Properties

Label 74.4.e.a
Level $74$
Weight $4$
Character orbit 74.e
Analytic conductor $4.366$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 74.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.36614134042\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 346 x^{18} + 50697 x^{16} + 4104768 x^{14} + 200532432 x^{12} + 6039270720 x^{10} + 109290291168 x^{8} + 1091662316544 x^{6} + 4894436404224 x^{4} + 5352885383680 x^{2} + 1118416232704\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + \beta_{14} q^{3} + 4 \beta_{6} q^{4} + ( -1 + \beta_{6} + \beta_{10} ) q^{5} + \beta_{1} q^{6} + ( -1 - \beta_{4} + \beta_{8} + \beta_{14} - \beta_{18} ) q^{7} + ( 4 \beta_{11} + 4 \beta_{12} ) q^{8} + ( -7 - \beta_{3} - 2 \beta_{4} + 6 \beta_{6} + \beta_{7} + \beta_{13} + 2 \beta_{14} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{9} +O(q^{10})\) \( q + \beta_{11} q^{2} + \beta_{14} q^{3} + 4 \beta_{6} q^{4} + ( -1 + \beta_{6} + \beta_{10} ) q^{5} + \beta_{1} q^{6} + ( -1 - \beta_{4} + \beta_{8} + \beta_{14} - \beta_{18} ) q^{7} + ( 4 \beta_{11} + 4 \beta_{12} ) q^{8} + ( -7 - \beta_{3} - 2 \beta_{4} + 6 \beta_{6} + \beta_{7} + \beta_{13} + 2 \beta_{14} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{9} + ( 2 \beta_{5} - \beta_{11} + \beta_{12} ) q^{10} + ( -3 + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{18} + \beta_{19} ) q^{11} + ( -4 \beta_{4} + 4 \beta_{14} ) q^{12} + ( -10 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + 4 \beta_{6} + \beta_{7} + \beta_{8} - 7 \beta_{12} + \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{13} + ( -1 + \beta_{3} + 2 \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{13} + \beta_{15} + \beta_{16} ) q^{14} + ( 4 + \beta_{2} - 2 \beta_{4} + 6 \beta_{6} - \beta_{8} + 3 \beta_{9} + 5 \beta_{11} + 2 \beta_{14} - \beta_{15} - 4 \beta_{18} ) q^{15} + ( -16 + 16 \beta_{6} ) q^{16} + ( 2 - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 11 \beta_{11} + 3 \beta_{13} + 3 \beta_{14} - \beta_{18} ) q^{17} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{10} + 7 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{16} + 2 \beta_{19} ) q^{18} + ( 11 - 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} + 2 \beta_{10} - 3 \beta_{12} + 2 \beta_{13} + \beta_{16} + 3 \beta_{17} - \beta_{19} ) q^{19} + ( -4 \beta_{6} - 4 \beta_{9} ) q^{20} + ( -1 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} - 3 \beta_{17} + \beta_{19} ) q^{21} + ( 2 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{13} + \beta_{15} + 2 \beta_{18} ) q^{22} + ( 10 + \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 20 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 4 \beta_{13} - 6 \beta_{14} - \beta_{15} - \beta_{16} + 4 \beta_{17} + 4 \beta_{18} + 4 \beta_{19} ) q^{23} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{24} + ( 53 - 6 \beta_{1} + 3 \beta_{2} - 5 \beta_{4} - 50 \beta_{6} - 6 \beta_{7} - 3 \beta_{9} - 6 \beta_{10} - 8 \beta_{11} - 16 \beta_{12} + 8 \beta_{14} + 4 \beta_{17} + 3 \beta_{18} ) q^{25} + ( 29 + 5 \beta_{3} - 3 \beta_{7} - 3 \beta_{8} - 4 \beta_{11} + 4 \beta_{12} + \beta_{15} - \beta_{16} + 2 \beta_{18} - 2 \beta_{19} ) q^{26} + ( -29 - 3 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + 9 \beta_{11} - 9 \beta_{12} + 3 \beta_{18} - 3 \beta_{19} ) q^{27} + ( 4 - 4 \beta_{7} + 4 \beta_{14} - 4 \beta_{19} ) q^{28} + ( -30 - 4 \beta_{1} + 3 \beta_{3} + 12 \beta_{4} + 60 \beta_{6} + \beta_{7} - \beta_{8} - 7 \beta_{11} - 7 \beta_{12} - 6 \beta_{13} - 24 \beta_{14} + 4 \beta_{18} + 4 \beta_{19} ) q^{29} + ( 4 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 6 \beta_{5} + 28 \beta_{6} - 4 \beta_{8} + 12 \beta_{11} + 6 \beta_{12} - 8 \beta_{13} - 8 \beta_{14} - 6 \beta_{17} - 2 \beta_{19} ) q^{30} + ( 23 + 8 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - 46 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} - 3 \beta_{12} - 6 \beta_{13} - 6 \beta_{14} - 4 \beta_{15} - 4 \beta_{16} + \beta_{18} + \beta_{19} ) q^{31} + 16 \beta_{12} q^{32} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{4} - 9 \beta_{5} - 62 \beta_{6} - 2 \beta_{9} - \beta_{10} + 28 \beta_{11} + 14 \beta_{12} - 4 \beta_{13} - 8 \beta_{14} - 9 \beta_{17} + 4 \beta_{18} + 2 \beta_{19} ) q^{33} + ( 6 \beta_{4} - 42 \beta_{6} + 10 \beta_{11} + 5 \beta_{12} - 2 \beta_{13} - 6 \beta_{14} + 12 \beta_{18} + 6 \beta_{19} ) q^{34} + ( -14 - 6 \beta_{2} - 5 \beta_{3} + 7 \beta_{4} - 3 \beta_{6} + \beta_{7} + 3 \beta_{8} + 7 \beta_{9} - 12 \beta_{11} - 5 \beta_{13} + 5 \beta_{14} + \beta_{15} + 3 \beta_{18} + \beta_{19} ) q^{35} + ( -32 - 4 \beta_{3} - 8 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} ) q^{36} + ( 23 - \beta_{1} - 8 \beta_{2} - \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - 42 \beta_{6} - \beta_{7} - 5 \beta_{8} + \beta_{9} + 7 \beta_{10} - 35 \beta_{11} - 11 \beta_{12} + 9 \beta_{13} + \beta_{14} - \beta_{15} - 4 \beta_{16} - \beta_{17} - 2 \beta_{18} - 6 \beta_{19} ) q^{37} + ( 6 - 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} + 6 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{18} + 2 \beta_{19} ) q^{38} + ( 31 + 12 \beta_{2} + 2 \beta_{3} - 9 \beta_{4} - 10 \beta_{5} + 36 \beta_{6} - \beta_{7} + \beta_{8} + 5 \beta_{9} - 37 \beta_{11} + 2 \beta_{13} - 8 \beta_{14} + 3 \beta_{15} - 5 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{39} + ( 8 \beta_{5} - 8 \beta_{11} - 4 \beta_{12} + 8 \beta_{17} ) q^{40} + ( 6 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - 5 \beta_{5} - 57 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 12 \beta_{9} - 6 \beta_{10} - 2 \beta_{13} + 8 \beta_{15} + 4 \beta_{16} - 5 \beta_{17} - 6 \beta_{18} - 5 \beta_{19} ) q^{41} + ( -9 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 12 \beta_{4} - 4 \beta_{5} + 7 \beta_{6} + \beta_{7} + 6 \beta_{10} + \beta_{13} - 8 \beta_{14} - \beta_{16} + 4 \beta_{17} + 4 \beta_{19} ) q^{42} + ( -3 + 6 \beta_{1} - 3 \beta_{3} + 17 \beta_{4} + \beta_{5} + 6 \beta_{6} + 8 \beta_{7} - 8 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 49 \beta_{11} + 49 \beta_{12} + 6 \beta_{13} - 34 \beta_{14} - 8 \beta_{15} - 8 \beta_{16} + 2 \beta_{17} + 3 \beta_{18} + 3 \beta_{19} ) q^{43} + ( -4 - 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} + 4 \beta_{13} - 4 \beta_{14} + 4 \beta_{17} - 4 \beta_{18} ) q^{44} + ( -5 + 3 \beta_{1} - 12 \beta_{4} + 3 \beta_{5} + 10 \beta_{6} - 13 \beta_{9} - 13 \beta_{10} - 60 \beta_{11} - 60 \beta_{12} + 24 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} + 6 \beta_{17} ) q^{45} + ( -6 \beta_{1} + 3 \beta_{2} - 9 \beta_{3} - 12 \beta_{4} - 9 \beta_{6} + 6 \beta_{7} - \beta_{8} - 4 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} - 16 \beta_{12} + 9 \beta_{13} + 6 \beta_{14} - \beta_{15} - 2 \beta_{16} - 6 \beta_{17} + 2 \beta_{18} + 8 \beta_{19} ) q^{46} + ( 34 + \beta_{1} - 2 \beta_{2} + 10 \beta_{3} + 22 \beta_{4} + 3 \beta_{5} - 5 \beta_{7} - 5 \beta_{8} - 8 \beta_{9} + 8 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} - 4 \beta_{15} + 4 \beta_{16} + 4 \beta_{18} - 4 \beta_{19} ) q^{47} -16 \beta_{4} q^{48} + ( 1 - 12 \beta_{1} + 6 \beta_{2} + 14 \beta_{3} + 20 \beta_{4} + 9 \beta_{6} - 17 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} - 6 \beta_{10} + 5 \beta_{11} + 10 \beta_{12} - 14 \beta_{13} - 15 \beta_{14} + 4 \beta_{15} + 8 \beta_{16} + 16 \beta_{17} + 4 \beta_{18} - \beta_{19} ) q^{49} + ( 62 + 2 \beta_{1} - 2 \beta_{2} + 12 \beta_{4} - 6 \beta_{5} - 32 \beta_{6} - 6 \beta_{7} - 8 \beta_{10} - 50 \beta_{12} - 6 \beta_{14} + 6 \beta_{16} + 6 \beta_{17} ) q^{50} + ( -56 - 24 \beta_{1} - 4 \beta_{3} - 24 \beta_{4} - 17 \beta_{5} + 112 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 34 \beta_{11} + 34 \beta_{12} + 8 \beta_{13} + 48 \beta_{14} + \beta_{15} + \beta_{16} - 34 \beta_{17} - 3 \beta_{18} - 3 \beta_{19} ) q^{51} + ( -24 - 8 \beta_{2} + 4 \beta_{4} - 16 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 28 \beta_{11} - 4 \beta_{15} + 8 \beta_{18} + 4 \beta_{19} ) q^{52} + ( -36 + 18 \beta_{1} - 9 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} + 40 \beta_{6} + 5 \beta_{7} - \beta_{8} + 8 \beta_{9} + 16 \beta_{10} + 8 \beta_{11} + 16 \beta_{12} - 8 \beta_{13} - 2 \beta_{14} - \beta_{15} - 2 \beta_{16} + 23 \beta_{17} + 5 \beta_{18} + 13 \beta_{19} ) q^{53} + ( 30 + 4 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} + 31 \beta_{6} - \beta_{8} + 2 \beta_{9} - 30 \beta_{11} + 5 \beta_{13} + 16 \beta_{14} - \beta_{15} - 6 \beta_{17} - 10 \beta_{18} ) q^{54} + ( -20 + 22 \beta_{1} - 22 \beta_{2} - 18 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 9 \beta_{6} - \beta_{7} - \beta_{8} - 4 \beta_{10} - 51 \beta_{12} + 9 \beta_{13} + 8 \beta_{14} - \beta_{16} + 3 \beta_{17} + \beta_{18} - 17 \beta_{19} ) q^{55} + ( -4 + 8 \beta_{3} + 4 \beta_{6} - 4 \beta_{7} - 4 \beta_{13} + 4 \beta_{16} ) q^{56} + ( 8 \beta_{2} + 8 \beta_{3} - 3 \beta_{4} + 38 \beta_{5} + 8 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 17 \beta_{11} + 8 \beta_{13} + 11 \beta_{14} - 4 \beta_{15} + 19 \beta_{17} - 11 \beta_{18} + 3 \beta_{19} ) q^{57} + ( 36 - 16 \beta_{1} + 8 \beta_{2} - 9 \beta_{3} + 16 \beta_{4} - 37 \beta_{6} + 2 \beta_{7} - \beta_{8} + 27 \beta_{11} + 54 \beta_{12} + 9 \beta_{13} - 10 \beta_{14} - \beta_{15} - 2 \beta_{16} - 6 \beta_{18} - 12 \beta_{19} ) q^{58} + ( 57 - 18 \beta_{2} - 2 \beta_{3} + 15 \beta_{4} + 28 \beta_{5} + 64 \beta_{6} - \beta_{7} + 6 \beta_{8} + 2 \beta_{9} - 27 \beta_{11} - 2 \beta_{13} + 29 \beta_{14} + 8 \beta_{15} + 14 \beta_{17} - 15 \beta_{18} - \beta_{19} ) q^{59} + ( -24 + 4 \beta_{1} - 8 \beta_{4} + 48 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 12 \beta_{9} + 12 \beta_{10} + 20 \beta_{11} + 20 \beta_{12} + 16 \beta_{14} - 4 \beta_{15} - 4 \beta_{16} - 16 \beta_{18} - 16 \beta_{19} ) q^{60} + ( -128 - \beta_{1} + \beta_{2} + 6 \beta_{3} - 74 \beta_{4} - 2 \beta_{5} + 64 \beta_{6} + 10 \beta_{7} + 4 \beta_{8} + 14 \beta_{10} + 66 \beta_{12} - 3 \beta_{13} + 31 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} - 4 \beta_{18} + 8 \beta_{19} ) q^{61} + ( -2 + 6 \beta_{1} - 3 \beta_{2} - \beta_{3} - 22 \beta_{4} - 13 \beta_{6} + 14 \beta_{7} + \beta_{8} - 26 \beta_{11} - 52 \beta_{12} + \beta_{13} + 20 \beta_{14} + \beta_{15} + 2 \beta_{16} - 2 \beta_{17} - 14 \beta_{18} - 12 \beta_{19} ) q^{62} + ( 72 - 6 \beta_{1} + 12 \beta_{2} - 14 \beta_{3} - 42 \beta_{4} - 4 \beta_{5} + 16 \beta_{7} + 16 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} + 51 \beta_{11} - 51 \beta_{12} - 11 \beta_{18} + 11 \beta_{19} ) q^{63} -64 q^{64} + ( -76 + 32 \beta_{1} - 16 \beta_{2} + 10 \beta_{3} - 43 \beta_{4} + 54 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} - 13 \beta_{9} - 26 \beta_{10} + 31 \beta_{11} + 62 \beta_{12} - 10 \beta_{13} + 33 \beta_{14} + 4 \beta_{15} + 8 \beta_{16} - 35 \beta_{17} - 3 \beta_{18} + 7 \beta_{19} ) q^{65} + ( -52 - 4 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} + 104 \beta_{6} + 18 \beta_{9} + 18 \beta_{10} - 66 \beta_{11} - 66 \beta_{12} + 8 \beta_{13} + 4 \beta_{17} - 8 \beta_{18} - 8 \beta_{19} ) q^{66} + ( -18 - 2 \beta_{4} - 2 \beta_{5} + 19 \beta_{6} - \beta_{7} + 20 \beta_{8} - 2 \beta_{9} - \beta_{10} + 54 \beta_{11} + 27 \beta_{12} + 3 \beta_{13} - 25 \beta_{14} + 2 \beta_{15} + \beta_{16} - 2 \beta_{17} + 14 \beta_{18} + 16 \beta_{19} ) q^{67} + ( -8 - 4 \beta_{1} - 12 \beta_{3} - 12 \beta_{4} + 16 \beta_{6} - 44 \beta_{11} - 44 \beta_{12} + 24 \beta_{13} + 24 \beta_{14} - 4 \beta_{18} - 4 \beta_{19} ) q^{68} + ( 45 + 24 \beta_{1} - 24 \beta_{2} + 28 \beta_{3} + 30 \beta_{4} + 13 \beta_{5} - 14 \beta_{6} - 3 \beta_{7} + 14 \beta_{10} + 97 \beta_{12} - 14 \beta_{13} - 23 \beta_{14} + 3 \beta_{16} - 13 \beta_{17} + 16 \beta_{19} ) q^{69} + ( -5 + 8 \beta_{1} + 8 \beta_{2} - 12 \beta_{4} - 14 \beta_{5} - 55 \beta_{6} - \beta_{7} + 6 \beta_{8} - 16 \beta_{11} - 8 \beta_{12} + 7 \beta_{13} - 10 \beta_{14} + 2 \beta_{15} + \beta_{16} - 14 \beta_{17} - 20 \beta_{18} - 8 \beta_{19} ) q^{70} + ( -2 + 27 \beta_{1} + 27 \beta_{2} - 16 \beta_{4} - 23 \beta_{5} - 37 \beta_{6} - \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 144 \beta_{11} + 72 \beta_{12} - 7 \beta_{13} + 20 \beta_{14} - 23 \beta_{17} - 33 \beta_{18} - 17 \beta_{19} ) q^{71} + ( -8 \beta_{2} + 4 \beta_{3} - 4 \beta_{6} + 4 \beta_{8} - 8 \beta_{9} - 28 \beta_{11} + 4 \beta_{13} + 8 \beta_{14} + 4 \beta_{15} - 8 \beta_{18} ) q^{72} + ( 8 - 10 \beta_{1} + 20 \beta_{2} - 12 \beta_{3} + 6 \beta_{4} + 21 \beta_{5} - 14 \beta_{7} - 14 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 60 \beta_{11} + 60 \beta_{12} + 13 \beta_{18} - 13 \beta_{19} ) q^{73} + ( 25 - 4 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} - 131 \beta_{6} + 15 \beta_{7} + 8 \beta_{8} - 10 \beta_{9} + 2 \beta_{10} - 23 \beta_{11} - 33 \beta_{12} - 9 \beta_{13} - 52 \beta_{14} - 4 \beta_{15} + \beta_{16} - 2 \beta_{17} + 8 \beta_{18} + 24 \beta_{19} ) q^{74} + ( -331 - 9 \beta_{1} + 18 \beta_{2} - 7 \beta_{3} + 63 \beta_{4} + 41 \beta_{5} + 16 \beta_{7} + 16 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} + 24 \beta_{11} - 24 \beta_{12} + 5 \beta_{18} - 5 \beta_{19} ) q^{75} + ( 28 + 12 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 24 \beta_{5} + 16 \beta_{6} - 4 \beta_{8} - 8 \beta_{9} + 12 \beta_{11} - 8 \beta_{13} - 4 \beta_{15} - 12 \beta_{17} + 4 \beta_{18} ) q^{76} + ( -18 + 7 \beta_{1} + 7 \beta_{2} - 8 \beta_{4} + 19 \beta_{5} + 375 \beta_{6} - 5 \beta_{7} + 14 \beta_{8} + 18 \beta_{9} + 9 \beta_{10} - 74 \beta_{11} - 37 \beta_{12} - 17 \beta_{13} - 6 \beta_{14} + 10 \beta_{15} + 5 \beta_{16} + 19 \beta_{17} - 12 \beta_{18} - 4 \beta_{19} ) q^{77} + ( -21 - 12 \beta_{1} - 12 \beta_{2} - 2 \beta_{4} - 10 \beta_{5} - 143 \beta_{6} + \beta_{7} + 10 \beta_{8} + 20 \beta_{9} + 10 \beta_{10} + 76 \beta_{11} + 38 \beta_{12} - 5 \beta_{13} + 52 \beta_{14} - 2 \beta_{15} - \beta_{16} - 10 \beta_{17} + 8 \beta_{18} + 10 \beta_{19} ) q^{78} + ( -243 - 22 \beta_{1} + 22 \beta_{2} + 6 \beta_{3} - 21 \beta_{4} - 10 \beta_{5} + 115 \beta_{6} + 4 \beta_{7} - 9 \beta_{10} - 30 \beta_{12} - 3 \beta_{13} + 14 \beta_{14} - 4 \beta_{16} + 10 \beta_{17} - 7 \beta_{19} ) q^{79} + ( 16 - 32 \beta_{6} - 16 \beta_{9} - 16 \beta_{10} ) q^{80} + ( 22 + 21 \beta_{1} + 21 \beta_{2} + 18 \beta_{4} - \beta_{5} + 220 \beta_{6} - 28 \beta_{8} + 12 \beta_{9} + 6 \beta_{10} + 126 \beta_{11} + 63 \beta_{12} + 19 \beta_{13} - 43 \beta_{14} - \beta_{17} + 8 \beta_{18} - 10 \beta_{19} ) q^{81} + ( -6 - 6 \beta_{1} + 6 \beta_{3} + 2 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} + 10 \beta_{9} + 10 \beta_{10} - 59 \beta_{11} - 59 \beta_{12} - 12 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} - 4 \beta_{16} + 24 \beta_{17} + 4 \beta_{18} + 4 \beta_{19} ) q^{82} + ( 114 + 6 \beta_{1} - 3 \beta_{2} - 7 \beta_{3} - 67 \beta_{4} - 91 \beta_{6} - 8 \beta_{7} - 8 \beta_{8} + 7 \beta_{9} + 14 \beta_{10} - 36 \beta_{11} - 72 \beta_{12} + 7 \beta_{13} + 86 \beta_{14} - 8 \beta_{15} - 16 \beta_{16} - 29 \beta_{17} + 5 \beta_{18} - 14 \beta_{19} ) q^{83} + ( -16 - 8 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} + 12 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 8 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} + 8 \beta_{12} - 4 \beta_{18} + 4 \beta_{19} ) q^{84} + ( -36 + 37 \beta_{1} - 74 \beta_{2} - 29 \beta_{3} - 28 \beta_{4} - 23 \beta_{5} - 13 \beta_{7} - 13 \beta_{8} - 15 \beta_{11} + 15 \beta_{12} + \beta_{15} - \beta_{16} + 14 \beta_{18} - 14 \beta_{19} ) q^{85} + ( -224 - 24 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} - 20 \beta_{4} + 190 \beta_{6} + 32 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} + 6 \beta_{11} + 12 \beta_{12} + 6 \beta_{13} - 2 \beta_{14} - 12 \beta_{17} - 10 \beta_{18} + 12 \beta_{19} ) q^{86} + ( 612 + 6 \beta_{1} - 6 \beta_{2} + 18 \beta_{3} + 6 \beta_{4} - 26 \beta_{5} - 287 \beta_{6} - 22 \beta_{7} - 9 \beta_{8} + 7 \beta_{10} - 184 \beta_{12} - 9 \beta_{13} + \beta_{14} + 4 \beta_{16} + 26 \beta_{17} + 9 \beta_{18} + \beta_{19} ) q^{87} + ( -4 - 12 \beta_{1} + 4 \beta_{3} + 8 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 8 \beta_{9} - 8 \beta_{10} - 8 \beta_{13} + 4 \beta_{15} + 4 \beta_{16} + 8 \beta_{18} + 8 \beta_{19} ) q^{88} + ( 50 - 49 \beta_{2} + 12 \beta_{3} + 23 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} - 12 \beta_{7} - 20 \beta_{8} - 11 \beta_{9} + 146 \beta_{11} + 12 \beta_{13} - 6 \beta_{14} + 4 \beta_{15} + 2 \beta_{17} + 17 \beta_{18} - 12 \beta_{19} ) q^{89} + ( 246 + 24 \beta_{1} - 12 \beta_{2} - 3 \beta_{3} - 12 \beta_{4} - 243 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} - 12 \beta_{10} + 5 \beta_{11} + 10 \beta_{12} + 3 \beta_{13} + 18 \beta_{14} - 3 \beta_{15} - 6 \beta_{16} + 26 \beta_{17} + 6 \beta_{18} ) q^{90} + ( 43 + 31 \beta_{2} - 19 \beta_{3} + 27 \beta_{4} - 46 \beta_{5} + 21 \beta_{6} - \beta_{7} - 10 \beta_{8} - 11 \beta_{9} - 139 \beta_{11} - 19 \beta_{13} - 8 \beta_{15} - 23 \beta_{17} + 26 \beta_{18} - \beta_{19} ) q^{91} + ( 76 + 4 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} - 8 \beta_{5} - 40 \beta_{6} + 12 \beta_{7} + 4 \beta_{8} + 12 \beta_{10} + 8 \beta_{13} - 12 \beta_{14} - 4 \beta_{16} + 8 \beta_{17} - 4 \beta_{18} + 20 \beta_{19} ) q^{92} + ( 95 + 47 \beta_{1} - 47 \beta_{2} + 28 \beta_{3} + 89 \beta_{4} - 18 \beta_{5} - 68 \beta_{6} + 11 \beta_{7} + 4 \beta_{8} - 26 \beta_{10} + 110 \beta_{12} - 14 \beta_{13} - 49 \beta_{14} - 3 \beta_{16} + 18 \beta_{17} - 4 \beta_{18} + 5 \beta_{19} ) q^{93} + ( 60 + 17 \beta_{2} + 7 \beta_{3} + 30 \beta_{4} + 32 \beta_{5} + 5 \beta_{6} - 16 \beta_{7} - 33 \beta_{8} - 6 \beta_{9} + 18 \beta_{11} + 7 \beta_{13} - 14 \beta_{14} - \beta_{15} + 16 \beta_{17} + 28 \beta_{18} - 16 \beta_{19} ) q^{94} + ( 418 - 94 \beta_{1} + 47 \beta_{2} + 6 \beta_{3} + 21 \beta_{4} - 406 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 6 \beta_{9} + 12 \beta_{10} + 184 \beta_{11} + 368 \beta_{12} - 6 \beta_{13} - 25 \beta_{14} - 4 \beta_{15} - 8 \beta_{16} - 19 \beta_{17} + 14 \beta_{18} + 18 \beta_{19} ) q^{95} -16 \beta_{2} q^{96} + ( 52 - 71 \beta_{1} + 7 \beta_{3} + 16 \beta_{4} + 18 \beta_{5} - 104 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} - 41 \beta_{9} - 41 \beta_{10} + 80 \beta_{11} + 80 \beta_{12} - 14 \beta_{13} - 32 \beta_{14} - 4 \beta_{15} - 4 \beta_{16} + 36 \beta_{17} + 20 \beta_{18} + 20 \beta_{19} ) q^{97} + ( -17 - 22 \beta_{1} + 22 \beta_{2} + 2 \beta_{3} + 52 \beta_{4} - 6 \beta_{5} + 21 \beta_{6} - 41 \beta_{7} - 16 \beta_{8} - 32 \beta_{10} - 5 \beta_{12} - \beta_{13} + 9 \beta_{16} + 6 \beta_{17} + 16 \beta_{18} - 36 \beta_{19} ) q^{98} + ( 212 + 6 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 14 \beta_{4} - 208 \beta_{6} + 5 \beta_{7} + 9 \beta_{9} + 18 \beta_{10} - 66 \beta_{11} - 132 \beta_{12} + 4 \beta_{13} - 28 \beta_{14} + 22 \beta_{17} + 9 \beta_{18} + 23 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 2q^{3} + 40q^{4} - 18q^{5} - 2q^{7} - 76q^{9} + O(q^{10}) \) \( 20q - 2q^{3} + 40q^{4} - 18q^{5} - 2q^{7} - 76q^{9} - 16q^{10} - 16q^{11} + 8q^{12} - 150q^{13} + 198q^{15} - 160q^{16} + 90q^{17} + 162q^{19} - 72q^{20} - 30q^{21} + 532q^{25} + 528q^{26} - 644q^{27} + 8q^{28} + 312q^{30} - 596q^{33} - 488q^{34} - 342q^{35} - 608q^{36} - 112q^{37} + 144q^{38} + 1146q^{39} - 32q^{40} - 498q^{41} - 120q^{42} - 32q^{44} + 424q^{47} + 64q^{48} + 84q^{49} + 1008q^{50} - 600q^{52} - 142q^{53} + 1080q^{54} - 540q^{55} + 138q^{57} + 224q^{58} + 1590q^{59} - 1542q^{61} + 8q^{62} + 1864q^{63} - 1280q^{64} - 694q^{65} + 62q^{67} + 708q^{69} - 368q^{70} - 178q^{71} - 528q^{73} - 560q^{74} - 7224q^{75} + 648q^{76} + 3468q^{77} - 1736q^{78} - 3474q^{79} + 2414q^{81} + 938q^{83} - 240q^{84} - 1100q^{85} - 2120q^{86} + 9420q^{87} + 510q^{89} + 2504q^{90} + 666q^{91} + 1344q^{92} + 1728q^{93} + 264q^{94} + 4126q^{95} - 816q^{98} + 2312q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 346 x^{18} + 50697 x^{16} + 4104768 x^{14} + 200532432 x^{12} + 6039270720 x^{10} + 109290291168 x^{8} + 1091662316544 x^{6} + 4894436404224 x^{4} + 5352885383680 x^{2} + 1118416232704\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(17184 \nu^{18} + 5549863 \nu^{16} + 741687905 \nu^{14} + 53039703547 \nu^{12} + 2184696639575 \nu^{10} + 51659416636942 \nu^{8} + 652451165998184 \nu^{6} + 3690779504633824 \nu^{4} + 6933170016056432 \nu^{2} + 856978891847424 \nu + 2406769808879200\)\()/ 856978891847424 \)
\(\beta_{3}\)\(=\)\((\)\(3111595001479 \nu^{18} + 968351882231062 \nu^{16} + 124062098214323507 \nu^{14} + 8457204023217404776 \nu^{12} + 329854532859681671540 \nu^{10} + 7321698777296356968160 \nu^{8} + 85507432431090080853392 \nu^{6} + 424864480413882791681920 \nu^{4} + 481873048665260986118336 \nu^{2} + 93195318018317307057152\)\()/ \)\(13\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(25858167 \nu^{18} + 8050739056 \nu^{16} + 1031897132831 \nu^{14} + 70375153812298 \nu^{12} + 2746038836451740 \nu^{10} + 60978891808349320 \nu^{8} + 712374391626167696 \nu^{6} + 3537657610769891680 \nu^{4} + 3973344591996331328 \nu^{2} + 832985252783591296\)\()/ 4634954077317120 \)
\(\beta_{5}\)\(=\)\((\)\(-8070670183116 \nu^{18} - 2511769129090493 \nu^{16} - 321813621320458618 \nu^{14} - 21938743978471096409 \nu^{12} - 855726346333897821460 \nu^{10} - 18996440174152195592180 \nu^{8} - 221887754630238074934208 \nu^{6} - 1102317048133946663903600 \nu^{4} - 1244026918119848973488704 \nu^{2} - 265203722277654617859008\)\()/ \)\(66\!\cdots\!60\)\( \)
\(\beta_{6}\)\(=\)\((\)\(1137896675 \nu^{19} + 384625762766 \nu^{17} + 54753313374787 \nu^{15} + 4278615095192120 \nu^{13} + 200139065319695128 \nu^{11} + 5716850921424945800 \nu^{9} + 97044799238233423408 \nu^{7} + 897198402466359348416 \nu^{5} + 3617767287023190635776 \nu^{3} + 2424936571335315376768 \nu + 453149870515513473024\)\()/ \)\(90\!\cdots\!48\)\( \)
\(\beta_{7}\)\(=\)\((\)\(847723382320573 \nu^{19} + 1350364270655614875 \nu^{18} + 182250871072061550 \nu^{17} + 420331755627576763920 \nu^{16} + 8374738375732916649 \nu^{15} + 53863141431252976440579 \nu^{14} - 953437773422540196252 \nu^{13} + 3672599592906993104365122 \nu^{12} - 131635554351631477502388 \nu^{11} + 143273416350617339361268428 \nu^{10} - 6585785453897818343216976 \nu^{9} + 3180954959475119938487383272 \nu^{8} - 164404657709766489547296336 \nu^{7} + 37157095665877826689774329168 \nu^{6} - 2006862912457326958069941312 \nu^{5} + 184565092469958141279333444576 \nu^{4} - 9497225618827725896424441024 \nu^{3} + 208005977985161028970436970048 \nu^{2} - 6140517809829441482113700096 \nu + 45475559394446851529385148800\)\()/ \)\(10\!\cdots\!28\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-847723382320573 \nu^{19} + 1350364270655614875 \nu^{18} - 182250871072061550 \nu^{17} + 420331755627576763920 \nu^{16} - 8374738375732916649 \nu^{15} + 53863141431252976440579 \nu^{14} + 953437773422540196252 \nu^{13} + 3672599592906993104365122 \nu^{12} + 131635554351631477502388 \nu^{11} + 143273416350617339361268428 \nu^{10} + 6585785453897818343216976 \nu^{9} + 3180954959475119938487383272 \nu^{8} + 164404657709766489547296336 \nu^{7} + 37157095665877826689774329168 \nu^{6} + 2006862912457326958069941312 \nu^{5} + 184565092469958141279333444576 \nu^{4} + 9497225618827725896424441024 \nu^{3} + 208005977985161028970436970048 \nu^{2} + 6140517809829441482113700096 \nu + 45475559394446851529385148800\)\()/ \)\(10\!\cdots\!28\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-6998820570413411 \nu^{19} - 58724066606289858 \nu^{18} - 2194246825443287220 \nu^{17} - 18172328904754120068 \nu^{16} - 284222985159927908571 \nu^{15} - 2312176685761886917818 \nu^{14} - 19704759085296906179238 \nu^{13} - 156316625481190911749376 \nu^{12} - 789697319641447570924884 \nu^{11} - 6037119168159189132417288 \nu^{10} - 18369283872870815970183816 \nu^{9} - 132476063528209320187234080 \nu^{8} - 234747540856578217405798416 \nu^{7} - 1526960924253905656016371296 \nu^{6} - 1438518147190193311894200864 \nu^{5} - 7480630823596250172327483648 \nu^{4} - 3255040995004501211928258240 \nu^{3} - 8525513169022257197382995328 \nu^{2} - 1779293518256524255533176192 \nu - 2489089750161529387723021824\)\()/ \)\(21\!\cdots\!56\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-6998820570413411 \nu^{19} + 58724066606289858 \nu^{18} - 2194246825443287220 \nu^{17} + 18172328904754120068 \nu^{16} - 284222985159927908571 \nu^{15} + 2312176685761886917818 \nu^{14} - 19704759085296906179238 \nu^{13} + 156316625481190911749376 \nu^{12} - 789697319641447570924884 \nu^{11} + 6037119168159189132417288 \nu^{10} - 18369283872870815970183816 \nu^{9} + 132476063528209320187234080 \nu^{8} - 234747540856578217405798416 \nu^{7} + 1526960924253905656016371296 \nu^{6} - 1438518147190193311894200864 \nu^{5} + 7480630823596250172327483648 \nu^{4} - 3255040995004501211928258240 \nu^{3} + 8525513169022257197382995328 \nu^{2} - 1779293518256524255533176192 \nu + 2489089750161529387723021824\)\()/ \)\(21\!\cdots\!56\)\( \)
\(\beta_{11}\)\(=\)\((\)\(7682877323108223 \nu^{19} + 78161731703545636 \nu^{18} + 2391535776860964024 \nu^{17} + 24329320222400499108 \nu^{16} + 306451477985486816919 \nu^{15} + 3117445556364804673068 \nu^{14} + 20891899886662263395202 \nu^{13} + 212523348214412258702604 \nu^{12} + 814711569169892779632540 \nu^{11} + 8287975462205121211227120 \nu^{10} + 18072227817706953041971080 \nu^{9} + 183885016509407911801864080 \nu^{8} + 210636482561331039124567824 \nu^{7} + 2144918003455144570941108288 \nu^{6} + 1038613607319799757794103520 \nu^{5} + 10613102808838704562862736960 \nu^{4} + 1110667179318409068139958592 \nu^{3} + 11715078975189317325707276544 \nu^{2} + 138547831517401784996774784 \nu + 2367583670871018412136013568\)\()/ \)\(47\!\cdots\!40\)\( \)
\(\beta_{12}\)\(=\)\((\)\(7682877323108223 \nu^{19} - 78161731703545636 \nu^{18} + 2391535776860964024 \nu^{17} - 24329320222400499108 \nu^{16} + 306451477985486816919 \nu^{15} - 3117445556364804673068 \nu^{14} + 20891899886662263395202 \nu^{13} - 212523348214412258702604 \nu^{12} + 814711569169892779632540 \nu^{11} - 8287975462205121211227120 \nu^{10} + 18072227817706953041971080 \nu^{9} - 183885016509407911801864080 \nu^{8} + 210636482561331039124567824 \nu^{7} - 2144918003455144570941108288 \nu^{6} + 1038613607319799757794103520 \nu^{5} - 10613102808838704562862736960 \nu^{4} + 1110667179318409068139958592 \nu^{3} - 11715078975189317325707276544 \nu^{2} + 138547831517401784996774784 \nu - 2367583670871018412136013568\)\()/ \)\(47\!\cdots\!40\)\( \)
\(\beta_{13}\)\(=\)\((\)\(121050143818653439 \nu^{19} + 411334189625514926 \nu^{18} + 37618308933791697032 \nu^{17} + 128010308719653010028 \nu^{16} + 4809190015531897327687 \nu^{15} + 16400265011344281684358 \nu^{14} + 326717273907370574715986 \nu^{13} + 1117991628645201606958544 \nu^{12} + 12668415350317937066858860 \nu^{11} + 43604790116852758887558760 \nu^{10} + 278085150493831543051808360 \nu^{9} + 967884648165914613048943040 \nu^{8} + 3167401967644652020711142032 \nu^{7} + 11303569522795522148333302048 \nu^{6} + 14560441058589718775389748960 \nu^{5} + 56164535123832821763599732480 \nu^{4} + 8839202541452905799355428416 \nu^{3} + 63700725795255510798927309184 \nu^{2} - 2295294303475031674079710848 \nu + 12319861870113438089113151488\)\()/ \)\(35\!\cdots\!60\)\( \)
\(\beta_{14}\)\(=\)\((\)\(591265350194 \nu^{19} + 2017790345511 \nu^{18} + 184042545216882 \nu^{17} + 628223320756848 \nu^{16} + 23582352878079222 \nu^{15} + 80522028966201423 \nu^{14} + 1607662588426193766 \nu^{13} + 5491584377435049834 \nu^{12} + 62695549436473071480 \nu^{11} + 214281648524838627420 \nu^{10} + 1391023923244685173320 \nu^{9} + 4758365864480922487560 \nu^{8} + 16225532198550195704352 \nu^{7} + 55588710901764743821968 \nu^{6} + 80284300413322121751840 \nu^{5} + 276054036341206957465440 \nu^{4} + 88620353232289796251776 \nu^{3} + 310051998547249722517824 \nu^{2} + 17909917778953798297472 \nu + 65000338230461979600768\)\()/ \)\(72\!\cdots\!20\)\( \)
\(\beta_{15}\)\(=\)\((\)\(46923190494141031 \nu^{19} + 113800607138337032 \nu^{18} + 14575454605113227688 \nu^{17} + 35365450605069936036 \nu^{16} + 1862320242455101574583 \nu^{15} + 4522711894335316175136 \nu^{14} + 126439740281669458545474 \nu^{13} + 307566713067154013512548 \nu^{12} + 4899754606008997576930740 \nu^{11} + 11954374745705226727770480 \nu^{10} + 107526303501384165237426360 \nu^{9} + 263840788333577182553956560 \nu^{8} + 1226090699673261796198281168 \nu^{7} + 3045973931447074122661481856 \nu^{6} + 5678585512853376074614969440 \nu^{5} + 14638431903751938599418279360 \nu^{4} + 3782581016298178200064891584 \nu^{3} + 13395934374510579339993093888 \nu^{2} - 1587449179675857197262799232 \nu + 1089784274706924341524918016\)\()/ \)\(28\!\cdots\!40\)\( \)
\(\beta_{16}\)\(=\)\((\)\(46923190494141031 \nu^{19} - 113800607138337032 \nu^{18} + 14575454605113227688 \nu^{17} - 35365450605069936036 \nu^{16} + 1862320242455101574583 \nu^{15} - 4522711894335316175136 \nu^{14} + 126439740281669458545474 \nu^{13} - 307566713067154013512548 \nu^{12} + 4899754606008997576930740 \nu^{11} - 11954374745705226727770480 \nu^{10} + 107526303501384165237426360 \nu^{9} - 263840788333577182553956560 \nu^{8} + 1226090699673261796198281168 \nu^{7} - 3045973931447074122661481856 \nu^{6} + 5678585512853376074614969440 \nu^{5} - 14638431903751938599418279360 \nu^{4} + 3782581016298178200064891584 \nu^{3} - 13395934374510579339993093888 \nu^{2} - 1587449179675857197262799232 \nu - 1089784274706924341524918016\)\()/ \)\(28\!\cdots\!40\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-1876318263801913681 \nu^{19} + 6401365045121019024 \nu^{18} - 583990639501122257808 \nu^{17} + 1992244849505931789852 \nu^{16} - 74829088275799721570673 \nu^{15} + 255250979141020239287352 \nu^{14} - 5101958027865246741835494 \nu^{13} + 17401021928940048712148076 \nu^{12} - 199047930993887746862049900 \nu^{11} + 678731331763579731660499440 \nu^{10} - 4420570902006857061692784360 \nu^{9} + 15067292474291252064675857520 \nu^{8} - 51682656031465583258010349488 \nu^{7} + 175993379013538152467116154112 \nu^{6} - 257447174091649333472825029920 \nu^{5} + 874318199166113671728434990400 \nu^{4} - 294880806467787523925439230784 \nu^{3} + 986717366483611891208194419456 \nu^{2} - 70955825940271741602977488768 \nu + 210350045176633647319522221312\)\()/ \)\(10\!\cdots\!80\)\( \)
\(\beta_{18}\)\(=\)\((\)\(2000269874855886615 \nu^{19} + 14543343343273244482 \nu^{18} + 622619185319719817340 \nu^{17} + 4526688157166118773076 \nu^{16} + 79783045543299366219615 \nu^{15} + 580039479385859637483546 \nu^{14} + 5439705018476709705082950 \nu^{13} + 39548047259550692917068048 \nu^{12} + 212198439817042536720071940 \nu^{11} + 1542843318350838455301416520 \nu^{10} + 4710845735738428372513529160 \nu^{9} + 34257993519900611693365023840 \nu^{8} + 55022714404527657389307049680 \nu^{7} + 400314399844585975343774530656 \nu^{6} + 273322951995571674895145241120 \nu^{5} + 1990637048537094821881881523200 \nu^{4} + 309231973754864725402272154560 \nu^{3} + 2254298596220350558248046025088 \nu^{2} + 73074975226234697843670460800 \nu + 482840339335893912162836179456\)\()/ \)\(10\!\cdots\!80\)\( \)
\(\beta_{19}\)\(=\)\((\)\(2000269874855886615 \nu^{19} - 14543343343273244482 \nu^{18} + 622619185319719817340 \nu^{17} - 4526688157166118773076 \nu^{16} + 79783045543299366219615 \nu^{15} - 580039479385859637483546 \nu^{14} + 5439705018476709705082950 \nu^{13} - 39548047259550692917068048 \nu^{12} + 212198439817042536720071940 \nu^{11} - 1542843318350838455301416520 \nu^{10} + 4710845735738428372513529160 \nu^{9} - 34257993519900611693365023840 \nu^{8} + 55022714404527657389307049680 \nu^{7} - 400314399844585975343774530656 \nu^{6} + 273322951995571674895145241120 \nu^{5} - 1990637048537094821881881523200 \nu^{4} + 309231973754864725402272154560 \nu^{3} - 2254298596220350558248046025088 \nu^{2} + 73074975226234697843670460800 \nu - 482840339335893912162836179456\)\()/ \)\(10\!\cdots\!80\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} - 35\)
\(\nu^{3}\)\(=\)\((\)\(-10 \beta_{19} - 10 \beta_{18} - 12 \beta_{17} - \beta_{16} - \beta_{15} + 32 \beta_{14} + 10 \beta_{13} - 30 \beta_{12} - 30 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + 62 \beta_{6} - 6 \beta_{5} - 16 \beta_{4} - 5 \beta_{3} - 50 \beta_{1} - 31\)\()/2\)
\(\nu^{4}\)\(=\)\(18 \beta_{19} - 18 \beta_{18} + 63 \beta_{12} - 63 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} - 53 \beta_{8} - 53 \beta_{7} - 80 \beta_{5} + 169 \beta_{4} + 62 \beta_{3} - 42 \beta_{2} + 21 \beta_{1} + 1842\)
\(\nu^{5}\)\(=\)\((\)\(1100 \beta_{19} + 1100 \beta_{18} + 1128 \beta_{17} + 17 \beta_{16} + 17 \beta_{15} - 3748 \beta_{14} - 1046 \beta_{13} + 3750 \beta_{12} + 3750 \beta_{11} - 88 \beta_{10} - 88 \beta_{9} + 77 \beta_{8} - 77 \beta_{7} - 10114 \beta_{6} + 564 \beta_{5} + 1874 \beta_{4} + 523 \beta_{3} + 2976 \beta_{1} + 5057\)\()/2\)
\(\nu^{6}\)\(=\)\(-1974 \beta_{19} + 1974 \beta_{18} + 60 \beta_{16} - 60 \beta_{15} - 8631 \beta_{12} + 8631 \beta_{11} - 750 \beta_{10} + 750 \beta_{9} + 3021 \beta_{8} + 3021 \beta_{7} + 5700 \beta_{5} - 14265 \beta_{4} - 4530 \beta_{3} + 5838 \beta_{2} - 2919 \beta_{1} - 111490\)
\(\nu^{7}\)\(=\)\((\)\(-97956 \beta_{19} - 97956 \beta_{18} - 90744 \beta_{17} + 2103 \beta_{16} + 2103 \beta_{15} + 353412 \beta_{14} + 93942 \beta_{13} - 371286 \beta_{12} - 371286 \beta_{11} - 5232 \beta_{10} - 5232 \beta_{9} - 6333 \beta_{8} + 6333 \beta_{7} + 1042722 \beta_{6} - 45372 \beta_{5} - 176706 \beta_{4} - 46971 \beta_{3} - 199300 \beta_{1} - 521361\)\()/2\)
\(\nu^{8}\)\(=\)\(179598 \beta_{19} - 179598 \beta_{18} - 9222 \beta_{16} + 9222 \beta_{15} + 895419 \beta_{12} - 895419 \beta_{11} + 77142 \beta_{10} - 77142 \beta_{9} - 182743 \beta_{8} - 182743 \beta_{7} - 405400 \beta_{5} + 1198325 \beta_{4} + 367132 \beta_{3} - 598326 \beta_{2} + 299163 \beta_{1} + 7419740\)
\(\nu^{9}\)\(=\)\((\)\(8250220 \beta_{19} + 8250220 \beta_{18} + 7048872 \beta_{17} - 351923 \beta_{16} - 351923 \beta_{15} - 31207316 \beta_{14} - 8059582 \beta_{13} + 33815622 \beta_{12} + 33815622 \beta_{11} + 1412056 \beta_{10} + 1412056 \beta_{9} + 499273 \beta_{8} - 499273 \beta_{7} - 93245210 \beta_{6} + 3524436 \beta_{5} + 15603658 \beta_{4} + 4029791 \beta_{3} + 14446484 \beta_{1} + 46622605\)\()/2\)
\(\nu^{10}\)\(=\)\(-15483054 \beta_{19} + 15483054 \beta_{18} + 1043268 \beta_{16} - 1043268 \beta_{15} - 84157299 \beta_{12} + 84157299 \beta_{11} - 7363710 \beta_{10} + 7363710 \beta_{9} + 11639237 \beta_{8} + 11639237 \beta_{7} + 29450720 \beta_{5} - 100383541 \beta_{4} - 30810710 \beta_{3} + 54858558 \beta_{2} - 27429279 \beta_{1} - 528154590\)
\(\nu^{11}\)\(=\)\((\)\(-681819932 \beta_{19} - 681819932 \beta_{18} - 544782600 \beta_{17} + 38507695 \beta_{16} + 38507695 \beta_{15} + 2680675060 \beta_{14} + 678771830 \beta_{13} - 2967706950 \beta_{12} - 2967706950 \beta_{11} - 180638312 \beta_{10} - 180638312 \beta_{9} - 37581509 \beta_{8} + 37581509 \beta_{7} + 7882682146 \beta_{6} - 272391300 \beta_{5} - 1340337530 \beta_{4} - 339385915 \beta_{3} - 1101052764 \beta_{1} - 3941341073\)\()/2\)
\(\nu^{12}\)\(=\)\(1305733614 \beta_{19} - 1305733614 \beta_{18} - 104415330 \beta_{16} + 104415330 \beta_{15} + 7540243515 \beta_{12} - 7540243515 \beta_{11} + 674256678 \beta_{10} - 674256678 \beta_{9} - 777491427 \beta_{8} - 777491427 \beta_{7} - 2193191592 \beta_{5} + 8390623941 \beta_{4} + 2600235312 \beta_{3} - 4779625926 \beta_{2} + 2389812963 \beta_{1} + 39393611536\)
\(\nu^{13}\)\(=\)\((\)\(55973553228 \beta_{19} + 55973553228 \beta_{18} + 42315645288 \beta_{17} - 3702226683 \beta_{16} - 3702226683 \beta_{15} - 227003545620 \beta_{14} - 56691373518 \beta_{13} + 255326171238 \beta_{12} + 255326171238 \beta_{11} + 18820408632 \beta_{10} + 18820408632 \beta_{9} + 2742337137 \beta_{8} - 2742337137 \beta_{7} - 650858536458 \beta_{6} + 21157822644 \beta_{5} + 113501772810 \beta_{4} + 28345686759 \beta_{3} + 86537507380 \beta_{1} + 325429268229\)\()/2\)
\(\nu^{14}\)\(=\)\(-108956680782 \beta_{19} + 108956680782 \beta_{18} + 9784553520 \beta_{16} - 9784553520 \beta_{15} - 657307017459 \beta_{12} + 657307017459 \beta_{11} - 60075860334 \beta_{10} + 60075860334 \beta_{9} + 54230318545 \beta_{8} + 54230318545 \beta_{7} + 167049003904 \beta_{5} - 699814808453 \beta_{4} - 218740170058 \beta_{3} + 405863060142 \beta_{2} - 202931530071 \beta_{1} - 3033275040866\)
\(\nu^{15}\)\(=\)\((\)\(-4586257361116 \beta_{19} - 4586257361116 \beta_{18} - 3314535601416 \beta_{17} + 335536331591 \beta_{16} + 335536331591 \beta_{15} + 19060532312180 \beta_{14} + 4715327528038 \beta_{13} - 21699753663558 \beta_{12} - 21699753663558 \beta_{11} - 1785382532968 \beta_{10} - 1785382532968 \beta_{9} - 196748133277 \beta_{8} + 196748133277 \beta_{7} + 53238713976242 \beta_{6} - 1657267800708 \beta_{5} - 9530266156090 \beta_{4} - 2357663764019 \beta_{3} - 6928948435628 \beta_{1} - 26619356988121\)\()/2\)
\(\nu^{16}\)\(=\)\(9041754045102 \beta_{19} - 9041754045102 \beta_{18} - 880632614766 \beta_{16} + 880632614766 \beta_{15} + 56317818830859 \beta_{12} - 56317818830859 \beta_{11} + 5249603574966 \beta_{10} - 5249603574966 \beta_{9} - 3927396956303 \beta_{8} - 3927396956303 \beta_{7} - 12963237581336 \beta_{5} + 58245495459109 \beta_{4} + 18314137994036 \beta_{3} - 33998723787222 \beta_{2} + 16999361893611 \beta_{1} + 238630955366916\)
\(\nu^{17}\)\(=\)\((\)\(375766067065196 \beta_{19} + 375766067065196 \beta_{18} + 261970896361128 \beta_{17} - 29446492548307 \beta_{16} - 29446492548307 \beta_{15} - 1591659506895700 \beta_{14} - 391275618683006 \beta_{13} + 1829118195403494 \beta_{12} + 1829118195403494 \beta_{11} + 160972937258264 \beta_{10} + 160972937258264 \beta_{9} + 14030010267209 \beta_{8} - 14030010267209 \beta_{7} - 4342155764191450 \beta_{6} + 130985448180564 \beta_{5} + 795829753447850 \beta_{4} + 195637809341503 \beta_{3} + 561041635179684 \beta_{1} + 2171077882095725\)\()/2\)
\(\nu^{18}\)\(=\)\(-748080482854734 \beta_{19} + 748080482854734 \beta_{18} + 77193638799228 \beta_{16} - 77193638799228 \beta_{15} - 4770231780528387 \beta_{12} + 4770231780528387 \beta_{11} - 452156686325118 \beta_{10} + 452156686325118 \beta_{9} + 293428566573885 \beta_{8} + 293428566573885 \beta_{7} + 1020914500008240 \beta_{5} - 4838593614065829 \beta_{4} - 1526989652431758 \beta_{3} + 2827118435526270 \beta_{2} - 1413559217763135 \beta_{1} - 19047337071126742\)
\(\nu^{19}\)\(=\)\((\)\(-30808515673522044 \beta_{19} - 30808515673522044 \beta_{18} - 20879550774187848 \beta_{17} + 2534355010929375 \beta_{16} + 2534355010929375 \beta_{15} + 132416817935418036 \beta_{14} + 32417550098814870 \beta_{13} - 153286623665057862 \beta_{12} - 153286623665057862 \beta_{11} - 14082375565754568 \beta_{10} - 14082375565754568 \beta_{9} - 1002480559625781 \beta_{8} + 1002480559625781 \beta_{7} + 354144191585896962 \beta_{6} - 10439775387093924 \beta_{5} - 66208408967709018 \beta_{4} - 16208775049407435 \beta_{3} - 45737159781279196 \beta_{1} - 177072095792948481\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
6.77277i
5.89468i
0.522574i
3.11082i
9.07921i
7.81115i
5.08624i
1.08183i
5.89146i
7.08776i
6.77277i
5.89468i
0.522574i
3.11082i
9.07921i
7.81115i
5.08624i
1.08183i
5.89146i
7.08776i
−1.73205 + 1.00000i −3.38639 + 5.86539i 2.00000 3.46410i 14.5934 + 8.42549i 13.5455i −7.56088 + 13.0958i 8.00000i −9.43523 16.3423i −33.7019
11.2 −1.73205 + 1.00000i −2.94734 + 5.10494i 2.00000 3.46410i −12.8210 7.40221i 11.7894i 7.99249 13.8434i 8.00000i −3.87362 6.70931i 29.6089
11.3 −1.73205 + 1.00000i −0.261287 + 0.452562i 2.00000 3.46410i 2.77235 + 1.60061i 1.04515i 8.06150 13.9629i 8.00000i 13.3635 + 23.1462i −6.40246
11.4 −1.73205 + 1.00000i 1.55541 2.69405i 2.00000 3.46410i −12.2401 7.06683i 6.22164i −13.5058 + 23.3927i 8.00000i 8.66140 + 15.0020i 28.2673
11.5 −1.73205 + 1.00000i 4.53960 7.86282i 2.00000 3.46410i 4.92746 + 2.84487i 18.1584i 4.51268 7.81619i 8.00000i −27.7160 48.0055i −11.3795
11.6 1.73205 1.00000i −3.90558 + 6.76466i 2.00000 3.46410i −18.8983 10.9109i 15.6223i −2.61919 + 4.53657i 8.00000i −17.0071 29.4571i −43.6437
11.7 1.73205 1.00000i −2.54312 + 4.40481i 2.00000 3.46410i 9.68300 + 5.59048i 10.1725i −10.2076 + 17.6800i 8.00000i 0.565092 + 0.978768i 22.3619
11.8 1.73205 1.00000i −0.540917 + 0.936896i 2.00000 3.46410i 2.09330 + 1.20857i 2.16367i 17.5687 30.4298i 8.00000i 12.9148 + 22.3691i 4.83427
11.9 1.73205 1.00000i 2.94573 5.10216i 2.00000 3.46410i −12.2374 7.06527i 11.7829i 0.0476865 0.0825955i 8.00000i −3.85468 6.67649i −28.2611
11.10 1.73205 1.00000i 3.54388 6.13818i 2.00000 3.46410i 13.1273 + 7.57908i 14.1755i −5.28959 + 9.16184i 8.00000i −11.6182 20.1233i 30.3163
27.1 −1.73205 1.00000i −3.38639 5.86539i 2.00000 + 3.46410i 14.5934 8.42549i 13.5455i −7.56088 13.0958i 8.00000i −9.43523 + 16.3423i −33.7019
27.2 −1.73205 1.00000i −2.94734 5.10494i 2.00000 + 3.46410i −12.8210 + 7.40221i 11.7894i 7.99249 + 13.8434i 8.00000i −3.87362 + 6.70931i 29.6089
27.3 −1.73205 1.00000i −0.261287 0.452562i 2.00000 + 3.46410i 2.77235 1.60061i 1.04515i 8.06150 + 13.9629i 8.00000i 13.3635 23.1462i −6.40246
27.4 −1.73205 1.00000i 1.55541 + 2.69405i 2.00000 + 3.46410i −12.2401 + 7.06683i 6.22164i −13.5058 23.3927i 8.00000i 8.66140 15.0020i 28.2673
27.5 −1.73205 1.00000i 4.53960 + 7.86282i 2.00000 + 3.46410i 4.92746 2.84487i 18.1584i 4.51268 + 7.81619i 8.00000i −27.7160 + 48.0055i −11.3795
27.6 1.73205 + 1.00000i −3.90558 6.76466i 2.00000 + 3.46410i −18.8983 + 10.9109i 15.6223i −2.61919 4.53657i 8.00000i −17.0071 + 29.4571i −43.6437
27.7 1.73205 + 1.00000i −2.54312 4.40481i 2.00000 + 3.46410i 9.68300 5.59048i 10.1725i −10.2076 17.6800i 8.00000i 0.565092 0.978768i 22.3619
27.8 1.73205 + 1.00000i −0.540917 0.936896i 2.00000 + 3.46410i 2.09330 1.20857i 2.16367i 17.5687 + 30.4298i 8.00000i 12.9148 22.3691i 4.83427
27.9 1.73205 + 1.00000i 2.94573 + 5.10216i 2.00000 + 3.46410i −12.2374 + 7.06527i 11.7829i 0.0476865 + 0.0825955i 8.00000i −3.85468 + 6.67649i −28.2611
27.10 1.73205 + 1.00000i 3.54388 + 6.13818i 2.00000 + 3.46410i 13.1273 7.57908i 14.1755i −5.28959 9.16184i 8.00000i −11.6182 + 20.1233i 30.3163
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.e.a 20
3.b odd 2 1 666.4.s.d 20
37.e even 6 1 inner 74.4.e.a 20
111.h odd 6 1 666.4.s.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.e.a 20 1.a even 1 1 trivial
74.4.e.a 20 37.e even 6 1 inner
666.4.s.d 20 3.b odd 2 1
666.4.s.d 20 111.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 4 T^{2} + T^{4} )^{5} \)
$3$ \( 1118416232704 + 3061621500416 T + 6866978351872 T^{2} + 5355016961536 T^{3} + 3940412787584 T^{4} + 490411982848 T^{5} + 589740706688 T^{6} + 80132629504 T^{7} + 47570253680 T^{8} + 4448347360 T^{9} + 2138449808 T^{10} + 169104448 T^{11} + 67210808 T^{12} + 3798544 T^{13} + 1370216 T^{14} + 57664 T^{15} + 20177 T^{16} + 490 T^{17} + 175 T^{18} + 2 T^{19} + T^{20} \)
$5$ \( 65813428156638380625 - 98227787402699237250 T + 62372477555676944775 T^{2} - 20154279362332952790 T^{3} + 2997629432184549879 T^{4} + 7485550660667412 T^{5} - 54394621214170446 T^{6} + 2760946778190996 T^{7} + 665949921488029 T^{8} - 47243361956166 T^{9} - 4956786956871 T^{10} + 387210722166 T^{11} + 29543086413 T^{12} - 2030710644 T^{13} - 120013486 T^{14} + 7025868 T^{15} + 383175 T^{16} - 15066 T^{17} - 729 T^{18} + 18 T^{19} + T^{20} \)
$7$ \( 12975993769784131584 - \)\(13\!\cdots\!08\)\( T + \)\(14\!\cdots\!16\)\( T^{2} + \)\(24\!\cdots\!20\)\( T^{3} + 74164308425166327808 T^{4} + 4252882214989774848 T^{5} + 1354340937446070272 T^{6} + 70073272653435904 T^{7} + 15388103102744960 T^{8} + 491142438948864 T^{9} + 96084713113856 T^{10} + 2614373191776 T^{11} + 419827203088 T^{12} + 8028386112 T^{13} + 1118150404 T^{14} + 18290960 T^{15} + 2037525 T^{16} + 16274 T^{17} + 1675 T^{18} + 2 T^{19} + T^{20} \)
$11$ \( ( -1734696000000 + 1033734960000 T - 29308791600 T^{2} - 22880148480 T^{3} - 460586208 T^{4} + 99382464 T^{5} + 3605368 T^{6} - 87968 T^{7} - 4216 T^{8} + 8 T^{9} + T^{10} )^{2} \)
$13$ \( \)\(32\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T + \)\(24\!\cdots\!00\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!36\)\( T^{4} + \)\(63\!\cdots\!76\)\( T^{5} + \)\(99\!\cdots\!36\)\( T^{6} - \)\(19\!\cdots\!76\)\( T^{7} - \)\(95\!\cdots\!00\)\( T^{8} + 38569799785787935488 T^{9} + 7601613815695111424 T^{10} + 86582743274186304 T^{11} - 2024281429187424 T^{12} - 38401644810816 T^{13} + 444309884704 T^{14} + 12585167316 T^{15} - 101679 T^{16} - 1571550 T^{17} - 2977 T^{18} + 150 T^{19} + T^{20} \)
$17$ \( \)\(15\!\cdots\!81\)\( + \)\(17\!\cdots\!78\)\( T + \)\(67\!\cdots\!51\)\( T^{2} + \)\(44\!\cdots\!94\)\( T^{3} + \)\(16\!\cdots\!03\)\( T^{4} - \)\(64\!\cdots\!16\)\( T^{5} - \)\(20\!\cdots\!14\)\( T^{6} + \)\(78\!\cdots\!48\)\( T^{7} + \)\(17\!\cdots\!41\)\( T^{8} - \)\(28\!\cdots\!58\)\( T^{9} - \)\(40\!\cdots\!87\)\( T^{10} - 1168090071485085822 T^{11} + 67308631445897773 T^{12} + 362757814130196 T^{13} - 6830918570526 T^{14} - 40527154428 T^{15} + 547317527 T^{16} + 2738610 T^{17} - 27729 T^{18} - 90 T^{19} + T^{20} \)
$19$ \( \)\(36\!\cdots\!00\)\( + \)\(11\!\cdots\!00\)\( T + \)\(14\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(36\!\cdots\!84\)\( T^{4} + \)\(50\!\cdots\!40\)\( T^{5} - \)\(69\!\cdots\!16\)\( T^{6} - \)\(27\!\cdots\!20\)\( T^{7} + \)\(47\!\cdots\!12\)\( T^{8} + \)\(79\!\cdots\!44\)\( T^{9} + \)\(11\!\cdots\!76\)\( T^{10} - 15324881925718435104 T^{11} - 4147331386810728 T^{12} + 1919701794530064 T^{13} - 2018348378720 T^{14} - 167086588512 T^{15} + 649420305 T^{16} + 5995458 T^{17} - 28261 T^{18} - 162 T^{19} + T^{20} \)
$23$ \( \)\(37\!\cdots\!56\)\( + \)\(10\!\cdots\!40\)\( T^{2} + \)\(98\!\cdots\!36\)\( T^{4} + \)\(47\!\cdots\!32\)\( T^{6} + \)\(12\!\cdots\!88\)\( T^{8} + \)\(21\!\cdots\!96\)\( T^{10} + 2178697850551291488 T^{12} + 140832851771200 T^{14} + 5477548512 T^{16} + 115688 T^{18} + T^{20} \)
$29$ \( \)\(59\!\cdots\!84\)\( + \)\(17\!\cdots\!72\)\( T^{2} + \)\(12\!\cdots\!45\)\( T^{4} + \)\(36\!\cdots\!68\)\( T^{6} + \)\(57\!\cdots\!36\)\( T^{8} + \)\(54\!\cdots\!56\)\( T^{10} + 30594335148070022374 T^{12} + 1051784643360136 T^{14} + 21223213500 T^{16} + 228376 T^{18} + T^{20} \)
$31$ \( \)\(80\!\cdots\!36\)\( + \)\(21\!\cdots\!20\)\( T^{2} + \)\(11\!\cdots\!24\)\( T^{4} + \)\(25\!\cdots\!80\)\( T^{6} + \)\(31\!\cdots\!72\)\( T^{8} + \)\(22\!\cdots\!16\)\( T^{10} + 96332217208955906400 T^{12} + 2451525320151296 T^{14} + 36674315424 T^{16} + 296664 T^{18} + T^{20} \)
$37$ \( \)\(11\!\cdots\!49\)\( + \)\(24\!\cdots\!96\)\( T + \)\(32\!\cdots\!54\)\( T^{2} + \)\(21\!\cdots\!92\)\( T^{3} + \)\(67\!\cdots\!27\)\( T^{4} + \)\(26\!\cdots\!92\)\( T^{5} + \)\(69\!\cdots\!68\)\( T^{6} - \)\(40\!\cdots\!60\)\( T^{7} - \)\(35\!\cdots\!91\)\( T^{8} - \)\(22\!\cdots\!92\)\( T^{9} - \)\(11\!\cdots\!70\)\( T^{10} - \)\(44\!\cdots\!64\)\( T^{11} - 13842912675786010499 T^{12} - 30789223370124980 T^{13} + 105001304342728 T^{14} + 788755444344 T^{15} + 3977580363 T^{16} + 25238916 T^{17} + 75114 T^{18} + 112 T^{19} + T^{20} \)
$41$ \( \)\(40\!\cdots\!09\)\( + \)\(13\!\cdots\!98\)\( T + \)\(32\!\cdots\!71\)\( T^{2} + \)\(48\!\cdots\!90\)\( T^{3} + \)\(61\!\cdots\!03\)\( T^{4} + \)\(62\!\cdots\!12\)\( T^{5} + \)\(58\!\cdots\!62\)\( T^{6} + \)\(45\!\cdots\!16\)\( T^{7} + \)\(33\!\cdots\!65\)\( T^{8} + \)\(19\!\cdots\!86\)\( T^{9} + \)\(10\!\cdots\!21\)\( T^{10} + \)\(43\!\cdots\!74\)\( T^{11} + \)\(17\!\cdots\!13\)\( T^{12} + 5396962545287593668 T^{13} + 17826980387396874 T^{14} + 43753468851900 T^{15} + 116451663135 T^{16} + 184185586 T^{17} + 445035 T^{18} + 498 T^{19} + T^{20} \)
$43$ \( \)\(55\!\cdots\!44\)\( + \)\(16\!\cdots\!80\)\( T^{2} + \)\(13\!\cdots\!28\)\( T^{4} + \)\(56\!\cdots\!08\)\( T^{6} + \)\(13\!\cdots\!40\)\( T^{8} + \)\(20\!\cdots\!16\)\( T^{10} + \)\(20\!\cdots\!76\)\( T^{12} + 126163298944722304 T^{14} + 489157749536 T^{16} + 1067392 T^{18} + T^{20} \)
$47$ \( ( \)\(49\!\cdots\!76\)\( + \)\(23\!\cdots\!52\)\( T + \)\(32\!\cdots\!92\)\( T^{2} + 26521876295925024 T^{3} - 10874476853713280 T^{4} - 9492724383056 T^{5} + 126513827864 T^{6} + 87780616 T^{7} - 600448 T^{8} - 212 T^{9} + T^{10} )^{2} \)
$53$ \( \)\(14\!\cdots\!16\)\( - \)\(25\!\cdots\!76\)\( T + \)\(47\!\cdots\!00\)\( T^{2} + \)\(70\!\cdots\!04\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} + \)\(55\!\cdots\!56\)\( T^{5} + \)\(82\!\cdots\!24\)\( T^{6} + \)\(19\!\cdots\!32\)\( T^{7} + \)\(38\!\cdots\!48\)\( T^{8} + \)\(23\!\cdots\!80\)\( T^{9} + \)\(49\!\cdots\!84\)\( T^{10} + \)\(23\!\cdots\!80\)\( T^{11} + \)\(46\!\cdots\!84\)\( T^{12} - 4699757062499524992 T^{13} + 197711171382399328 T^{14} + 12706997636736 T^{15} + 600741504769 T^{16} + 31656302 T^{17} + 939891 T^{18} + 142 T^{19} + T^{20} \)
$59$ \( \)\(19\!\cdots\!04\)\( + \)\(10\!\cdots\!96\)\( T + \)\(17\!\cdots\!36\)\( T^{2} - \)\(14\!\cdots\!68\)\( T^{3} - \)\(20\!\cdots\!40\)\( T^{4} + \)\(38\!\cdots\!96\)\( T^{5} + \)\(56\!\cdots\!80\)\( T^{6} - \)\(43\!\cdots\!20\)\( T^{7} - \)\(51\!\cdots\!56\)\( T^{8} + \)\(31\!\cdots\!96\)\( T^{9} + \)\(37\!\cdots\!60\)\( T^{10} - \)\(14\!\cdots\!80\)\( T^{11} - \)\(12\!\cdots\!64\)\( T^{12} + \)\(48\!\cdots\!20\)\( T^{13} + 288789897442932864 T^{14} - 1157218599549552 T^{15} - 160371388255 T^{16} + 1678634550 T^{17} - 213045 T^{18} - 1590 T^{19} + T^{20} \)
$61$ \( \)\(65\!\cdots\!69\)\( - \)\(19\!\cdots\!06\)\( T + \)\(18\!\cdots\!15\)\( T^{2} + \)\(14\!\cdots\!42\)\( T^{3} - \)\(10\!\cdots\!73\)\( T^{4} + \)\(11\!\cdots\!36\)\( T^{5} + \)\(41\!\cdots\!38\)\( T^{6} - \)\(87\!\cdots\!00\)\( T^{7} - \)\(83\!\cdots\!31\)\( T^{8} + \)\(13\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!77\)\( T^{10} + \)\(28\!\cdots\!86\)\( T^{11} - \)\(80\!\cdots\!59\)\( T^{12} - 58528050970692962460 T^{13} + 369433033308215386 T^{14} + 629443128432612 T^{15} - 235160948929 T^{16} - 1054051062 T^{17} + 109027 T^{18} + 1542 T^{19} + T^{20} \)
$67$ \( \)\(61\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( T + \)\(25\!\cdots\!00\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!44\)\( T^{4} - \)\(54\!\cdots\!24\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(70\!\cdots\!08\)\( T^{7} + \)\(39\!\cdots\!36\)\( T^{8} - \)\(55\!\cdots\!84\)\( T^{9} + \)\(30\!\cdots\!44\)\( T^{10} - \)\(25\!\cdots\!56\)\( T^{11} + \)\(17\!\cdots\!60\)\( T^{12} - 71400073576609095856 T^{13} + 546160346706020648 T^{14} - 81004231935424 T^{15} + 1255203988817 T^{16} - 65728054 T^{17} + 1332415 T^{18} - 62 T^{19} + T^{20} \)
$71$ \( \)\(17\!\cdots\!04\)\( - \)\(27\!\cdots\!12\)\( T + \)\(34\!\cdots\!32\)\( T^{2} - \)\(20\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!76\)\( T^{4} - \)\(31\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!64\)\( T^{6} - \)\(28\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!28\)\( T^{8} - \)\(15\!\cdots\!16\)\( T^{9} + \)\(47\!\cdots\!40\)\( T^{10} - \)\(46\!\cdots\!16\)\( T^{11} + \)\(15\!\cdots\!32\)\( T^{12} - \)\(88\!\cdots\!48\)\( T^{13} + 2697714218832074924 T^{14} - 548704510153904 T^{15} + 3336864448333 T^{16} - 236789998 T^{17} + 2233459 T^{18} + 178 T^{19} + T^{20} \)
$73$ \( ( \)\(60\!\cdots\!00\)\( + \)\(24\!\cdots\!40\)\( T + \)\(11\!\cdots\!76\)\( T^{2} - 18220827366103113728 T^{3} - 38970079624631552 T^{4} + 205531195220608 T^{5} + 455929160608 T^{6} - 497165200 T^{7} - 1295940 T^{8} + 264 T^{9} + T^{10} )^{2} \)
$79$ \( \)\(35\!\cdots\!04\)\( + \)\(76\!\cdots\!96\)\( T - \)\(61\!\cdots\!72\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!36\)\( T^{4} + \)\(16\!\cdots\!84\)\( T^{5} + \)\(10\!\cdots\!36\)\( T^{6} + \)\(40\!\cdots\!24\)\( T^{7} + \)\(78\!\cdots\!40\)\( T^{8} + \)\(34\!\cdots\!76\)\( T^{9} - \)\(15\!\cdots\!72\)\( T^{10} - \)\(11\!\cdots\!88\)\( T^{11} + \)\(86\!\cdots\!88\)\( T^{12} + \)\(22\!\cdots\!16\)\( T^{13} + 1704005149098008816 T^{14} - 1076432023263732 T^{15} - 1975192795287 T^{16} + 1298216430 T^{17} + 4396587 T^{18} + 3474 T^{19} + T^{20} \)
$83$ \( \)\(55\!\cdots\!84\)\( - \)\(45\!\cdots\!36\)\( T + \)\(64\!\cdots\!60\)\( T^{2} - \)\(31\!\cdots\!24\)\( T^{3} + \)\(37\!\cdots\!68\)\( T^{4} - \)\(17\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} - \)\(21\!\cdots\!16\)\( T^{7} + \)\(70\!\cdots\!32\)\( T^{8} - \)\(90\!\cdots\!24\)\( T^{9} + \)\(26\!\cdots\!00\)\( T^{10} - \)\(27\!\cdots\!64\)\( T^{11} + \)\(56\!\cdots\!16\)\( T^{12} - \)\(40\!\cdots\!68\)\( T^{13} + 7390000134370529816 T^{14} - 4302252360046640 T^{15} + 6417179206417 T^{16} - 2390340802 T^{17} + 3303799 T^{18} - 938 T^{19} + T^{20} \)
$89$ \( \)\(12\!\cdots\!25\)\( - \)\(17\!\cdots\!50\)\( T - \)\(79\!\cdots\!25\)\( T^{2} + \)\(12\!\cdots\!30\)\( T^{3} + \)\(21\!\cdots\!11\)\( T^{4} - \)\(56\!\cdots\!92\)\( T^{5} - \)\(45\!\cdots\!86\)\( T^{6} + \)\(15\!\cdots\!68\)\( T^{7} + \)\(38\!\cdots\!37\)\( T^{8} - \)\(66\!\cdots\!22\)\( T^{9} - \)\(26\!\cdots\!35\)\( T^{10} + \)\(24\!\cdots\!06\)\( T^{11} + \)\(13\!\cdots\!29\)\( T^{12} - \)\(23\!\cdots\!60\)\( T^{13} - 2810016880717663014 T^{14} + 19568232844428 T^{15} + 4401009150911 T^{16} + 1225179630 T^{17} - 2315613 T^{18} - 510 T^{19} + T^{20} \)
$97$ \( \)\(12\!\cdots\!04\)\( + \)\(22\!\cdots\!80\)\( T^{2} + \)\(11\!\cdots\!57\)\( T^{4} + \)\(16\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{8} + \)\(38\!\cdots\!64\)\( T^{10} + \)\(71\!\cdots\!54\)\( T^{12} + 70473180382883834400 T^{14} + 37052000690068 T^{16} + 9730832 T^{18} + T^{20} \)
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