4} - \beta_{6} - 3 \beta_{7} - 4 \beta_{8} + 12 \beta_{9} - 7 \beta_{10} - 7 \beta_{13} ) q^{53} + ( -3 - 15 \beta_{2} - 7 \beta_{3} - 42 \beta_{5} + 4 \beta_{8} - 6 \beta_{11} + 34 \beta_{14} - 7 \beta_{15} - \beta_{16} + 5 \beta_{17} - 4 \beta_{18} - 6 \beta_{19} ) q^{54} + ( 17 \beta_{1} - 18 \beta_{3} - 139 \beta_{4} - 29 \beta_{6} - 20 \beta_{7} + 5 \beta_{8} - 19 \beta_{9} - 3 \beta_{10} + 6 \beta_{12} + 5 \beta_{13} ) q^{55} + ( -109 - 120 \beta_{1} + 3 \beta_{2} + 109 \beta_{4} - 120 \beta_{5} + 25 \beta_{6} + 13 \beta_{7} - 3 \beta_{9} + 12 \beta_{10} - 12 \beta_{11} - \beta_{12} - 4 \beta_{13} + 25 \beta_{14} + 6 \beta_{15} + 4 \beta_{16} - 13 \beta_{17} - 7 \beta_{18} + \beta_{19} ) q^{56} + ( -61 - \beta_{1} - 41 \beta_{2} + 61 \beta_{4} - \beta_{5} - 32 \beta_{6} - 4 \beta_{7} + 41 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} - 5 \beta_{12} + 4 \beta_{13} - 32 \beta_{14} + 13 \beta_{15} - 4 \beta_{16} + 4 \beta_{17} - 5 \beta_{18} + 5 \beta_{19} ) q^{57} + ( 160 + 13 \beta_{1} + 44 \beta_{2} - 160 \beta_{4} + 13 \beta_{5} - 40 \beta_{6} + 6 \beta_{7} - 44 \beta_{9} - 8 \beta_{10} + 8 \beta_{11} - 5 \beta_{12} - 5 \beta_{13} - 40 \beta_{14} + 5 \beta_{16} - 6 \beta_{17} - 5 \beta_{18} + 5 \beta_{19} ) q^{58} + ( -139 - 55 \beta_{2} + 10 \beta_{3} + 2 \beta_{5} - 8 \beta_{8} + 11 \beta_{11} - 35 \beta_{14} + 10 \beta_{15} - 10 \beta_{16} - 25 \beta_{17} + 8 \beta_{18} - 6 \beta_{19} ) q^{59} + ( -150 + 102 \beta_{1} - 47 \beta_{2} + 150 \beta_{4} + 102 \beta_{5} + 84 \beta_{6} + 3 \beta_{7} + 47 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} + \beta_{12} + 9 \beta_{13} + 84 \beta_{14} - 4 \beta_{15} - 9 \beta_{16} - 3 \beta_{17} + 10 \beta_{18} - \beta_{19} ) q^{60} + ( 21 + \beta_{1} + 30 \beta_{2} - 21 \beta_{4} + \beta_{5} + 41 \beta_{6} - 2 \beta_{7} - 30 \beta_{9} - 7 \beta_{10} + 7 \beta_{11} - 5 \beta_{12} + 10 \beta_{13} + 41 \beta_{14} + \beta_{15} - 10 \beta_{16} + 2 \beta_{17} + 7 \beta_{18} + 5 \beta_{19} ) q^{61} + ( -11 \beta_{1} - 3 \beta_{3} + 197 \beta_{4} + 72 \beta_{6} - 20 \beta_{7} + 7 \beta_{8} + 28 \beta_{9} - 7 \beta_{10} + 12 \beta_{12} + 3 \beta_{13} ) q^{62} + ( -93 \beta_{1} - 5 \beta_{3} - 188 \beta_{4} + 45 \beta_{6} - 22 \beta_{7} + 8 \beta_{8} - 38 \beta_{9} - 9 \beta_{10} + 4 \beta_{12} - 5 \beta_{13} ) q^{63} + ( -44 + 2 \beta_{2} + 11 \beta_{3} - 45 \beta_{5} + 5 \beta_{8} - 8 \beta_{11} - 61 \beta_{14} + 11 \beta_{15} - 5 \beta_{16} + 6 \beta_{17} - 5 \beta_{18} - \beta_{19} ) q^{64} + ( -101 + 5 \beta_{2} - 17 \beta_{3} - 45 \beta_{5} + \beta_{8} - 2 \beta_{11} - 18 \beta_{14} - 17 \beta_{15} + 12 \beta_{16} - \beta_{18} + \beta_{19} ) q^{65} + ( 112 \beta_{1} + \beta_{3} + 109 \beta_{4} - 76 \beta_{6} - 30 \beta_{7} + 10 \beta_{8} + 32 \beta_{9} + \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{66} + ( 46 \beta_{1} + 5 \beta_{3} - 57 \beta_{4} + 46 \beta_{6} + 3 \beta_{7} - 21 \beta_{8} - 4 \beta_{9} + 13 \beta_{10} - 6 \beta_{12} ) q^{67} + ( -111 - 4 \beta_{1} + 17 \beta_{2} + 111 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} - 20 \beta_{7} - 17 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 5 \beta_{12} + 6 \beta_{14} - 15 \beta_{15} + 20 \beta_{17} - 5 \beta_{19} ) q^{68} + ( -358 + 41 \beta_{1} + 13 \beta_{2} + 358 \beta_{4} + 41 \beta_{5} + 25 \beta_{6} - 4 \beta_{7} - 13 \beta_{9} + \beta_{10} - \beta_{11} - 7 \beta_{12} - 16 \beta_{13} + 25 \beta_{14} - 4 \beta_{15} + 16 \beta_{16} + 4 \beta_{17} + 4 \beta_{18} + 7 \beta_{19} ) q^{69} + ( -7 - 16 \beta_{2} + 10 \beta_{3} + 101 \beta_{5} - 15 \beta_{8} + 12 \beta_{11} - 7 \beta_{14} + 10 \beta_{15} - 2 \beta_{16} + 29 \beta_{17} + 15 \beta_{18} - 5 \beta_{19} ) q^{70} + ( -171 - 118 \beta_{1} - 28 \beta_{2} + 171 \beta_{4} - 118 \beta_{5} + 2 \beta_{6} - 13 \beta_{7} + 28 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} - 7 \beta_{12} - \beta_{13} + 2 \beta_{14} + 5 \beta_{15} + \beta_{16} + 13 \beta_{17} - 4 \beta_{18} + 7 \beta_{19} ) q^{71} + ( 411 - 85 \beta_{1} + 17 \beta_{2} - 411 \beta_{4} - 85 \beta_{5} - 36 \beta_{6} + 40 \beta_{7} - 17 \beta_{9} + 21 \beta_{10} - 21 \beta_{11} - 6 \beta_{12} + 5 \beta_{13} - 36 \beta_{14} - 5 \beta_{15} - 5 \beta_{16} - 40 \beta_{17} - 23 \beta_{18} + 6 \beta_{19} ) q^{72} + ( 116 + 59 \beta_{1} + 28 \beta_{2} - 116 \beta_{4} + 59 \beta_{5} + 39 \beta_{6} + 11 \beta_{7} - 28 \beta_{9} - 7 \beta_{10} + 7 \beta_{11} + 11 \beta_{12} - 12 \beta_{13} + 39 \beta_{14} + 3 \beta_{15} + 12 \beta_{16} - 11 \beta_{17} + 9 \beta_{18} - 11 \beta_{19} ) q^{73} + ( -31 \beta_{1} - 3 \beta_{3} - 235 \beta_{4} - 16 \beta_{6} + 45 \beta_{7} - 14 \beta_{8} + 6 \beta_{9} - 4 \beta_{10} - 4 \beta_{12} + 21 \beta_{13} ) q^{74} + ( -9 + 27 \beta_{2} - 3 \beta_{3} - 184 \beta_{5} - 3 \beta_{8} + 16 \beta_{11} - 10 \beta_{14} - 3 \beta_{15} + 16 \beta_{16} + 34 \beta_{17} + 3 \beta_{18} - 6 \beta_{19} ) q^{75} + ( 143 \beta_{1} - 2 \beta_{3} - 83 \beta_{4} - 101 \beta_{6} + 5 \beta_{7} + 18 \beta_{8} - 32 \beta_{9} - 18 \beta_{10} - 6 \beta_{12} + \beta_{13} ) q^{76} + ( 142 - 92 \beta_{1} - 53 \beta_{2} - 142 \beta_{4} - 92 \beta_{5} - 30 \beta_{6} + 11 \beta_{7} + 53 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 5 \beta_{12} + 21 \beta_{13} - 30 \beta_{14} - 3 \beta_{15} - 21 \beta_{16} - 11 \beta_{17} + 4 \beta_{18} - 5 \beta_{19} ) q^{77} + ( 275 + 28 \beta_{2} + 11 \beta_{3} + 151 \beta_{5} - 12 \beta_{8} + 20 \beta_{11} + 36 \beta_{14} + 11 \beta_{15} - 11 \beta_{16} - 17 \beta_{17} + 12 \beta_{18} + 2 \beta_{19} ) q^{78} + ( -129 + 58 \beta_{1} + 4 \beta_{2} + 129 \beta_{4} + 58 \beta_{5} + 2 \beta_{6} - 15 \beta_{7} - 4 \beta_{9} + 11 \beta_{10} - 11 \beta_{11} + 7 \beta_{12} - 7 \beta_{13} + 2 \beta_{14} - 15 \beta_{15} + 7 \beta_{16} + 15 \beta_{17} - 16 \beta_{18} - 7 \beta_{19} ) q^{79} + ( -130 \beta_{1} + 4 \beta_{3} - 358 \beta_{4} - 92 \beta_{6} + \beta_{7} - 4 \beta_{8} - 43 \beta_{9} + 5 \beta_{10} + 5 \beta_{12} - 27 \beta_{13} ) q^{80} + ( -60 \beta_{1} + 12 \beta_{3} + 116 \beta_{4} + 24 \beta_{6} + 7 \beta_{7} + 11 \beta_{8} + 6 \beta_{9} + 18 \beta_{10} + 7 \beta_{12} - 13 \beta_{13} ) q^{81} + ( 342 + 39 \beta_{2} - 18 \beta_{3} + 47 \beta_{5} + 16 \beta_{8} - 17 \beta_{11} + 14 \beta_{14} - 18 \beta_{15} + 7 \beta_{16} - 13 \beta_{17} - 16 \beta_{18} - \beta_{19} ) q^{82} + ( 101 \beta_{1} + 4 \beta_{3} + 62 \beta_{4} + 58 \beta_{6} + 16 \beta_{7} - 21 \beta_{8} - 25 \beta_{9} + \beta_{10} + 11 \beta_{13} ) q^{83} + ( 452 + 58 \beta_{2} - 3 \beta_{3} + 103 \beta_{5} + 19 \beta_{8} - 11 \beta_{11} - 117 \beta_{14} - 3 \beta_{15} + 8 \beta_{16} - 19 \beta_{17} - 19 \beta_{18} + 18 \beta_{19} ) q^{84} + ( 93 + 13 \beta_{2} + 22 \beta_{3} + 111 \beta_{5} + 15 \beta_{8} - 19 \beta_{11} + 59 \beta_{14} + 22 \beta_{15} - 17 \beta_{16} - 25 \beta_{17} - 15 \beta_{18} + 8 \beta_{19} ) q^{85} + ( -137 - 37 \beta_{1} - 12 \beta_{2} + \beta_{3} + 524 \beta_{4} + 75 \beta_{5} + 16 \beta_{6} - 5 \beta_{7} + 2 \beta_{8} + 65 \beta_{9} + 17 \beta_{10} - 5 \beta_{11} - 5 \beta_{12} + \beta_{13} + 52 \beta_{14} + 11 \beta_{15} + 4 \beta_{16} + 10 \beta_{17} + 16 \beta_{18} - \beta_{19} ) q^{86} + ( 164 + 5 \beta_{2} - \beta_{3} - 143 \beta_{5} + 21 \beta_{8} - 26 \beta_{11} - 43 \beta_{14} - \beta_{15} - 8 \beta_{16} - 27 \beta_{17} - 21 \beta_{18} - 5 \beta_{19} ) q^{87} + ( -311 + 32 \beta_{2} - 16 \beta_{3} + 108 \beta_{5} - 21 \beta_{8} + 6 \beta_{11} + 89 \beta_{14} - 16 \beta_{15} + 16 \beta_{16} + 23 \beta_{17} + 21 \beta_{18} - \beta_{19} ) q^{88} + ( -110 \beta_{1} - 25 \beta_{3} - 211 \beta_{4} - 20 \beta_{6} + 11 \beta_{7} + 12 \beta_{8} + 29 \beta_{9} - 14 \beta_{10} + \beta_{12} + 7 \beta_{13} ) q^{89} + ( -86 - 60 \beta_{2} + 32 \beta_{3} + 246 \beta_{5} - 30 \beta_{8} + 14 \beta_{11} + 30 \beta_{14} + 32 \beta_{15} - 14 \beta_{16} + 24 \beta_{17} + 30 \beta_{18} - 8 \beta_{19} ) q^{90} + ( 48 \beta_{1} + 14 \beta_{3} - 373 \beta_{4} - 41 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} - 22 \beta_{9} - 8 \beta_{10} - 6 \beta_{12} + 2 \beta_{13} ) q^{91} + ( -35 \beta_{1} - 7 \beta_{3} + 242 \beta_{4} - 35 \beta_{6} + 12 \beta_{7} - 4 \beta_{8} - 8 \beta_{9} + 6 \beta_{10} - 14 \beta_{13} ) q^{92} + ( -33 + 259 \beta_{1} - 53 \beta_{2} + 33 \beta_{4} + 259 \beta_{5} - 121 \beta_{6} - 23 \beta_{7} + 53 \beta_{9} - 7 \beta_{10} + 7 \beta_{11} + 6 \beta_{12} + 23 \beta_{13} - 121 \beta_{14} + 18 \beta_{15} - 23 \beta_{16} + 23 \beta_{17} + 29 \beta_{18} - 6 \beta_{19} ) q^{93} + ( 519 - \beta_{2} + 14 \beta_{3} - 139 \beta_{5} - 4 \beta_{8} - 6 \beta_{11} - 79 \beta_{14} + 14 \beta_{15} - 13 \beta_{16} + \beta_{17} + 4 \beta_{18} + 6 \beta_{19} ) q^{94} + ( -46 - 50 \beta_{1} - 21 \beta_{2} + 46 \beta_{4} - 50 \beta_{5} + 66 \beta_{6} + 33 \beta_{7} + 21 \beta_{9} + 5 \beta_{10} - 5 \beta_{11} - 11 \beta_{12} + 5 \beta_{13} + 66 \beta_{14} - 16 \beta_{15} - 5 \beta_{16} - 33 \beta_{17} - 17 \beta_{18} + 11 \beta_{19} ) q^{95} + ( 320 \beta_{1} - 13 \beta_{3} + 335 \beta_{4} + 4 \beta_{6} + 15 \beta_{7} + 16 \beta_{8} + 91 \beta_{9} + 18 \beta_{10} - 2 \beta_{12} + 19 \beta_{13} ) q^{96} + ( -67 - 11 \beta_{2} + 9 \beta_{3} + 15 \beta_{5} - 32 \beta_{8} + 24 \beta_{11} + 18 \beta_{14} + 9 \beta_{15} + 17 \beta_{16} + 9 \beta_{17} + 32 \beta_{18} + 10 \beta_{19} ) q^{97} + ( 117 \beta_{1} + 25 \beta_{3} - 71 \beta_{4} - 70 \beta_{6} + 5 \beta_{7} - 4 \beta_{8} - 45 \beta_{9} - 10 \beta_{10} - 10 \beta_{12} - 9 \beta_{13} ) q^{98} + ( -397 - 176 \beta_{1} + 44 \beta_{2} + 397 \beta_{4} - 176 \beta_{5} + 79 \beta_{6} - 42 \beta_{7} - 44 \beta_{9} - 45 \beta_{10} + 45 \beta_{11} + 5 \beta_{12} + 3 \beta_{13} + 79 \beta_{14} - 7 \beta_{15} - 3 \beta_{16} + 42 \beta_{17} + 30 \beta_{18} - 5 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 2q^{2} - 5q^{3} + 78q^{4} - 19q^{5} + 15q^{6} - 51q^{7} - 72q^{8} - 117q^{9} + O(q^{10}) \) \( 20q - 2q^{2} - 5q^{3} + 78q^{4} - 19q^{5} + 15q^{6} - 51q^{7} - 72q^{8} - 117q^{9} + 27q^{10} + 54q^{11} - 72q^{12} - 15q^{13} + 96q^{14} + 65q^{15} + 134q^{16} - 82q^{17} + 247q^{18} + 78q^{19} - 495q^{20} - 18q^{21} + 380q^{22} - 61q^{23} + 202q^{24} - 151q^{25} - 21q^{26} - 194q^{27} - 794q^{28} - 53q^{29} + 627q^{30} + 253q^{31} - 798q^{32} - 424q^{33} - 231q^{34} + 710q^{35} - 1092q^{36} - 129q^{37} - 854q^{38} + 1382q^{39} + 1345q^{40} + 782q^{41} + 62q^{42} + 1025q^{43} + 754q^{44} + 1888q^{45} - 40q^{46} - 668q^{47} - 2401q^{48} - 115q^{49} + 424q^{50} + 1590q^{51} - 564q^{52} + 773q^{53} + 364q^{54} - 1242q^{55} - 923q^{56} - 765q^{57} + 1328q^{58} - 2966q^{59} - 1075q^{60} + 437q^{61} + 1509q^{62} - 2222q^{63} - 1476q^{64} - 2126q^{65} + 1483q^{66} - 642q^{67} - 1052q^{68} - 3503q^{69} - 170q^{70} - 1545q^{71} + 3834q^{72} + 1292q^{73} - 2232q^{74} + 164q^{75} - 252q^{76} + 1448q^{77} + 5644q^{78} - 1405q^{79} - 3157q^{80} + 974q^{81} + 6608q^{82} + 543q^{83} + 7304q^{84} + 1946q^{85} + 2776q^{86} + 2818q^{87} - 5372q^{88} - 2196q^{89} - 1484q^{90} - 3513q^{91} + 2629q^{92} - 983q^{93} + 9878q^{94} - 149q^{95} + 3540q^{96} - 850q^{97} - 213q^{98} - 3181q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - x^{19} + 60 x^{18} - 25 x^{17} + 2336 x^{16} - 645 x^{15} + 52478 x^{14} - 2415 x^{13} + 850704 x^{12} - 4147 x^{11} + 8670544 x^{10} + 873865 x^{9} + 62344097 x^{8} + 3655316 x^{7} + 215661012 x^{6} + 43840208 x^{5} + 507687824 x^{4} - 3907840 x^{3} + 17517568 x^{2} + 245760 x + 589824\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(20\!\cdots\!63\)\( \nu^{19} + \)\(75\!\cdots\!39\)\( \nu^{18} - \)\(12\!\cdots\!60\)\( \nu^{17} + \)\(36\!\cdots\!23\)\( \nu^{16} - \)\(47\!\cdots\!00\)\( \nu^{15} + \)\(13\!\cdots\!83\)\( \nu^{14} - \)\(10\!\cdots\!22\)\( \nu^{13} + \)\(25\!\cdots\!77\)\( \nu^{12} - \)\(16\!\cdots\!64\)\( \nu^{11} + \)\(39\!\cdots\!17\)\( \nu^{10} - \)\(15\!\cdots\!92\)\( \nu^{9} + \)\(34\!\cdots\!09\)\( \nu^{8} - \)\(10\!\cdots\!43\)\( \nu^{7} + \)\(23\!\cdots\!84\)\( \nu^{6} - \)\(33\!\cdots\!80\)\( \nu^{5} + \)\(27\!\cdots\!96\)\( \nu^{4} - \)\(60\!\cdots\!44\)\( \nu^{3} + \)\(92\!\cdots\!56\)\( \nu^{2} - \)\(28\!\cdots\!64\)\( \nu - \)\(21\!\cdots\!60\)\(\)\()/ \)\(18\!\cdots\!08\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(62\!\cdots\!89\)\( \nu^{19} + \)\(38\!\cdots\!77\)\( \nu^{18} - \)\(42\!\cdots\!16\)\( \nu^{17} + \)\(20\!\cdots\!21\)\( \nu^{16} - \)\(16\!\cdots\!04\)\( \nu^{15} + \)\(79\!\cdots\!21\)\( \nu^{14} - \)\(38\!\cdots\!26\)\( \nu^{13} + \)\(17\!\cdots\!27\)\( \nu^{12} - \)\(63\!\cdots\!68\)\( \nu^{11} + \)\(27\!\cdots\!11\)\( \nu^{10} - \)\(68\!\cdots\!16\)\( \nu^{9} + \)\(27\!\cdots\!79\)\( \nu^{8} - \)\(50\!\cdots\!25\)\( \nu^{7} + \)\(19\!\cdots\!96\)\( \nu^{6} - \)\(22\!\cdots\!88\)\( \nu^{5} + \)\(68\!\cdots\!68\)\( \nu^{4} - \)\(52\!\cdots\!92\)\( \nu^{3} + \)\(15\!\cdots\!88\)\( \nu^{2} - \)\(61\!\cdots\!48\)\( \nu + \)\(54\!\cdots\!60\)\(\)\()/ \)\(25\!\cdots\!92\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(28\!\cdots\!69\)\( \nu^{19} + \)\(96\!\cdots\!65\)\( \nu^{18} - \)\(17\!\cdots\!04\)\( \nu^{17} + \)\(47\!\cdots\!89\)\( \nu^{16} - \)\(68\!\cdots\!44\)\( \nu^{15} + \)\(17\!\cdots\!89\)\( \nu^{14} - \)\(15\!\cdots\!02\)\( \nu^{13} + \)\(36\!\cdots\!11\)\( \nu^{12} - \)\(24\!\cdots\!08\)\( \nu^{11} + \)\(57\!\cdots\!99\)\( \nu^{10} - \)\(24\!\cdots\!52\)\( \nu^{9} + \)\(56\!\cdots\!51\)\( \nu^{8} - \)\(17\!\cdots\!65\)\( \nu^{7} + \)\(41\!\cdots\!32\)\( \nu^{6} - \)\(58\!\cdots\!40\)\( \nu^{5} + \)\(13\!\cdots\!24\)\( \nu^{4} - \)\(11\!\cdots\!68\)\( \nu^{3} + \)\(34\!\cdots\!20\)\( \nu^{2} - \)\(38\!\cdots\!16\)\( \nu + \)\(11\!\cdots\!04\)\(\)\()/ \)\(11\!\cdots\!12\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(17\!\cdots\!99\)\( \nu^{19} - \)\(16\!\cdots\!91\)\( \nu^{18} + \)\(10\!\cdots\!16\)\( \nu^{17} - \)\(40\!\cdots\!15\)\( \nu^{16} + \)\(39\!\cdots\!96\)\( \nu^{15} - \)\(10\!\cdots\!55\)\( \nu^{14} + \)\(89\!\cdots\!94\)\( \nu^{13} - \)\(24\!\cdots\!33\)\( \nu^{12} + \)\(14\!\cdots\!64\)\( \nu^{11} + \)\(18\!\cdots\!71\)\( \nu^{10} + \)\(14\!\cdots\!84\)\( \nu^{9} + \)\(17\!\cdots\!07\)\( \nu^{8} + \)\(10\!\cdots\!59\)\( \nu^{7} + \)\(79\!\cdots\!72\)\( \nu^{6} + \)\(36\!\cdots\!44\)\( \nu^{5} + \)\(80\!\cdots\!72\)\( \nu^{4} + \)\(86\!\cdots\!40\)\( \nu^{3} + \)\(30\!\cdots\!44\)\( \nu^{2} + \)\(45\!\cdots\!08\)\( \nu + \)\(42\!\cdots\!64\)\(\)\()/ \)\(29\!\cdots\!28\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(16\!\cdots\!97\)\( \nu^{19} - \)\(67\!\cdots\!21\)\( \nu^{18} + \)\(10\!\cdots\!44\)\( \nu^{17} - \)\(35\!\cdots\!69\)\( \nu^{16} + \)\(41\!\cdots\!84\)\( \nu^{15} - \)\(13\!\cdots\!01\)\( \nu^{14} + \)\(93\!\cdots\!54\)\( \nu^{13} - \)\(27\!\cdots\!79\)\( \nu^{12} + \)\(14\!\cdots\!84\)\( \nu^{11} - \)\(43\!\cdots\!87\)\( \nu^{10} + \)\(15\!\cdots\!80\)\( \nu^{9} - \)\(43\!\cdots\!27\)\( \nu^{8} + \)\(10\!\cdots\!93\)\( \nu^{7} - \)\(31\!\cdots\!48\)\( \nu^{6} + \)\(40\!\cdots\!00\)\( \nu^{5} - \)\(10\!\cdots\!56\)\( \nu^{4} + \)\(85\!\cdots\!84\)\( \nu^{3} - \)\(25\!\cdots\!56\)\( \nu^{2} + \)\(54\!\cdots\!64\)\( \nu - \)\(88\!\cdots\!28\)\(\)\()/ \)\(25\!\cdots\!92\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(91\!\cdots\!75\)\( \nu^{19} - \)\(89\!\cdots\!51\)\( \nu^{18} + \)\(53\!\cdots\!00\)\( \nu^{17} - \)\(20\!\cdots\!91\)\( \nu^{16} + \)\(20\!\cdots\!12\)\( \nu^{15} - \)\(51\!\cdots\!31\)\( \nu^{14} + \)\(43\!\cdots\!46\)\( \nu^{13} - \)\(55\!\cdots\!01\)\( \nu^{12} + \)\(67\!\cdots\!72\)\( \nu^{11} + \)\(60\!\cdots\!19\)\( \nu^{10} + \)\(62\!\cdots\!76\)\( \nu^{9} + \)\(77\!\cdots\!87\)\( \nu^{8} + \)\(39\!\cdots\!75\)\( \nu^{7} + \)\(96\!\cdots\!00\)\( \nu^{6} + \)\(73\!\cdots\!92\)\( \nu^{5} + \)\(21\!\cdots\!92\)\( \nu^{4} + \)\(35\!\cdots\!28\)\( \nu^{3} - \)\(10\!\cdots\!80\)\( \nu^{2} - \)\(94\!\cdots\!04\)\( \nu - \)\(35\!\cdots\!88\)\(\)\()/ \)\(12\!\cdots\!96\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(42\!\cdots\!61\)\( \nu^{19} + \)\(20\!\cdots\!65\)\( \nu^{18} - \)\(27\!\cdots\!80\)\( \nu^{17} + \)\(10\!\cdots\!29\)\( \nu^{16} - \)\(10\!\cdots\!52\)\( \nu^{15} + \)\(39\!\cdots\!49\)\( \nu^{14} - \)\(24\!\cdots\!70\)\( \nu^{13} + \)\(84\!\cdots\!63\)\( \nu^{12} - \)\(39\!\cdots\!36\)\( \nu^{11} + \)\(13\!\cdots\!55\)\( \nu^{10} - \)\(41\!\cdots\!08\)\( \nu^{9} + \)\(13\!\cdots\!47\)\( \nu^{8} - \)\(29\!\cdots\!21\)\( \nu^{7} + \)\(96\!\cdots\!84\)\( \nu^{6} - \)\(11\!\cdots\!92\)\( \nu^{5} + \)\(31\!\cdots\!20\)\( \nu^{4} - \)\(25\!\cdots\!32\)\( \nu^{3} + \)\(77\!\cdots\!84\)\( \nu^{2} - \)\(26\!\cdots\!32\)\( \nu + \)\(26\!\cdots\!84\)\(\)\()/ \)\(25\!\cdots\!92\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(85\!\cdots\!99\)\( \nu^{19} - \)\(28\!\cdots\!71\)\( \nu^{18} + \)\(53\!\cdots\!52\)\( \nu^{17} - \)\(14\!\cdots\!99\)\( \nu^{16} + \)\(20\!\cdots\!32\)\( \nu^{15} - \)\(53\!\cdots\!39\)\( \nu^{14} + \)\(46\!\cdots\!54\)\( \nu^{13} - \)\(10\!\cdots\!01\)\( \nu^{12} + \)\(73\!\cdots\!00\)\( \nu^{11} - \)\(17\!\cdots\!25\)\( \nu^{10} + \)\(74\!\cdots\!84\)\( \nu^{9} - \)\(16\!\cdots\!09\)\( \nu^{8} + \)\(51\!\cdots\!07\)\( \nu^{7} - \)\(12\!\cdots\!52\)\( \nu^{6} + \)\(17\!\cdots\!40\)\( \nu^{5} - \)\(39\!\cdots\!36\)\( \nu^{4} + \)\(33\!\cdots\!00\)\( \nu^{3} - \)\(10\!\cdots\!36\)\( \nu^{2} + \)\(11\!\cdots\!24\)\( \nu - \)\(35\!\cdots\!36\)\(\)\()/ \)\(29\!\cdots\!28\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(87\!\cdots\!51\)\( \nu^{19} + \)\(30\!\cdots\!47\)\( \nu^{18} - \)\(54\!\cdots\!88\)\( \nu^{17} + \)\(15\!\cdots\!75\)\( \nu^{16} - \)\(20\!\cdots\!44\)\( \nu^{15} + \)\(56\!\cdots\!99\)\( \nu^{14} - \)\(47\!\cdots\!58\)\( \nu^{13} + \)\(11\!\cdots\!57\)\( \nu^{12} - \)\(75\!\cdots\!36\)\( \nu^{11} + \)\(18\!\cdots\!33\)\( \nu^{10} - \)\(76\!\cdots\!84\)\( \nu^{9} + \)\(18\!\cdots\!05\)\( \nu^{8} - \)\(53\!\cdots\!39\)\( \nu^{7} + \)\(13\!\cdots\!08\)\( \nu^{6} - \)\(18\!\cdots\!36\)\( \nu^{5} + \)\(42\!\cdots\!40\)\( \nu^{4} - \)\(36\!\cdots\!04\)\( \nu^{3} + \)\(11\!\cdots\!44\)\( \nu^{2} - \)\(29\!\cdots\!84\)\( \nu + \)\(37\!\cdots\!28\)\(\)\()/ \)\(25\!\cdots\!92\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(21\!\cdots\!73\)\( \nu^{19} - \)\(21\!\cdots\!45\)\( \nu^{18} + \)\(12\!\cdots\!48\)\( \nu^{17} - \)\(54\!\cdots\!09\)\( \nu^{16} + \)\(49\!\cdots\!88\)\( \nu^{15} - \)\(14\!\cdots\!45\)\( \nu^{14} + \)\(11\!\cdots\!06\)\( \nu^{13} - \)\(63\!\cdots\!99\)\( \nu^{12} + \)\(17\!\cdots\!76\)\( \nu^{11} - \)\(27\!\cdots\!15\)\( \nu^{10} + \)\(18\!\cdots\!64\)\( \nu^{9} + \)\(17\!\cdots\!25\)\( \nu^{8} + \)\(13\!\cdots\!41\)\( \nu^{7} + \)\(69\!\cdots\!64\)\( \nu^{6} + \)\(45\!\cdots\!48\)\( \nu^{5} + \)\(94\!\cdots\!68\)\( \nu^{4} + \)\(10\!\cdots\!96\)\( \nu^{3} + \)\(36\!\cdots\!68\)\( \nu^{2} + \)\(56\!\cdots\!24\)\( \nu + \)\(75\!\cdots\!08\)\(\)\()/ \)\(51\!\cdots\!84\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(45\!\cdots\!29\)\( \nu^{19} + \)\(19\!\cdots\!17\)\( \nu^{18} - \)\(29\!\cdots\!40\)\( \nu^{17} + \)\(98\!\cdots\!81\)\( \nu^{16} - \)\(11\!\cdots\!00\)\( \nu^{15} + \)\(36\!\cdots\!89\)\( \nu^{14} - \)\(25\!\cdots\!78\)\( \nu^{13} + \)\(77\!\cdots\!31\)\( \nu^{12} - \)\(41\!\cdots\!32\)\( \nu^{11} + \)\(12\!\cdots\!07\)\( \nu^{10} - \)\(42\!\cdots\!16\)\( \nu^{9} + \)\(12\!\cdots\!07\)\( \nu^{8} - \)\(30\!\cdots\!05\)\( \nu^{7} + \)\(88\!\cdots\!12\)\( \nu^{6} - \)\(11\!\cdots\!80\)\( \nu^{5} + \)\(29\!\cdots\!16\)\( \nu^{4} - \)\(23\!\cdots\!92\)\( \nu^{3} + \)\(72\!\cdots\!72\)\( \nu^{2} - \)\(17\!\cdots\!24\)\( \nu + \)\(24\!\cdots\!48\)\(\)\()/ \)\(80\!\cdots\!56\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(66\!\cdots\!41\)\( \nu^{19} + \)\(21\!\cdots\!13\)\( \nu^{18} - \)\(41\!\cdots\!20\)\( \nu^{17} + \)\(10\!\cdots\!05\)\( \nu^{16} - \)\(15\!\cdots\!72\)\( \nu^{15} + \)\(38\!\cdots\!25\)\( \nu^{14} - \)\(35\!\cdots\!98\)\( \nu^{13} + \)\(78\!\cdots\!07\)\( \nu^{12} - \)\(56\!\cdots\!24\)\( \nu^{11} + \)\(12\!\cdots\!79\)\( \nu^{10} - \)\(56\!\cdots\!12\)\( \nu^{9} + \)\(12\!\cdots\!15\)\( \nu^{8} - \)\(39\!\cdots\!29\)\( \nu^{7} + \)\(89\!\cdots\!32\)\( \nu^{6} - \)\(13\!\cdots\!52\)\( \nu^{5} + \)\(28\!\cdots\!88\)\( \nu^{4} - \)\(24\!\cdots\!80\)\( \nu^{3} + \)\(75\!\cdots\!32\)\( \nu^{2} + \)\(49\!\cdots\!72\)\( \nu + \)\(25\!\cdots\!56\)\(\)\()/ \)\(85\!\cdots\!64\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(60\!\cdots\!55\)\( \nu^{19} - \)\(60\!\cdots\!55\)\( \nu^{18} + \)\(36\!\cdots\!28\)\( \nu^{17} - \)\(14\!\cdots\!75\)\( \nu^{16} + \)\(14\!\cdots\!80\)\( \nu^{15} - \)\(38\!\cdots\!19\)\( \nu^{14} + \)\(31\!\cdots\!54\)\( \nu^{13} - \)\(12\!\cdots\!53\)\( \nu^{12} + \)\(51\!\cdots\!20\)\( \nu^{11} + \)\(17\!\cdots\!95\)\( \nu^{10} + \)\(52\!\cdots\!36\)\( \nu^{9} + \)\(57\!\cdots\!75\)\( \nu^{8} + \)\(37\!\cdots\!67\)\( \nu^{7} + \)\(25\!\cdots\!04\)\( \nu^{6} + \)\(13\!\cdots\!16\)\( \nu^{5} + \)\(28\!\cdots\!76\)\( \nu^{4} + \)\(30\!\cdots\!60\)\( \nu^{3} + \)\(10\!\cdots\!68\)\( \nu^{2} + \)\(16\!\cdots\!96\)\( \nu + \)\(19\!\cdots\!92\)\(\)\()/ \)\(51\!\cdots\!84\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(79\!\cdots\!09\)\( \nu^{19} - \)\(86\!\cdots\!65\)\( \nu^{18} + \)\(48\!\cdots\!72\)\( \nu^{17} - \)\(24\!\cdots\!89\)\( \nu^{16} + \)\(18\!\cdots\!56\)\( \nu^{15} - \)\(67\!\cdots\!89\)\( \nu^{14} + \)\(42\!\cdots\!98\)\( \nu^{13} - \)\(54\!\cdots\!11\)\( \nu^{12} + \)\(68\!\cdots\!48\)\( \nu^{11} - \)\(59\!\cdots\!63\)\( \nu^{10} + \)\(69\!\cdots\!48\)\( \nu^{9} + \)\(12\!\cdots\!13\)\( \nu^{8} + \)\(49\!\cdots\!77\)\( \nu^{7} - \)\(11\!\cdots\!24\)\( \nu^{6} + \)\(17\!\cdots\!80\)\( \nu^{5} + \)\(21\!\cdots\!32\)\( \nu^{4} + \)\(40\!\cdots\!92\)\( \nu^{3} - \)\(34\!\cdots\!00\)\( \nu^{2} + \)\(13\!\cdots\!44\)\( \nu + \)\(17\!\cdots\!20\)\(\)\()/ \)\(57\!\cdots\!76\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(22\!\cdots\!03\)\( \nu^{19} + \)\(22\!\cdots\!67\)\( \nu^{18} - \)\(13\!\cdots\!24\)\( \nu^{17} + \)\(52\!\cdots\!67\)\( \nu^{16} - \)\(52\!\cdots\!84\)\( \nu^{15} + \)\(13\!\cdots\!47\)\( \nu^{14} - \)\(11\!\cdots\!94\)\( \nu^{13} + \)\(24\!\cdots\!13\)\( \nu^{12} - \)\(19\!\cdots\!72\)\( \nu^{11} - \)\(38\!\cdots\!91\)\( \nu^{10} - \)\(19\!\cdots\!56\)\( \nu^{9} - \)\(24\!\cdots\!67\)\( \nu^{8} - \)\(14\!\cdots\!03\)\( \nu^{7} - \)\(11\!\cdots\!96\)\( \nu^{6} - \)\(48\!\cdots\!56\)\( \nu^{5} - \)\(10\!\cdots\!64\)\( \nu^{4} - \)\(11\!\cdots\!36\)\( \nu^{3} - \)\(40\!\cdots\!72\)\( \nu^{2} - \)\(60\!\cdots\!08\)\( \nu + \)\(16\!\cdots\!80\)\(\)\()/ \)\(57\!\cdots\!76\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(11\!\cdots\!27\)\( \nu^{19} - \)\(10\!\cdots\!43\)\( \nu^{18} + \)\(66\!\cdots\!08\)\( \nu^{17} - \)\(26\!\cdots\!03\)\( \nu^{16} + \)\(26\!\cdots\!24\)\( \nu^{15} - \)\(67\!\cdots\!79\)\( \nu^{14} + \)\(58\!\cdots\!22\)\( \nu^{13} - \)\(17\!\cdots\!41\)\( \nu^{12} + \)\(94\!\cdots\!08\)\( \nu^{11} + \)\(10\!\cdots\!19\)\( \nu^{10} + \)\(96\!\cdots\!88\)\( \nu^{9} + \)\(11\!\cdots\!43\)\( \nu^{8} + \)\(69\!\cdots\!15\)\( \nu^{7} + \)\(51\!\cdots\!00\)\( \nu^{6} + \)\(23\!\cdots\!68\)\( \nu^{5} + \)\(52\!\cdots\!40\)\( \nu^{4} + \)\(55\!\cdots\!28\)\( \nu^{3} + \)\(19\!\cdots\!60\)\( \nu^{2} + \)\(29\!\cdots\!36\)\( \nu + \)\(20\!\cdots\!24\)\(\)\()/ \)\(25\!\cdots\!92\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(30\!\cdots\!17\)\( \nu^{19} + \)\(33\!\cdots\!73\)\( \nu^{18} - \)\(18\!\cdots\!36\)\( \nu^{17} + \)\(93\!\cdots\!53\)\( \nu^{16} - \)\(72\!\cdots\!12\)\( \nu^{15} + \)\(26\!\cdots\!61\)\( \nu^{14} - \)\(16\!\cdots\!98\)\( \nu^{13} + \)\(21\!\cdots\!63\)\( \nu^{12} - \)\(26\!\cdots\!12\)\( \nu^{11} + \)\(24\!\cdots\!15\)\( \nu^{10} - \)\(26\!\cdots\!44\)\( \nu^{9} - \)\(33\!\cdots\!41\)\( \nu^{8} - \)\(19\!\cdots\!01\)\( \nu^{7} + \)\(56\!\cdots\!00\)\( \nu^{6} - \)\(66\!\cdots\!76\)\( \nu^{5} - \)\(77\!\cdots\!12\)\( \nu^{4} - \)\(15\!\cdots\!12\)\( \nu^{3} + \)\(15\!\cdots\!72\)\( \nu^{2} - \)\(53\!\cdots\!16\)\( \nu - \)\(68\!\cdots\!00\)\(\)\()/ \)\(51\!\cdots\!84\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(66\!\cdots\!41\)\( \nu^{19} - \)\(65\!\cdots\!81\)\( \nu^{18} + \)\(39\!\cdots\!80\)\( \nu^{17} - \)\(16\!\cdots\!01\)\( \nu^{16} + \)\(15\!\cdots\!80\)\( \nu^{15} - \)\(40\!\cdots\!81\)\( \nu^{14} + \)\(34\!\cdots\!34\)\( \nu^{13} - \)\(11\!\cdots\!59\)\( \nu^{12} + \)\(56\!\cdots\!28\)\( \nu^{11} + \)\(37\!\cdots\!61\)\( \nu^{10} + \)\(57\!\cdots\!20\)\( \nu^{9} + \)\(64\!\cdots\!97\)\( \nu^{8} + \)\(41\!\cdots\!81\)\( \nu^{7} + \)\(28\!\cdots\!12\)\( \nu^{6} + \)\(14\!\cdots\!60\)\( \nu^{5} + \)\(30\!\cdots\!68\)\( \nu^{4} + \)\(33\!\cdots\!16\)\( \nu^{3} + \)\(11\!\cdots\!08\)\( \nu^{2} + \)\(17\!\cdots\!28\)\( \nu + \)\(18\!\cdots\!52\)\(\)\()/ \)\(53\!\cdots\!04\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 12 \beta_{4} - \beta_{2} - 12\)
\(\nu^{3}\)\(=\)\(\beta_{18} + \beta_{17} + \beta_{11} - \beta_{8} + 19 \beta_{5} - 4\)
\(\nu^{4}\)\(=\)\(-\beta_{13} - \beta_{12} - \beta_{10} - 26 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - 5 \beta_{6} - 234 \beta_{4} + \beta_{3} - 3 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-\beta_{19} - 33 \beta_{18} - 31 \beta_{17} + 2 \beta_{16} + 7 \beta_{15} - 10 \beta_{14} - 2 \beta_{13} + \beta_{12} - 30 \beta_{11} + 30 \beta_{10} - 8 \beta_{9} + 31 \beta_{7} - 10 \beta_{6} - 411 \beta_{5} - 175 \beta_{4} + 8 \beta_{2} - 411 \beta_{1} + 175\)
\(\nu^{6}\)\(=\)\(39 \beta_{19} - 85 \beta_{18} - 114 \beta_{17} + 35 \beta_{16} - 29 \beta_{15} - 261 \beta_{14} + 32 \beta_{11} + 85 \beta_{8} - 165 \beta_{5} - 29 \beta_{3} + 658 \beta_{2} + 5220\)
\(\nu^{7}\)\(=\)\(89 \beta_{13} - 26 \beta_{12} - 797 \beta_{10} + 423 \beta_{9} + 941 \beta_{8} - 858 \beta_{7} + 573 \beta_{6} + 6161 \beta_{4} + 328 \beta_{3} + 9604 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-1214 \beta_{19} + 2809 \beta_{18} + 3510 \beta_{17} - 1071 \beta_{16} + 602 \beta_{15} + 9313 \beta_{14} + 1071 \beta_{13} + 1214 \beta_{12} - 731 \beta_{11} + 731 \beta_{10} + 16880 \beta_{9} - 3510 \beta_{7} + 9313 \beta_{6} + 6767 \beta_{5} + 125541 \beta_{4} - 16880 \beta_{2} + 6767 \beta_{1} - 125541\)
\(\nu^{9}\)\(=\)\(262 \beta_{19} + 25994 \beta_{18} + 23623 \beta_{17} - 3051 \beta_{16} - 11118 \beta_{15} + 23121 \beta_{14} + 20842 \beta_{11} - 25994 \beta_{8} + 236018 \beta_{5} - 11118 \beta_{3} - 16559 \beta_{2} - 200385\)
\(\nu^{10}\)\(=\)\(-31428 \beta_{13} - 35011 \beta_{12} - 13116 \beta_{10} - 439433 \beta_{9} - 85759 \beta_{8} + 102117 \beta_{7} - 291070 \beta_{6} - 3164039 \beta_{4} + 9355 \beta_{3} - 245466 \beta_{1}\)
\(\nu^{11}\)\(=\)\(8957 \beta_{19} - 713246 \beta_{18} - 653525 \beta_{17} + 96345 \beta_{16} + 334769 \beta_{15} - 808103 \beta_{14} - 96345 \beta_{13} - 8957 \beta_{12} - 545087 \beta_{11} + 545087 \beta_{10} - 580083 \beta_{9} + 653525 \beta_{7} - 808103 \beta_{6} - 5997894 \beta_{5} - 6271036 \beta_{4} + 580083 \beta_{2} - 5997894 \beta_{1} + 6271036\)
\(\nu^{12}\)\(=\)\(976201 \beta_{19} - 2537170 \beta_{18} - 2919128 \beta_{17} + 900906 \beta_{16} - 56975 \beta_{15} - 8580962 \beta_{14} + 148099 \beta_{11} + 2537170 \beta_{8} - 8322224 \beta_{5} - 56975 \beta_{3} + 11575149 \beta_{2} + 82182493\)
\(\nu^{13}\)\(=\)\(2937448 \beta_{13} + 695832 \beta_{12} - 14300942 \beta_{10} + 19184426 \beta_{9} + 19584446 \beta_{8} - 18152612 \beta_{7} + 26213562 \beta_{6} + 191920346 \beta_{4} + 9548318 \beta_{3} + 155903951 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-26786708 \beta_{19} + 73994922 \beta_{18} + 83012388 \beta_{17} - 25459314 \beta_{16} - 3537576 \beta_{15} + 245837586 \beta_{14} + 25459314 \beta_{13} + 26786708 \beta_{12} + 1647306 \beta_{11} - 1647306 \beta_{10} + 307772079 \beta_{9} - 83012388 \beta_{7} + 245837586 \beta_{6} + 270312142 \beta_{5} + 2177138374 \beta_{4} - 307772079 \beta_{2} + 270312142 \beta_{1} - 2177138374\)
\(\nu^{15}\)\(=\)\(-30644196 \beta_{19} + 539337187 \beta_{18} + 505673069 \beta_{17} - 87877358 \beta_{16} - 265107060 \beta_{15} + 814166530 \beta_{14} + 376728163 \beta_{11} - 539337187 \beta_{8} + 4116217857 \beta_{5} - 265107060 \beta_{3} - 611970494 \beta_{2} - 5786632798\)
\(\nu^{16}\)\(=\)\(-713546967 \beta_{13} - 729712581 \beta_{12} + 192190169 \beta_{10} - 8246507944 \beta_{9} - 2141929996 \beta_{8} + 2357594071 \beta_{7} - 6940850999 \beta_{6} - 58448689252 \beta_{4} - 223196347 \beta_{3} - 8522961601 \beta_{1}\)
\(\nu^{17}\)\(=\)\(1128933483 \beta_{19} - 14902802571 \beta_{18} - 14117320811 \beta_{17} + 2596956032 \beta_{16} + 7262745291 \beta_{15} - 24610174900 \beta_{14} - 2596956032 \beta_{13} - 1128933483 \beta_{12} - 9968804928 \beta_{11} + 9968804928 \beta_{10} - 19035251646 \beta_{9} + 14117320811 \beta_{7} - 24610174900 \beta_{6} - 109907746577 \beta_{5} - 172553617129 \beta_{4} + 19035251646 \beta_{2} - 109907746577 \beta_{1} + 172553617129\)
\(\nu^{18}\)\(=\)\(19828506251 \beta_{19} - 61720140511 \beta_{18} - 66939618878 \beta_{17} + 19912043553 \beta_{16} + 9213865279 \beta_{15} - 194516947503 \beta_{14} - 8991634730 \beta_{11} + 61720140511 \beta_{8} - 262941343995 \beta_{5} + 9213865279 \beta_{3} + 222397949320 \beta_{2} + 1583906721350\)
\(\nu^{19}\)\(=\)\(76069946855 \beta_{13} + 38117543562 \beta_{12} - 265039818431 \beta_{10} + 581108305805 \beta_{9} + 413083617071 \beta_{8} - 394821026782 \beta_{7} + 730782979719 \beta_{6} + 5100570582163 \beta_{4} + 197761025652 \beta_{3} + 2959713591578 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
2.65668 4.60150i
2.11736 3.66738i
1.77920 3.08166i
1.07390 1.86006i
0.0954556 0.165334i
−0.0898836 + 0.155683i
−0.961392 + 1.66518i
−1.79682 + 3.11219i
−1.91170 + 3.31117i
−2.46280 + 4.26569i
2.65668 + 4.60150i
2.11736 + 3.66738i
1.77920 + 3.08166i
1.07390 + 1.86006i
0.0954556 + 0.165334i
−0.0898836 0.155683i
−0.961392 1.66518i
−1.79682 3.11219i
−1.91170 3.31117i
−2.46280 4.26569i
−5.31336 −4.02821 + 6.97706i 20.2317 −7.95967 + 13.7866i 21.4033 37.0716i −8.31614 14.4040i −64.9916 −18.9529 32.8274i 42.2926 73.2529i
6.2 −4.23473 4.42929 7.67176i 9.93292 −2.98244 + 5.16574i −18.7568 + 32.4878i −11.2528 19.4903i −8.18537 −25.7372 44.5782i 12.6298 21.8755i
6.3 −3.55840 0.194544 0.336960i 4.66219 0.863899 1.49632i −0.692264 + 1.19904i 10.5931 + 18.3477i 11.8773 13.4243 + 23.2516i −3.07410 + 5.32449i
6.4 −2.14781 −3.42441 + 5.93126i −3.38692 8.56817 14.8405i 7.35498 12.7392i −11.4893 19.9001i 24.4569 −9.95322 17.2395i −18.4028 + 31.8746i
6.5 −0.190911 −0.719182 + 1.24566i −7.96355 −8.17312 + 14.1563i 0.137300 0.237810i −3.11997 5.40394i 3.04762 12.4656 + 21.5910i 1.56034 2.70259i
6.6 0.179767 3.01124 5.21562i −7.96768 5.48517 9.50060i 0.541322 0.937597i −4.39183 7.60688i −2.87047 −4.63513 8.02828i 0.986054 1.70790i
6.7 1.92278 −4.18916 + 7.25584i −4.30290 −0.0351129 + 0.0608173i −8.05486 + 13.9514i 11.7213 + 20.3018i −23.6558 −21.5982 37.4091i −0.0675145 + 0.116939i
6.8 3.59365 0.838961 1.45312i 4.91431 4.86091 8.41934i 3.01493 5.22201i 7.88013 + 13.6488i −11.0889 12.0923 + 20.9445i 17.4684 30.2561i
6.9 3.82341 3.88258 6.72483i 6.61843 −9.06615 + 15.7030i 14.8447 25.7118i −2.59898 4.50156i −5.28231 −16.6489 28.8368i −34.6635 + 60.0390i
6.10 4.92559 −2.49565 + 4.32260i 16.2615 −1.06165 + 1.83883i −12.2926 + 21.2914i −14.5255 25.1588i 40.6926 1.04342 + 1.80726i −5.22925 + 9.05732i
36.1 −5.31336 −4.02821 6.97706i 20.2317 −7.95967 13.7866i 21.4033 + 37.0716i −8.31614 + 14.4040i −64.9916 −18.9529 + 32.8274i 42.2926 + 73.2529i
36.2 −4.23473 4.42929 + 7.67176i 9.93292 −2.98244 5.16574i −18.7568 32.4878i −11.2528 + 19.4903i −8.18537 −25.7372 + 44.5782i 12.6298 + 21.8755i
36.3 −3.55840 0.194544 + 0.336960i 4.66219 0.863899 + 1.49632i −0.692264 1.19904i 10.5931 18.3477i 11.8773 13.4243 23.2516i −3.07410 5.32449i
36.4 −2.14781 −3.42441 5.93126i −3.38692 8.56817 + 14.8405i 7.35498 + 12.7392i −11.4893 + 19.9001i 24.4569 −9.95322 + 17.2395i −18.4028 31.8746i
36.5 −0.190911 −0.719182 1.24566i −7.96355 −8.17312 14.1563i 0.137300 + 0.237810i −3.11997 + 5.40394i 3.04762 12.4656 21.5910i 1.56034 + 2.70259i
36.6 0.179767 3.01124 + 5.21562i −7.96768 5.48517 + 9.50060i 0.541322 + 0.937597i −4.39183 + 7.60688i −2.87047 −4.63513 + 8.02828i 0.986054 + 1.70790i
36.7 1.92278 −4.18916 7.25584i −4.30290 −0.0351129 0.0608173i −8.05486 13.9514i 11.7213 20.3018i −23.6558 −21.5982 + 37.4091i −0.0675145 0.116939i
36.8 3.59365 0.838961 + 1.45312i 4.91431 4.86091 + 8.41934i 3.01493 + 5.22201i 7.88013 13.6488i −11.0889 12.0923 20.9445i 17.4684 + 30.2561i
36.9 3.82341 3.88258 + 6.72483i 6.61843 −9.06615 15.7030i 14.8447 + 25.7118i −2.59898 + 4.50156i −5.28231 −16.6489 + 28.8368i −34.6635 60.0390i
36.10 4.92559 −2.49565 4.32260i 16.2615 −1.06165 1.83883i −12.2926 21.2914i −14.5255 + 25.1588i 40.6926 1.04342 1.80726i −5.22925 9.05732i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.4.c.a 20
43.c even 3 1 inner 43.4.c.a 20
43.c even 3 1 1849.4.a.d 10
43.d odd 6 1 1849.4.a.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.c.a 20 1.a even 1 1 trivial
43.4.c.a 20 43.c even 3 1 inner
1849.4.a.d 10 43.c even 3 1
1849.4.a.f 10 43.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(43, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + 21 T^{2} + 30 T^{3} + 291 T^{4} + 493 T^{5} + 3299 T^{6} + 6020 T^{7} + 30260 T^{8} + 57696 T^{9} + 252608 T^{10} + 461568 T^{11} + 1936640 T^{12} + 3082240 T^{13} + 13512704 T^{14} + 16154624 T^{15} + 76283904 T^{16} + 62914560 T^{17} + 352321536 T^{18} + 134217728 T^{19} + 1073741824 T^{20} )^{2} \)
$3$ \( 1 + 5 T - 64 T^{2} - 207 T^{3} + 1736 T^{4} - 2493 T^{5} + 14940 T^{6} + 502453 T^{7} - 2029707 T^{8} - 12348180 T^{9} + 49889273 T^{10} - 155791671 T^{11} + 111657687 T^{12} + 13413963364 T^{13} - 31402447776 T^{14} - 104284718676 T^{15} + 597172338392 T^{16} - 7718563954902 T^{17} + 6795103553147 T^{18} + 149929321126415 T^{19} - 522136902551354 T^{20} + 4048091670413205 T^{21} + 4953630490244163 T^{22} - 151924494324336066 T^{23} + 317361864687382872 T^{24} - 1496371729803087132 T^{25} - 12165951673174882464 T^{26} + \)\(14\!\cdots\!92\)\( T^{27} + 31535428783950579447 T^{28} - \)\(11\!\cdots\!77\)\( T^{29} + \)\(10\!\cdots\!77\)\( T^{30} - \)\(68\!\cdots\!40\)\( T^{31} - \)\(30\!\cdots\!47\)\( T^{32} + \)\(20\!\cdots\!51\)\( T^{33} + \)\(16\!\cdots\!60\)\( T^{34} - \)\(73\!\cdots\!99\)\( T^{35} + \)\(13\!\cdots\!96\)\( T^{36} - \)\(44\!\cdots\!29\)\( T^{37} - \)\(37\!\cdots\!16\)\( T^{38} + \)\(78\!\cdots\!15\)\( T^{39} + \)\(42\!\cdots\!01\)\( T^{40} \)
$5$ \( 1 + 19 T - 369 T^{2} - 9420 T^{3} + 46555 T^{4} + 2017871 T^{5} + 1496842 T^{6} - 191282201 T^{7} - 899992568 T^{8} - 8445628937 T^{9} - 69732522494 T^{10} + 4381874197151 T^{11} + 59385793792057 T^{12} - 382392294295462 T^{13} - 10614686309241807 T^{14} - 29318878428412277 T^{15} + 668588555926177131 T^{16} + 9697248609861628750 T^{17} + 75906644519801998580 T^{18} - \)\(63\!\cdots\!54\)\( T^{19} - \)\(19\!\cdots\!04\)\( T^{20} - \)\(79\!\cdots\!50\)\( T^{21} + \)\(11\!\cdots\!00\)\( T^{22} + \)\(18\!\cdots\!50\)\( T^{23} + \)\(16\!\cdots\!75\)\( T^{24} - \)\(89\!\cdots\!25\)\( T^{25} - \)\(40\!\cdots\!75\)\( T^{26} - \)\(18\!\cdots\!50\)\( T^{27} + \)\(35\!\cdots\!25\)\( T^{28} + \)\(32\!\cdots\!75\)\( T^{29} - \)\(64\!\cdots\!50\)\( T^{30} - \)\(98\!\cdots\!25\)\( T^{31} - \)\(13\!\cdots\!00\)\( T^{32} - \)\(34\!\cdots\!25\)\( T^{33} + \)\(34\!\cdots\!50\)\( T^{34} + \)\(57\!\cdots\!75\)\( T^{35} + \)\(16\!\cdots\!75\)\( T^{36} - \)\(41\!\cdots\!00\)\( T^{37} - \)\(20\!\cdots\!25\)\( T^{38} + \)\(13\!\cdots\!75\)\( T^{39} + \)\(86\!\cdots\!25\)\( T^{40} \)
$7$ \( 1 + 51 T - 357 T^{2} - 45708 T^{3} + 199158 T^{4} + 23478106 T^{5} - 285493259 T^{6} - 11658162295 T^{7} + 146829104946 T^{8} + 4094897966023 T^{9} - 71194687182651 T^{10} - 1461672246803082 T^{11} + 35394258687371444 T^{12} + 643809247546209966 T^{13} - 13649568920422493291 T^{14} - \)\(16\!\cdots\!91\)\( T^{15} + \)\(61\!\cdots\!77\)\( T^{16} + \)\(32\!\cdots\!32\)\( T^{17} - \)\(23\!\cdots\!26\)\( T^{18} - \)\(58\!\cdots\!98\)\( T^{19} + \)\(71\!\cdots\!44\)\( T^{20} - \)\(20\!\cdots\!14\)\( T^{21} - \)\(27\!\cdots\!74\)\( T^{22} + \)\(13\!\cdots\!24\)\( T^{23} + \)\(84\!\cdots\!77\)\( T^{24} - \)\(77\!\cdots\!13\)\( T^{25} - \)\(22\!\cdots\!59\)\( T^{26} + \)\(35\!\cdots\!62\)\( T^{27} + \)\(67\!\cdots\!44\)\( T^{28} - \)\(96\!\cdots\!26\)\( T^{29} - \)\(16\!\cdots\!99\)\( T^{30} + \)\(31\!\cdots\!61\)\( T^{31} + \)\(38\!\cdots\!46\)\( T^{32} - \)\(10\!\cdots\!85\)\( T^{33} - \)\(89\!\cdots\!91\)\( T^{34} + \)\(25\!\cdots\!42\)\( T^{35} + \)\(73\!\cdots\!58\)\( T^{36} - \)\(57\!\cdots\!44\)\( T^{37} - \)\(15\!\cdots\!93\)\( T^{38} + \)\(75\!\cdots\!57\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$11$ \( ( 1 - 27 T + 5077 T^{2} - 61984 T^{3} + 15043654 T^{4} - 162363748 T^{5} + 35128969454 T^{6} - 264446154846 T^{7} + 61397630693497 T^{8} - 405185649214743 T^{9} + 91897922249805306 T^{10} - 539302099104822933 T^{11} + \)\(10\!\cdots\!17\)\( T^{12} - \)\(62\!\cdots\!86\)\( T^{13} + \)\(11\!\cdots\!34\)\( T^{14} - \)\(67\!\cdots\!48\)\( T^{15} + \)\(83\!\cdots\!74\)\( T^{16} - \)\(45\!\cdots\!24\)\( T^{17} + \)\(50\!\cdots\!57\)\( T^{18} - \)\(35\!\cdots\!17\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} )^{2} \)
$13$ \( 1 + 15 T - 15836 T^{2} - 84609 T^{3} + 136428704 T^{4} - 211873241 T^{5} - 798195901414 T^{6} + 5549492881873 T^{7} + 3492359466060033 T^{8} - 39363659442734576 T^{9} - 11971166560261837535 T^{10} + \)\(17\!\cdots\!97\)\( T^{11} + \)\(33\!\cdots\!29\)\( T^{12} - \)\(59\!\cdots\!52\)\( T^{13} - \)\(74\!\cdots\!72\)\( T^{14} + \)\(14\!\cdots\!74\)\( T^{15} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(26\!\cdots\!76\)\( T^{17} - \)\(23\!\cdots\!67\)\( T^{18} + \)\(22\!\cdots\!07\)\( T^{19} + \)\(44\!\cdots\!78\)\( T^{20} + \)\(48\!\cdots\!79\)\( T^{21} - \)\(11\!\cdots\!03\)\( T^{22} - \)\(27\!\cdots\!48\)\( T^{23} + \)\(32\!\cdots\!50\)\( T^{24} + \)\(75\!\cdots\!18\)\( T^{25} - \)\(83\!\cdots\!88\)\( T^{26} - \)\(14\!\cdots\!76\)\( T^{27} + \)\(17\!\cdots\!69\)\( T^{28} + \)\(21\!\cdots\!49\)\( T^{29} - \)\(31\!\cdots\!15\)\( T^{30} - \)\(22\!\cdots\!28\)\( T^{31} + \)\(44\!\cdots\!53\)\( T^{32} + \)\(15\!\cdots\!21\)\( T^{33} - \)\(48\!\cdots\!66\)\( T^{34} - \)\(28\!\cdots\!13\)\( T^{35} + \)\(40\!\cdots\!84\)\( T^{36} - \)\(54\!\cdots\!33\)\( T^{37} - \)\(22\!\cdots\!04\)\( T^{38} + \)\(46\!\cdots\!95\)\( T^{39} + \)\(68\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 + 82 T - 24818 T^{2} - 2873008 T^{3} + 275626616 T^{4} + 48442716340 T^{5} - 1590978438438 T^{6} - 545228525674854 T^{7} + 42616778692923 T^{8} + 4700974927774535486 T^{9} + \)\(11\!\cdots\!15\)\( T^{10} - \)\(32\!\cdots\!40\)\( T^{11} - \)\(15\!\cdots\!83\)\( T^{12} + \)\(18\!\cdots\!16\)\( T^{13} + \)\(13\!\cdots\!50\)\( T^{14} - \)\(88\!\cdots\!82\)\( T^{15} - \)\(99\!\cdots\!98\)\( T^{16} + \)\(31\!\cdots\!96\)\( T^{17} + \)\(60\!\cdots\!95\)\( T^{18} - \)\(56\!\cdots\!84\)\( T^{19} - \)\(31\!\cdots\!82\)\( T^{20} - \)\(27\!\cdots\!92\)\( T^{21} + \)\(14\!\cdots\!55\)\( T^{22} + \)\(37\!\cdots\!12\)\( T^{23} - \)\(57\!\cdots\!78\)\( T^{24} - \)\(25\!\cdots\!26\)\( T^{25} + \)\(19\!\cdots\!50\)\( T^{26} + \)\(13\!\cdots\!72\)\( T^{27} - \)\(52\!\cdots\!43\)\( T^{28} - \)\(54\!\cdots\!20\)\( T^{29} + \)\(92\!\cdots\!35\)\( T^{30} + \)\(18\!\cdots\!82\)\( T^{31} + \)\(84\!\cdots\!63\)\( T^{32} - \)\(52\!\cdots\!62\)\( T^{33} - \)\(75\!\cdots\!82\)\( T^{34} + \)\(11\!\cdots\!80\)\( T^{35} + \)\(31\!\cdots\!56\)\( T^{36} - \)\(16\!\cdots\!64\)\( T^{37} - \)\(69\!\cdots\!22\)\( T^{38} + \)\(11\!\cdots\!14\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 - 78 T - 30307 T^{2} + 1643314 T^{3} + 557392853 T^{4} - 19116976304 T^{5} - 6385440198566 T^{6} + 80081143266136 T^{7} + 48128038162948702 T^{8} + 395871623097331272 T^{9} - \)\(15\!\cdots\!30\)\( T^{10} - \)\(76\!\cdots\!40\)\( T^{11} - \)\(95\!\cdots\!37\)\( T^{12} + \)\(90\!\cdots\!74\)\( T^{13} + \)\(17\!\cdots\!19\)\( T^{14} + \)\(54\!\cdots\!06\)\( T^{15} - \)\(11\!\cdots\!65\)\( T^{16} - \)\(60\!\cdots\!84\)\( T^{17} + \)\(38\!\cdots\!88\)\( T^{18} + \)\(19\!\cdots\!44\)\( T^{19} - \)\(94\!\cdots\!24\)\( T^{20} + \)\(13\!\cdots\!96\)\( T^{21} + \)\(18\!\cdots\!28\)\( T^{22} - \)\(19\!\cdots\!36\)\( T^{23} - \)\(25\!\cdots\!65\)\( T^{24} + \)\(82\!\cdots\!94\)\( T^{25} + \)\(18\!\cdots\!79\)\( T^{26} + \)\(64\!\cdots\!06\)\( T^{27} - \)\(46\!\cdots\!77\)\( T^{28} - \)\(25\!\cdots\!60\)\( T^{29} - \)\(36\!\cdots\!30\)\( T^{30} + \)\(62\!\cdots\!48\)\( T^{31} + \)\(52\!\cdots\!62\)\( T^{32} + \)\(59\!\cdots\!44\)\( T^{33} - \)\(32\!\cdots\!26\)\( T^{34} - \)\(66\!\cdots\!96\)\( T^{35} + \)\(13\!\cdots\!73\)\( T^{36} + \)\(27\!\cdots\!66\)\( T^{37} - \)\(34\!\cdots\!47\)\( T^{38} - \)\(60\!\cdots\!42\)\( T^{39} + \)\(53\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 + 61 T - 81824 T^{2} - 6253131 T^{3} + 3389445188 T^{4} + 298260670973 T^{5} - 95613835472806 T^{6} - 9095183914407171 T^{7} + 2101707189884822683 T^{8} + \)\(20\!\cdots\!68\)\( T^{9} - \)\(39\!\cdots\!71\)\( T^{10} - \)\(34\!\cdots\!69\)\( T^{11} + \)\(65\!\cdots\!49\)\( T^{12} + \)\(47\!\cdots\!72\)\( T^{13} - \)\(10\!\cdots\!88\)\( T^{14} - \)\(51\!\cdots\!20\)\( T^{15} + \)\(15\!\cdots\!62\)\( T^{16} + \)\(43\!\cdots\!14\)\( T^{17} - \)\(21\!\cdots\!19\)\( T^{18} - \)\(18\!\cdots\!05\)\( T^{19} + \)\(27\!\cdots\!22\)\( T^{20} - \)\(22\!\cdots\!35\)\( T^{21} - \)\(31\!\cdots\!91\)\( T^{22} + \)\(77\!\cdots\!82\)\( T^{23} + \)\(33\!\cdots\!02\)\( T^{24} - \)\(13\!\cdots\!40\)\( T^{25} - \)\(33\!\cdots\!72\)\( T^{26} + \)\(18\!\cdots\!56\)\( T^{27} + \)\(31\!\cdots\!09\)\( T^{28} - \)\(20\!\cdots\!43\)\( T^{29} - \)\(27\!\cdots\!79\)\( T^{30} + \)\(17\!\cdots\!44\)\( T^{31} + \)\(22\!\cdots\!63\)\( T^{32} - \)\(11\!\cdots\!77\)\( T^{33} - \)\(14\!\cdots\!74\)\( T^{34} + \)\(56\!\cdots\!39\)\( T^{35} + \)\(78\!\cdots\!28\)\( T^{36} - \)\(17\!\cdots\!37\)\( T^{37} - \)\(27\!\cdots\!16\)\( T^{38} + \)\(25\!\cdots\!83\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 + 53 T - 138329 T^{2} - 8566904 T^{3} + 9770438424 T^{4} + 711713671522 T^{5} - 454180068094483 T^{6} - 40428004977832863 T^{7} + 15194742238715729146 T^{8} + \)\(18\!\cdots\!33\)\( T^{9} - \)\(37\!\cdots\!31\)\( T^{10} - \)\(67\!\cdots\!46\)\( T^{11} + \)\(61\!\cdots\!62\)\( T^{12} + \)\(21\!\cdots\!70\)\( T^{13} - \)\(27\!\cdots\!19\)\( T^{14} - \)\(55\!\cdots\!63\)\( T^{15} - \)\(25\!\cdots\!79\)\( T^{16} + \)\(10\!\cdots\!00\)\( T^{17} + \)\(12\!\cdots\!66\)\( T^{18} - \)\(10\!\cdots\!10\)\( T^{19} - \)\(34\!\cdots\!60\)\( T^{20} - \)\(24\!\cdots\!90\)\( T^{21} + \)\(72\!\cdots\!86\)\( T^{22} + \)\(15\!\cdots\!00\)\( T^{23} - \)\(89\!\cdots\!39\)\( T^{24} - \)\(47\!\cdots\!87\)\( T^{25} - \)\(57\!\cdots\!59\)\( T^{26} + \)\(10\!\cdots\!30\)\( T^{27} + \)\(77\!\cdots\!22\)\( T^{28} - \)\(20\!\cdots\!14\)\( T^{29} - \)\(27\!\cdots\!31\)\( T^{30} + \)\(32\!\cdots\!37\)\( T^{31} + \)\(67\!\cdots\!66\)\( T^{32} - \)\(43\!\cdots\!47\)\( T^{33} - \)\(11\!\cdots\!03\)\( T^{34} + \)\(45\!\cdots\!78\)\( T^{35} + \)\(15\!\cdots\!64\)\( T^{36} - \)\(32\!\cdots\!16\)\( T^{37} - \)\(12\!\cdots\!49\)\( T^{38} + \)\(12\!\cdots\!77\)\( T^{39} + \)\(55\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 - 253 T - 41875 T^{2} + 7234722 T^{3} + 561602045 T^{4} + 650618176597 T^{5} - 97721636388434 T^{6} - 29167394163008097 T^{7} + 4500921270624132994 T^{8} + \)\(12\!\cdots\!49\)\( T^{9} + \)\(95\!\cdots\!38\)\( T^{10} - \)\(12\!\cdots\!41\)\( T^{11} - \)\(81\!\cdots\!49\)\( T^{12} + \)\(10\!\cdots\!88\)\( T^{13} + \)\(81\!\cdots\!75\)\( T^{14} - \)\(39\!\cdots\!15\)\( T^{15} + \)\(17\!\cdots\!87\)\( T^{16} - \)\(94\!\cdots\!70\)\( T^{17} + \)\(34\!\cdots\!84\)\( T^{18} + \)\(16\!\cdots\!10\)\( T^{19} - \)\(33\!\cdots\!96\)\( T^{20} + \)\(49\!\cdots\!10\)\( T^{21} + \)\(31\!\cdots\!04\)\( T^{22} - \)\(25\!\cdots\!70\)\( T^{23} + \)\(13\!\cdots\!07\)\( T^{24} - \)\(93\!\cdots\!65\)\( T^{25} + \)\(56\!\cdots\!75\)\( T^{26} + \)\(22\!\cdots\!28\)\( T^{27} - \)\(50\!\cdots\!29\)\( T^{28} - \)\(23\!\cdots\!51\)\( T^{29} + \)\(52\!\cdots\!38\)\( T^{30} + \)\(19\!\cdots\!59\)\( T^{31} + \)\(21\!\cdots\!14\)\( T^{32} - \)\(42\!\cdots\!87\)\( T^{33} - \)\(42\!\cdots\!74\)\( T^{34} + \)\(84\!\cdots\!47\)\( T^{35} + \)\(21\!\cdots\!45\)\( T^{36} + \)\(82\!\cdots\!82\)\( T^{37} - \)\(14\!\cdots\!75\)\( T^{38} - \)\(25\!\cdots\!83\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 129 T - 268978 T^{2} - 43370241 T^{3} + 34974292754 T^{4} + 6620200735309 T^{5} - 2909843471321326 T^{6} - 614683056380244321 T^{7} + \)\(17\!\cdots\!39\)\( T^{8} + \)\(39\!\cdots\!18\)\( T^{9} - \)\(91\!\cdots\!59\)\( T^{10} - \)\(18\!\cdots\!75\)\( T^{11} + \)\(44\!\cdots\!41\)\( T^{12} + \)\(78\!\cdots\!22\)\( T^{13} - \)\(19\!\cdots\!38\)\( T^{14} - \)\(35\!\cdots\!54\)\( T^{15} + \)\(53\!\cdots\!04\)\( T^{16} + \)\(14\!\cdots\!10\)\( T^{17} + \)\(31\!\cdots\!45\)\( T^{18} - \)\(31\!\cdots\!85\)\( T^{19} - \)\(90\!\cdots\!42\)\( T^{20} - \)\(15\!\cdots\!05\)\( T^{21} + \)\(81\!\cdots\!05\)\( T^{22} + \)\(19\!\cdots\!70\)\( T^{23} + \)\(35\!\cdots\!24\)\( T^{24} - \)\(11\!\cdots\!22\)\( T^{25} - \)\(32\!\cdots\!02\)\( T^{26} + \)\(67\!\cdots\!14\)\( T^{27} + \)\(19\!\cdots\!01\)\( T^{28} - \)\(40\!\cdots\!75\)\( T^{29} - \)\(10\!\cdots\!91\)\( T^{30} + \)\(21\!\cdots\!46\)\( T^{31} + \)\(50\!\cdots\!99\)\( T^{32} - \)\(88\!\cdots\!33\)\( T^{33} - \)\(21\!\cdots\!94\)\( T^{34} + \)\(24\!\cdots\!13\)\( T^{35} + \)\(65\!\cdots\!34\)\( T^{36} - \)\(41\!\cdots\!33\)\( T^{37} - \)\(12\!\cdots\!42\)\( T^{38} + \)\(31\!\cdots\!93\)\( T^{39} + \)\(12\!\cdots\!01\)\( T^{40} \)
$41$ \( ( 1 - 391 T + 466858 T^{2} - 159315937 T^{3} + 107799629054 T^{4} - 32107569594125 T^{5} + 16028988094146372 T^{6} - 4181844238111783617 T^{7} + \)\(16\!\cdots\!81\)\( T^{8} - \)\(38\!\cdots\!70\)\( T^{9} + \)\(13\!\cdots\!96\)\( T^{10} - \)\(26\!\cdots\!70\)\( T^{11} + \)\(80\!\cdots\!21\)\( T^{12} - \)\(13\!\cdots\!37\)\( T^{13} + \)\(36\!\cdots\!32\)\( T^{14} - \)\(49\!\cdots\!25\)\( T^{15} + \)\(11\!\cdots\!34\)\( T^{16} - \)\(11\!\cdots\!17\)\( T^{17} + \)\(23\!\cdots\!38\)\( T^{18} - \)\(13\!\cdots\!71\)\( T^{19} + \)\(24\!\cdots\!01\)\( T^{20} )^{2} \)
$43$ \( 1 - 1025 T + 585952 T^{2} - 190558261 T^{3} + 20265475934 T^{4} + 14357877818201 T^{5} - 8972906466293482 T^{6} + 2452051846789185023 T^{7} - \)\(15\!\cdots\!87\)\( T^{8} - \)\(14\!\cdots\!18\)\( T^{9} + \)\(66\!\cdots\!40\)\( T^{10} - \)\(11\!\cdots\!26\)\( T^{11} - \)\(10\!\cdots\!63\)\( T^{12} + \)\(12\!\cdots\!89\)\( T^{13} - \)\(35\!\cdots\!82\)\( T^{14} + \)\(45\!\cdots\!07\)\( T^{15} + \)\(51\!\cdots\!66\)\( T^{16} - \)\(38\!\cdots\!23\)\( T^{17} + \)\(93\!\cdots\!52\)\( T^{18} - \)\(13\!\cdots\!75\)\( T^{19} + \)\(10\!\cdots\!49\)\( T^{20} \)
$47$ \( ( 1 + 334 T + 469033 T^{2} + 125686554 T^{3} + 97290890628 T^{4} + 20127660597266 T^{5} + 12996286592743642 T^{6} + 2230581953572455798 T^{7} + \)\(14\!\cdots\!19\)\( T^{8} + \)\(24\!\cdots\!68\)\( T^{9} + \)\(16\!\cdots\!34\)\( T^{10} + \)\(25\!\cdots\!64\)\( T^{11} + \)\(15\!\cdots\!51\)\( T^{12} + \)\(24\!\cdots\!66\)\( T^{13} + \)\(15\!\cdots\!22\)\( T^{14} + \)\(24\!\cdots\!38\)\( T^{15} + \)\(12\!\cdots\!92\)\( T^{16} + \)\(16\!\cdots\!38\)\( T^{17} + \)\(63\!\cdots\!73\)\( T^{18} + \)\(46\!\cdots\!42\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \)
$53$ \( 1 - 773 T - 541704 T^{2} + 559887799 T^{3} + 134971478324 T^{4} - 208241369369157 T^{5} - 17263563843724798 T^{6} + 53174144863295438605 T^{7} - \)\(60\!\cdots\!27\)\( T^{8} - \)\(10\!\cdots\!04\)\( T^{9} + \)\(82\!\cdots\!89\)\( T^{10} + \)\(14\!\cdots\!17\)\( T^{11} - \)\(18\!\cdots\!03\)\( T^{12} - \)\(14\!\cdots\!72\)\( T^{13} + \)\(16\!\cdots\!32\)\( T^{14} + \)\(81\!\cdots\!34\)\( T^{15} + \)\(21\!\cdots\!18\)\( T^{16} + \)\(38\!\cdots\!40\)\( T^{17} - \)\(10\!\cdots\!15\)\( T^{18} - \)\(28\!\cdots\!09\)\( T^{19} + \)\(21\!\cdots\!62\)\( T^{20} - \)\(42\!\cdots\!93\)\( T^{21} - \)\(23\!\cdots\!35\)\( T^{22} + \)\(12\!\cdots\!20\)\( T^{23} + \)\(10\!\cdots\!38\)\( T^{24} + \)\(59\!\cdots\!38\)\( T^{25} + \)\(17\!\cdots\!48\)\( T^{26} - \)\(22\!\cdots\!16\)\( T^{27} - \)\(44\!\cdots\!43\)\( T^{28} + \)\(51\!\cdots\!29\)\( T^{29} + \)\(44\!\cdots\!61\)\( T^{30} - \)\(80\!\cdots\!92\)\( T^{31} - \)\(71\!\cdots\!67\)\( T^{32} + \)\(93\!\cdots\!85\)\( T^{33} - \)\(45\!\cdots\!82\)\( T^{34} - \)\(81\!\cdots\!01\)\( T^{35} + \)\(78\!\cdots\!64\)\( T^{36} + \)\(48\!\cdots\!03\)\( T^{37} - \)\(69\!\cdots\!76\)\( T^{38} - \)\(14\!\cdots\!49\)\( T^{39} + \)\(28\!\cdots\!01\)\( T^{40} \)
$59$ \( ( 1 + 1483 T + 1378157 T^{2} + 969130766 T^{3} + 686264212828 T^{4} + 451249739385536 T^{5} + 271319879713111662 T^{6} + \)\(14\!\cdots\!74\)\( T^{7} + \)\(73\!\cdots\!15\)\( T^{8} + \)\(36\!\cdots\!33\)\( T^{9} + \)\(17\!\cdots\!86\)\( T^{10} + \)\(74\!\cdots\!07\)\( T^{11} + \)\(30\!\cdots\!15\)\( T^{12} + \)\(12\!\cdots\!86\)\( T^{13} + \)\(48\!\cdots\!22\)\( T^{14} + \)\(16\!\cdots\!64\)\( T^{15} + \)\(51\!\cdots\!88\)\( T^{16} + \)\(14\!\cdots\!94\)\( T^{17} + \)\(43\!\cdots\!77\)\( T^{18} + \)\(96\!\cdots\!77\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \)
$61$ \( 1 - 437 T - 744916 T^{2} + 424927067 T^{3} + 139480796216 T^{4} - 122519571726057 T^{5} + 19509236640919974 T^{6} - 8885281027720171051 T^{7} + \)\(17\!\cdots\!49\)\( T^{8} + \)\(92\!\cdots\!28\)\( T^{9} - \)\(74\!\cdots\!75\)\( T^{10} + \)\(52\!\cdots\!53\)\( T^{11} + \)\(19\!\cdots\!81\)\( T^{12} - \)\(90\!\cdots\!44\)\( T^{13} + \)\(54\!\cdots\!48\)\( T^{14} + \)\(67\!\cdots\!10\)\( T^{15} - \)\(60\!\cdots\!46\)\( T^{16} + \)\(50\!\cdots\!76\)\( T^{17} - \)\(13\!\cdots\!43\)\( T^{18} - \)\(85\!\cdots\!77\)\( T^{19} + \)\(79\!\cdots\!02\)\( T^{20} - \)\(19\!\cdots\!37\)\( T^{21} - \)\(70\!\cdots\!23\)\( T^{22} + \)\(58\!\cdots\!16\)\( T^{23} - \)\(16\!\cdots\!66\)\( T^{24} + \)\(40\!\cdots\!10\)\( T^{25} + \)\(74\!\cdots\!88\)\( T^{26} - \)\(28\!\cdots\!84\)\( T^{27} + \)\(13\!\cdots\!21\)\( T^{28} + \)\(83\!\cdots\!13\)\( T^{29} - \)\(26\!\cdots\!75\)\( T^{30} + \)\(76\!\cdots\!68\)\( T^{31} + \)\(32\!\cdots\!89\)\( T^{32} - \)\(37\!\cdots\!91\)\( T^{33} + \)\(18\!\cdots\!54\)\( T^{34} - \)\(26\!\cdots\!57\)\( T^{35} + \)\(69\!\cdots\!96\)\( T^{36} + \)\(47\!\cdots\!87\)\( T^{37} - \)\(19\!\cdots\!56\)\( T^{38} - \)\(25\!\cdots\!77\)\( T^{39} + \)\(13\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 + 642 T - 1655222 T^{2} - 1351264924 T^{3} + 1395720548382 T^{4} + 1507860208990244 T^{5} - 730390077743505158 T^{6} - \)\(11\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!15\)\( T^{8} + \)\(69\!\cdots\!46\)\( T^{9} + \)\(32\!\cdots\!85\)\( T^{10} - \)\(33\!\cdots\!96\)\( T^{11} - \)\(78\!\cdots\!53\)\( T^{12} + \)\(12\!\cdots\!70\)\( T^{13} + \)\(56\!\cdots\!78\)\( T^{14} - \)\(39\!\cdots\!80\)\( T^{15} - \)\(28\!\cdots\!94\)\( T^{16} + \)\(89\!\cdots\!68\)\( T^{17} + \)\(11\!\cdots\!05\)\( T^{18} - \)\(10\!\cdots\!22\)\( T^{19} - \)\(36\!\cdots\!78\)\( T^{20} - \)\(30\!\cdots\!86\)\( T^{21} + \)\(10\!\cdots\!45\)\( T^{22} + \)\(24\!\cdots\!96\)\( T^{23} - \)\(22\!\cdots\!34\)\( T^{24} - \)\(96\!\cdots\!40\)\( T^{25} + \)\(41\!\cdots\!02\)\( T^{26} + \)\(28\!\cdots\!90\)\( T^{27} - \)\(52\!\cdots\!13\)\( T^{28} - \)\(66\!\cdots\!08\)\( T^{29} + \)\(19\!\cdots\!65\)\( T^{30} + \)\(12\!\cdots\!02\)\( T^{31} + \)\(11\!\cdots\!15\)\( T^{32} - \)\(19\!\cdots\!20\)\( T^{33} - \)\(36\!\cdots\!62\)\( T^{34} + \)\(22\!\cdots\!08\)\( T^{35} + \)\(62\!\cdots\!62\)\( T^{36} - \)\(18\!\cdots\!92\)\( T^{37} - \)\(67\!\cdots\!38\)\( T^{38} + \)\(78\!\cdots\!34\)\( T^{39} + \)\(36\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 + 1545 T - 164612 T^{2} - 823498495 T^{3} + 510816804584 T^{4} + 421399428301951 T^{5} - 410556697843186024 T^{6} + 14010431784563891953 T^{7} + \)\(29\!\cdots\!89\)\( T^{8} - \)\(96\!\cdots\!56\)\( T^{9} - \)\(10\!\cdots\!59\)\( T^{10} + \)\(85\!\cdots\!53\)\( T^{11} + \)\(18\!\cdots\!67\)\( T^{12} - \)\(42\!\cdots\!72\)\( T^{13} + \)\(64\!\cdots\!76\)\( T^{14} + \)\(14\!\cdots\!32\)\( T^{15} - \)\(71\!\cdots\!48\)\( T^{16} - \)\(37\!\cdots\!38\)\( T^{17} + \)\(33\!\cdots\!67\)\( T^{18} + \)\(36\!\cdots\!55\)\( T^{19} - \)\(13\!\cdots\!22\)\( T^{20} + \)\(13\!\cdots\!05\)\( T^{21} + \)\(42\!\cdots\!07\)\( T^{22} - \)\(16\!\cdots\!78\)\( T^{23} - \)\(11\!\cdots\!68\)\( T^{24} + \)\(83\!\cdots\!32\)\( T^{25} + \)\(13\!\cdots\!36\)\( T^{26} - \)\(31\!\cdots\!12\)\( T^{27} + \)\(49\!\cdots\!27\)\( T^{28} + \)\(82\!\cdots\!23\)\( T^{29} - \)\(37\!\cdots\!59\)\( T^{30} - \)\(11\!\cdots\!16\)\( T^{31} + \)\(13\!\cdots\!69\)\( T^{32} + \)\(22\!\cdots\!43\)\( T^{33} - \)\(23\!\cdots\!84\)\( T^{34} + \)\(85\!\cdots\!01\)\( T^{35} + \)\(37\!\cdots\!24\)\( T^{36} - \)\(21\!\cdots\!45\)\( T^{37} - \)\(15\!\cdots\!72\)\( T^{38} + \)\(51\!\cdots\!95\)\( T^{39} + \)\(11\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 - 1292 T - 800760 T^{2} + 1888773084 T^{3} - 294654454086 T^{4} - 930615313063966 T^{5} + 430214695631874962 T^{6} + \)\(18\!\cdots\!12\)\( T^{7} - \)\(10\!\cdots\!67\)\( T^{8} - \)\(52\!\cdots\!64\)\( T^{9} - \)\(10\!\cdots\!33\)\( T^{10} + \)\(39\!\cdots\!20\)\( T^{11} + \)\(82\!\cdots\!53\)\( T^{12} - \)\(16\!\cdots\!46\)\( T^{13} - \)\(29\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!50\)\( T^{15} + \)\(21\!\cdots\!44\)\( T^{16} + \)\(12\!\cdots\!62\)\( T^{17} - \)\(27\!\cdots\!53\)\( T^{18} - \)\(51\!\cdots\!40\)\( T^{19} + \)\(16\!\cdots\!58\)\( T^{20} - \)\(19\!\cdots\!80\)\( T^{21} - \)\(41\!\cdots\!17\)\( T^{22} + \)\(75\!\cdots\!06\)\( T^{23} + \)\(48\!\cdots\!24\)\( T^{24} + \)\(24\!\cdots\!50\)\( T^{25} - \)\(10\!\cdots\!00\)\( T^{26} - \)\(21\!\cdots\!58\)\( T^{27} + \)\(43\!\cdots\!73\)\( T^{28} + \)\(80\!\cdots\!40\)\( T^{29} - \)\(81\!\cdots\!17\)\( T^{30} - \)\(16\!\cdots\!12\)\( T^{31} - \)\(12\!\cdots\!87\)\( T^{32} + \)\(84\!\cdots\!44\)\( T^{33} + \)\(78\!\cdots\!98\)\( T^{34} - \)\(65\!\cdots\!38\)\( T^{35} - \)\(81\!\cdots\!66\)\( T^{36} + \)\(20\!\cdots\!68\)\( T^{37} - \)\(33\!\cdots\!40\)\( T^{38} - \)\(20\!\cdots\!76\)\( T^{39} + \)\(63\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 + 1405 T - 2150268 T^{2} - 3116848211 T^{3} + 3360157724444 T^{4} + 4151188953835363 T^{5} - 3817398869736906676 T^{6} - \)\(37\!\cdots\!15\)\( T^{7} + \)\(34\!\cdots\!89\)\( T^{8} + \)\(24\!\cdots\!40\)\( T^{9} - \)\(24\!\cdots\!47\)\( T^{10} - \)\(12\!\cdots\!11\)\( T^{11} + \)\(14\!\cdots\!03\)\( T^{12} + \)\(47\!\cdots\!60\)\( T^{13} - \)\(67\!\cdots\!72\)\( T^{14} - \)\(13\!\cdots\!52\)\( T^{15} + \)\(28\!\cdots\!56\)\( T^{16} + \)\(29\!\cdots\!90\)\( T^{17} - \)\(11\!\cdots\!77\)\( T^{18} - \)\(35\!\cdots\!53\)\( T^{19} + \)\(54\!\cdots\!18\)\( T^{20} - \)\(17\!\cdots\!67\)\( T^{21} - \)\(29\!\cdots\!17\)\( T^{22} + \)\(34\!\cdots\!10\)\( T^{23} + \)\(17\!\cdots\!96\)\( T^{24} - \)\(40\!\cdots\!48\)\( T^{25} - \)\(97\!\cdots\!92\)\( T^{26} + \)\(33\!\cdots\!40\)\( T^{27} + \)\(49\!\cdots\!43\)\( T^{28} - \)\(21\!\cdots\!49\)\( T^{29} - \)\(20\!\cdots\!47\)\( T^{30} + \)\(10\!\cdots\!60\)\( T^{31} + \)\(70\!\cdots\!69\)\( T^{32} - \)\(38\!\cdots\!85\)\( T^{33} - \)\(19\!\cdots\!16\)\( T^{34} + \)\(10\!\cdots\!37\)\( T^{35} + \)\(40\!\cdots\!84\)\( T^{36} - \)\(18\!\cdots\!69\)\( T^{37} - \)\(63\!\cdots\!08\)\( T^{38} + \)\(20\!\cdots\!95\)\( T^{39} + \)\(72\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - 543 T - 3210972 T^{2} + 2149693073 T^{3} + 5106187705928 T^{4} - 4006405925115629 T^{5} - 5191885555542057060 T^{6} + \)\(46\!\cdots\!33\)\( T^{7} + \)\(37\!\cdots\!57\)\( T^{8} - \)\(37\!\cdots\!40\)\( T^{9} - \)\(22\!\cdots\!35\)\( T^{10} + \)\(23\!\cdots\!85\)\( T^{11} + \)\(11\!\cdots\!11\)\( T^{12} - \)\(13\!\cdots\!48\)\( T^{13} - \)\(56\!\cdots\!52\)\( T^{14} + \)\(70\!\cdots\!28\)\( T^{15} + \)\(18\!\cdots\!76\)\( T^{16} - \)\(31\!\cdots\!22\)\( T^{17} - \)\(49\!\cdots\!25\)\( T^{18} + \)\(72\!\cdots\!87\)\( T^{19} - \)\(25\!\cdots\!94\)\( T^{20} + \)\(41\!\cdots\!69\)\( T^{21} - \)\(16\!\cdots\!25\)\( T^{22} - \)\(59\!\cdots\!66\)\( T^{23} + \)\(19\!\cdots\!36\)\( T^{24} + \)\(43\!\cdots\!96\)\( T^{25} - \)\(19\!\cdots\!68\)\( T^{26} - \)\(26\!\cdots\!84\)\( T^{27} + \)\(13\!\cdots\!31\)\( T^{28} + \)\(15\!\cdots\!95\)\( T^{29} - \)\(83\!\cdots\!15\)\( T^{30} - \)\(80\!\cdots\!20\)\( T^{31} + \)\(46\!\cdots\!17\)\( T^{32} + \)\(32\!\cdots\!51\)\( T^{33} - \)\(20\!\cdots\!40\)\( T^{34} - \)\(91\!\cdots\!47\)\( T^{35} + \)\(66\!\cdots\!48\)\( T^{36} + \)\(16\!\cdots\!91\)\( T^{37} - \)\(13\!\cdots\!88\)\( T^{38} - \)\(13\!\cdots\!89\)\( T^{39} + \)\(13\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 + 2196 T - 2279256 T^{2} - 6744108292 T^{3} + 4590931714074 T^{4} + 12293000410375634 T^{5} - 9448115145172476374 T^{6} - \)\(17\!\cdots\!88\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} + \)\(20\!\cdots\!52\)\( T^{9} - \)\(21\!\cdots\!85\)\( T^{10} - \)\(19\!\cdots\!24\)\( T^{11} + \)\(25\!\cdots\!53\)\( T^{12} + \)\(15\!\cdots\!82\)\( T^{13} - \)\(26\!\cdots\!20\)\( T^{14} - \)\(10\!\cdots\!82\)\( T^{15} + \)\(24\!\cdots\!12\)\( T^{16} + \)\(53\!\cdots\!26\)\( T^{17} - \)\(20\!\cdots\!41\)\( T^{18} - \)\(13\!\cdots\!08\)\( T^{19} + \)\(15\!\cdots\!66\)\( T^{20} - \)\(96\!\cdots\!52\)\( T^{21} - \)\(10\!\cdots\!01\)\( T^{22} + \)\(18\!\cdots\!34\)\( T^{23} + \)\(61\!\cdots\!52\)\( T^{24} - \)\(18\!\cdots\!18\)\( T^{25} - \)\(33\!\cdots\!20\)\( T^{26} + \)\(13\!\cdots\!98\)\( T^{27} + \)\(15\!\cdots\!73\)\( T^{28} - \)\(83\!\cdots\!96\)\( T^{29} - \)\(66\!\cdots\!85\)\( T^{30} + \)\(43\!\cdots\!88\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} - \)\(18\!\cdots\!92\)\( T^{33} - \)\(70\!\cdots\!54\)\( T^{34} + \)\(64\!\cdots\!66\)\( T^{35} + \)\(17\!\cdots\!94\)\( T^{36} - \)\(17\!\cdots\!88\)\( T^{37} - \)\(42\!\cdots\!96\)\( T^{38} + \)\(28\!\cdots\!84\)\( T^{39} + \)\(91\!\cdots\!01\)\( T^{40} \)
$97$ \( ( 1 + 425 T + 4855100 T^{2} + 2674736785 T^{3} + 12413935569789 T^{4} + 7844309405033348 T^{5} + 21709474743257501488 T^{6} + \)\(14\!\cdots\!84\)\( T^{7} + \)\(28\!\cdots\!94\)\( T^{8} + \)\(18\!\cdots\!26\)\( T^{9} + \)\(29\!\cdots\!88\)\( T^{10} + \)\(16\!\cdots\!98\)\( T^{11} + \)\(23\!\cdots\!26\)\( T^{12} + \)\(10\!\cdots\!28\)\( T^{13} + \)\(15\!\cdots\!08\)\( T^{14} + \)\(49\!\cdots\!64\)\( T^{15} + \)\(71\!\cdots\!21\)\( T^{16} + \)\(14\!\cdots\!45\)\( T^{17} + \)\(23\!\cdots\!00\)\( T^{18} + \)\(18\!\cdots\!25\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} )^{2} \)
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