Properties

Label 63.4.f.b
Level $63$
Weight $4$
Character orbit 63.f
Analytic conductor $3.717$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + 599392 x^{8} - 1089732 x^{7} + 4808401 x^{6} - 7939134 x^{5} + 26225236 x^{4} - 39450864 x^{3} + 62254768 x^{2} - 39660672 x + 21307456\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{7} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -\beta_{3} - \beta_{7} ) q^{3} + ( 5 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{4} + ( 4 \beta_{3} - \beta_{8} - \beta_{13} ) q^{5} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{14} ) q^{6} + ( 7 + 7 \beta_{3} ) q^{7} + ( -2 + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} + 2 \beta_{14} ) q^{8} + ( -8 - 2 \beta_{1} + 3 \beta_{2} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -\beta_{3} - \beta_{7} ) q^{3} + ( 5 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{4} + ( 4 \beta_{3} - \beta_{8} - \beta_{13} ) q^{5} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{14} ) q^{6} + ( 7 + 7 \beta_{3} ) q^{7} + ( -2 + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} + 2 \beta_{14} ) q^{8} + ( -8 - 2 \beta_{1} + 3 \beta_{2} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{9} + ( -3 + 6 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{10} + ( -3 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{11} + ( 15 - \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - \beta_{6} + 5 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{13} - \beta_{15} ) q^{12} + ( -1 + 7 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{13} + ( -7 \beta_{1} + 7 \beta_{2} ) q^{14} + ( 7 + 5 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - \beta_{4} + 5 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} + \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{15} + ( -8 - 5 \beta_{1} + \beta_{2} - 8 \beta_{3} - 2 \beta_{4} + 11 \beta_{5} + 4 \beta_{7} + \beta_{8} - 5 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{16} + ( 19 - 4 \beta_{2} - \beta_{3} - \beta_{4} - 7 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} - \beta_{8} - \beta_{9} + 5 \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{17} + ( -29 + 14 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + 5 \beta_{5} + 8 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} - 4 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{18} + ( 14 - 4 \beta_{1} + 11 \beta_{2} - 5 \beta_{5} - 3 \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{19} + ( -47 + 3 \beta_{1} + 2 \beta_{2} - 52 \beta_{3} + \beta_{4} - 13 \beta_{5} - 5 \beta_{6} - 18 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} - 7 \beta_{10} + 5 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{20} + ( 7 + 7 \beta_{5} ) q^{21} + ( -4 + 4 \beta_{1} - 8 \beta_{2} + 20 \beta_{3} - 5 \beta_{5} + 2 \beta_{6} + 10 \beta_{7} - 4 \beta_{8} + \beta_{9} - 3 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - 5 \beta_{13} + \beta_{14} ) q^{22} + ( 3 + 2 \beta_{1} - 3 \beta_{2} + 26 \beta_{3} + 4 \beta_{4} + 7 \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + 8 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{23} + ( 16 - 4 \beta_{1} - 18 \beta_{2} + 33 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 13 \beta_{6} - 6 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} + 11 \beta_{10} + 5 \beta_{11} - 7 \beta_{12} + 4 \beta_{13} - \beta_{14} + \beta_{15} ) q^{24} + ( -30 - 15 \beta_{1} - 32 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - \beta_{11} + 3 \beta_{12} + 10 \beta_{13} + 4 \beta_{15} ) q^{25} + ( 93 - 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 13 \beta_{6} + 22 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} + \beta_{10} + 3 \beta_{12} + 4 \beta_{13} - 8 \beta_{14} ) q^{26} + ( 4 + 15 \beta_{1} - 7 \beta_{2} - 20 \beta_{3} - \beta_{4} + 8 \beta_{6} + 7 \beta_{7} + \beta_{8} + 3 \beta_{9} + 13 \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} - 7 \beta_{14} + 3 \beta_{15} ) q^{27} + ( -35 + 7 \beta_{6} - 7 \beta_{7} ) q^{28} + ( -75 + \beta_{1} + 3 \beta_{2} - 74 \beta_{3} + 9 \beta_{5} + \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + 8 \beta_{9} + 5 \beta_{10} - 6 \beta_{11} + 9 \beta_{12} - \beta_{13} - 4 \beta_{14} ) q^{29} + ( 78 - 48 \beta_{1} + 18 \beta_{2} + 28 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} - 2 \beta_{7} + 3 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 9 \beta_{11} - 12 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} + 6 \beta_{15} ) q^{30} + ( -1 + 5 \beta_{1} - 16 \beta_{2} + 7 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 10 \beta_{6} + 9 \beta_{7} + \beta_{8} + 6 \beta_{9} + 3 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} + 5 \beta_{15} ) q^{31} + ( 6 + 3 \beta_{1} - \beta_{2} + 92 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 16 \beta_{7} + 6 \beta_{8} + \beta_{9} + 7 \beta_{10} + 6 \beta_{11} - 4 \beta_{12} + 7 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{32} + ( -45 + 20 \beta_{1} - 38 \beta_{2} - 10 \beta_{3} - \beta_{4} + 7 \beta_{5} - 9 \beta_{6} + 10 \beta_{7} - 8 \beta_{8} - \beta_{9} - 8 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{33} + ( 60 - 29 \beta_{1} - 2 \beta_{2} + 69 \beta_{3} - 2 \beta_{4} + 41 \beta_{5} + 9 \beta_{6} + 34 \beta_{7} + 7 \beta_{8} + 12 \beta_{9} + \beta_{10} - 12 \beta_{11} + 8 \beta_{12} - 12 \beta_{13} - 5 \beta_{14} - 4 \beta_{15} ) q^{34} + ( -28 - 7 \beta_{8} ) q^{35} + ( -5 + 42 \beta_{1} + 8 \beta_{2} - 75 \beta_{3} + 6 \beta_{4} - 16 \beta_{5} - 12 \beta_{6} - 22 \beta_{7} - 9 \beta_{8} - 12 \beta_{9} + 3 \beta_{11} - 12 \beta_{12} + 3 \beta_{13} + \beta_{14} + 4 \beta_{15} ) q^{36} + ( 31 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} + 15 \beta_{6} - \beta_{7} + 4 \beta_{8} + \beta_{9} + 3 \beta_{10} + 10 \beta_{11} - \beta_{12} + \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{37} + ( -116 - 18 \beta_{1} + 15 \beta_{2} - 119 \beta_{3} - 9 \beta_{4} + 21 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} + 12 \beta_{8} + 15 \beta_{9} - 15 \beta_{10} - 6 \beta_{11} + 12 \beta_{12} + 11 \beta_{13} - 3 \beta_{14} - 18 \beta_{15} ) q^{38} + ( -8 - 21 \beta_{1} + 24 \beta_{2} + 4 \beta_{3} + 19 \beta_{5} + 10 \beta_{7} - 12 \beta_{8} - 6 \beta_{9} - 9 \beta_{10} - 15 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} - 9 \beta_{15} ) q^{39} + ( -3 + 57 \beta_{1} - 42 \beta_{2} - 63 \beta_{3} - 18 \beta_{4} + 13 \beta_{5} - 16 \beta_{6} + 25 \beta_{7} + 12 \beta_{8} - 3 \beta_{9} - 16 \beta_{10} - 3 \beta_{11} + 15 \beta_{12} + 6 \beta_{13} + 6 \beta_{14} - 9 \beta_{15} ) q^{40} + ( -6 + 62 \beta_{3} - 6 \beta_{4} + 12 \beta_{5} - 18 \beta_{6} + 21 \beta_{7} - 8 \beta_{8} - 6 \beta_{9} - 18 \beta_{10} - 6 \beta_{11} - 5 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} ) q^{41} + ( -14 \beta_{2} + 14 \beta_{3} - 14 \beta_{5} - 7 \beta_{7} - 7 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} - 7 \beta_{12} + 7 \beta_{15} ) q^{42} + ( -8 - 4 \beta_{1} - 16 \beta_{2} - 11 \beta_{3} + 11 \beta_{4} - 40 \beta_{5} - 3 \beta_{6} - 16 \beta_{7} - 19 \beta_{8} - 27 \beta_{9} + 4 \beta_{10} + 15 \beta_{11} - 20 \beta_{12} - 21 \beta_{13} + 8 \beta_{14} + 22 \beta_{15} ) q^{43} + ( 43 - 3 \beta_{1} + 28 \beta_{2} - 5 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} + 15 \beta_{6} - 20 \beta_{7} + 9 \beta_{8} + 4 \beta_{9} - 9 \beta_{10} - 14 \beta_{11} + 3 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - 10 \beta_{15} ) q^{44} + ( 8 + 4 \beta_{1} + 38 \beta_{2} - 145 \beta_{3} + 16 \beta_{5} + 10 \beta_{6} - \beta_{7} + 20 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 25 \beta_{13} - 10 \beta_{15} ) q^{45} + ( 14 - 3 \beta_{1} + 60 \beta_{2} - 17 \beta_{3} - 10 \beta_{4} - 40 \beta_{5} - 26 \beta_{6} - 11 \beta_{7} - 7 \beta_{8} - 12 \beta_{9} - 5 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 12 \beta_{13} + 14 \beta_{14} + 10 \beta_{15} ) q^{46} + ( -86 + 37 \beta_{1} - 2 \beta_{2} - 89 \beta_{3} - \beta_{4} - 11 \beta_{5} - 3 \beta_{6} - 7 \beta_{7} - 5 \beta_{8} - 11 \beta_{9} - 3 \beta_{10} + 4 \beta_{11} - 7 \beta_{12} - 11 \beta_{13} + 6 \beta_{14} - 2 \beta_{15} ) q^{47} + ( 138 - 23 \beta_{1} + 56 \beta_{2} + 70 \beta_{3} - \beta_{4} - 63 \beta_{5} + \beta_{6} - 33 \beta_{7} - 11 \beta_{8} - 15 \beta_{9} + 20 \beta_{10} + 18 \beta_{11} - 3 \beta_{12} - 13 \beta_{13} + 21 \beta_{14} + 10 \beta_{15} ) q^{48} + 49 \beta_{3} q^{49} + ( -6 + 57 \beta_{1} - 23 \beta_{2} + 133 \beta_{3} - 40 \beta_{4} + 27 \beta_{5} - 31 \beta_{6} + 39 \beta_{7} - 4 \beta_{8} - 14 \beta_{9} - 23 \beta_{10} - 6 \beta_{11} + 28 \beta_{12} - 10 \beta_{13} + 6 \beta_{14} - 20 \beta_{15} ) q^{50} + ( -166 + 67 \beta_{1} - 48 \beta_{2} + 3 \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{6} - 18 \beta_{7} - 5 \beta_{8} - 11 \beta_{9} - 5 \beta_{10} - 2 \beta_{11} + 7 \beta_{12} - 10 \beta_{13} + 7 \beta_{14} - 7 \beta_{15} ) q^{51} + ( 113 - 130 \beta_{1} - \beta_{2} + 124 \beta_{3} + 11 \beta_{5} + 11 \beta_{6} + 16 \beta_{7} + 10 \beta_{8} + 20 \beta_{9} - 3 \beta_{10} + 7 \beta_{11} - 8 \beta_{12} - 13 \beta_{13} - 10 \beta_{14} ) q^{52} + ( 106 + 12 \beta_{1} - 23 \beta_{2} + 14 \beta_{3} - 6 \beta_{4} + 33 \beta_{5} + 35 \beta_{6} - 14 \beta_{7} + 18 \beta_{8} - 2 \beta_{9} + 16 \beta_{10} + 15 \beta_{11} - 12 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 6 \beta_{15} ) q^{53} + ( 59 - 24 \beta_{1} + 25 \beta_{2} - 127 \beta_{3} + 7 \beta_{4} - 12 \beta_{5} - 17 \beta_{6} + 32 \beta_{7} - 4 \beta_{8} + 9 \beta_{9} - \beta_{10} - 4 \beta_{11} - 7 \beta_{12} - 29 \beta_{13} + 4 \beta_{14} + 6 \beta_{15} ) q^{54} + ( -115 + 6 \beta_{1} + 54 \beta_{2} + 5 \beta_{3} + 7 \beta_{4} + 49 \beta_{5} - 15 \beta_{6} + 7 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - 23 \beta_{11} - 6 \beta_{12} + 3 \beta_{13} + \beta_{14} - 7 \beta_{15} ) q^{55} + ( -14 + 28 \beta_{1} - 21 \beta_{3} - 21 \beta_{5} - 7 \beta_{6} - 14 \beta_{7} - 7 \beta_{8} - 14 \beta_{9} + 7 \beta_{10} + 7 \beta_{14} ) q^{56} + ( 152 - 47 \beta_{1} + 17 \beta_{2} + 285 \beta_{3} + 2 \beta_{4} + 23 \beta_{5} + 13 \beta_{6} - 16 \beta_{7} + 7 \beta_{8} + 6 \beta_{9} + 11 \beta_{10} - 12 \beta_{11} - 3 \beta_{12} + 23 \beta_{13} - 30 \beta_{14} - 2 \beta_{15} ) q^{57} + ( 7 + 47 \beta_{1} - 40 \beta_{2} - 31 \beta_{3} - 25 \beta_{5} + 7 \beta_{6} - 55 \beta_{7} + 13 \beta_{8} + 8 \beta_{9} + 6 \beta_{10} + 7 \beta_{11} - 17 \beta_{12} + 5 \beta_{13} + 8 \beta_{14} ) q^{58} + ( 9 - 10 \beta_{1} + 19 \beta_{2} - 21 \beta_{3} + 6 \beta_{5} - 17 \beta_{6} - 13 \beta_{7} + 17 \beta_{8} + 14 \beta_{9} - 22 \beta_{10} + 9 \beta_{11} - 6 \beta_{12} + 3 \beta_{13} + 14 \beta_{14} ) q^{59} + ( -249 - 59 \beta_{1} - 13 \beta_{2} + 286 \beta_{3} - 2 \beta_{4} - 31 \beta_{5} + 21 \beta_{6} - 13 \beta_{7} - 40 \beta_{8} + 7 \beta_{9} + 5 \beta_{10} - 4 \beta_{12} - 29 \beta_{13} + 3 \beta_{14} + 4 \beta_{15} ) q^{60} + ( -82 - \beta_{1} + 20 \beta_{2} - 97 \beta_{3} - 7 \beta_{4} - 19 \beta_{5} - 15 \beta_{6} - 43 \beta_{7} + 5 \beta_{8} + 3 \beta_{9} - 11 \beta_{10} + 13 \beta_{12} + 18 \beta_{13} + 2 \beta_{14} - 14 \beta_{15} ) q^{61} + ( 108 - 15 \beta_{1} + 32 \beta_{2} - 21 \beta_{3} + 4 \beta_{4} - 41 \beta_{5} - 16 \beta_{6} - 28 \beta_{7} - 32 \beta_{8} - \beta_{9} - 20 \beta_{10} - 26 \beta_{11} + 15 \beta_{12} - \beta_{13} + 6 \beta_{14} - 4 \beta_{15} ) q^{62} + ( -56 + 14 \beta_{2} - 63 \beta_{3} - 7 \beta_{5} - 7 \beta_{7} + 7 \beta_{14} + 7 \beta_{15} ) q^{63} + ( -28 + 7 \beta_{1} + 22 \beta_{2} - 11 \beta_{3} - 10 \beta_{4} - 26 \beta_{5} + 2 \beta_{6} - 15 \beta_{7} - 35 \beta_{8} - 14 \beta_{9} + 3 \beta_{10} + 22 \beta_{11} - 7 \beta_{12} - 14 \beta_{13} + 18 \beta_{14} + 10 \beta_{15} ) q^{64} + ( 68 - 90 \beta_{1} - 18 \beta_{2} + 87 \beta_{3} + 9 \beta_{4} - 24 \beta_{5} + 19 \beta_{6} + 38 \beta_{7} + \beta_{8} + 11 \beta_{9} + 44 \beta_{10} + 9 \beta_{11} - 18 \beta_{12} + 15 \beta_{13} - 10 \beta_{14} + 18 \beta_{15} ) q^{65} + ( 477 + 44 \beta_{1} - 50 \beta_{2} + 214 \beta_{3} + 4 \beta_{4} + 27 \beta_{5} - 7 \beta_{6} + 29 \beta_{7} + 11 \beta_{8} + 12 \beta_{9} + 13 \beta_{10} - 15 \beta_{11} + 12 \beta_{12} + 7 \beta_{13} - 15 \beta_{14} - 16 \beta_{15} ) q^{66} + ( 2 + 5 \beta_{1} - \beta_{2} - 94 \beta_{3} - 2 \beta_{4} + 48 \beta_{5} - 29 \beta_{6} + 24 \beta_{7} - 32 \beta_{8} - 18 \beta_{9} - 9 \beta_{10} + 2 \beta_{11} + \beta_{12} - 15 \beta_{13} - 17 \beta_{14} - \beta_{15} ) q^{67} + ( 9 - 80 \beta_{1} + 29 \beta_{2} + 308 \beta_{3} + 60 \beta_{4} - 75 \beta_{5} + 88 \beta_{6} - 94 \beta_{7} + 9 \beta_{8} + 17 \beta_{9} + 80 \beta_{10} + 9 \beta_{11} - 39 \beta_{12} + 22 \beta_{13} - 13 \beta_{14} + 30 \beta_{15} ) q^{68} + ( -125 - 19 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} + 37 \beta_{5} - 20 \beta_{6} + 45 \beta_{7} + 26 \beta_{8} + 11 \beta_{9} - 43 \beta_{10} - \beta_{11} + 17 \beta_{12} - 17 \beta_{13} + 14 \beta_{14} + 4 \beta_{15} ) q^{69} + ( -21 + 49 \beta_{1} + 7 \beta_{2} - 28 \beta_{3} - 7 \beta_{4} + 28 \beta_{5} - 7 \beta_{6} - 7 \beta_{9} - 14 \beta_{10} - 14 \beta_{11} + 14 \beta_{12} - 7 \beta_{13} + 7 \beta_{14} - 14 \beta_{15} ) q^{70} + ( 129 + 6 \beta_{1} - 19 \beta_{2} - 26 \beta_{4} - 11 \beta_{5} - 7 \beta_{6} - 22 \beta_{7} - 2 \beta_{8} - 16 \beta_{9} + 16 \beta_{10} + 13 \beta_{11} - 6 \beta_{12} - 16 \beta_{13} + 6 \beta_{14} + 26 \beta_{15} ) q^{71} + ( -175 - 11 \beta_{1} + 5 \beta_{2} - 519 \beta_{3} - 22 \beta_{4} + 17 \beta_{5} - 54 \beta_{6} + 52 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} - 51 \beta_{10} + 24 \beta_{11} + 30 \beta_{12} - 18 \beta_{13} + 16 \beta_{14} - 8 \beta_{15} ) q^{72} + ( 85 - 5 \beta_{1} + 13 \beta_{2} - 14 \beta_{3} + 11 \beta_{4} - 59 \beta_{5} - 4 \beta_{6} + 33 \beta_{7} - 2 \beta_{8} + \beta_{9} - 15 \beta_{10} + 31 \beta_{11} + 5 \beta_{12} + \beta_{13} + 9 \beta_{14} - 11 \beta_{15} ) q^{73} + ( 1 + 55 \beta_{1} - 11 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} - 26 \beta_{5} + 4 \beta_{6} + 16 \beta_{7} - 7 \beta_{8} - 6 \beta_{9} + 44 \beta_{10} - 5 \beta_{11} + 2 \beta_{12} - 27 \beta_{13} - \beta_{14} + 16 \beta_{15} ) q^{74} + ( 27 - 38 \beta_{1} - 28 \beta_{2} + 244 \beta_{3} - 22 \beta_{4} - 45 \beta_{5} + 58 \beta_{6} - 33 \beta_{7} + 10 \beta_{8} + 2 \beta_{10} + 12 \beta_{11} + 24 \beta_{12} - 10 \beta_{13} + 27 \beta_{14} + 7 \beta_{15} ) q^{75} + ( 13 + 118 \beta_{1} - 95 \beta_{2} + 100 \beta_{3} - 10 \beta_{4} - 34 \beta_{5} + 45 \beta_{6} - 100 \beta_{7} + 26 \beta_{8} - 15 \beta_{9} + 73 \beta_{10} + 13 \beta_{11} - 7 \beta_{12} + 36 \beta_{13} - 10 \beta_{14} - 5 \beta_{15} ) q^{76} + ( -14 \beta_{1} - 14 \beta_{3} + 14 \beta_{4} - 7 \beta_{7} - 14 \beta_{8} - 14 \beta_{12} - 7 \beta_{13} - 7 \beta_{14} + 7 \beta_{15} ) q^{77} + ( -377 + 31 \beta_{1} + 11 \beta_{2} - 83 \beta_{3} + 10 \beta_{4} + 36 \beta_{5} - 8 \beta_{6} - 69 \beta_{7} + 15 \beta_{8} - 21 \beta_{9} - 20 \beta_{10} - 18 \beta_{11} + 14 \beta_{12} + 57 \beta_{13} - 16 \beta_{14} - 20 \beta_{15} ) q^{78} + ( -34 + 67 \beta_{1} + 19 \beta_{2} - 38 \beta_{3} - 13 \beta_{4} - 57 \beta_{5} - 4 \beta_{6} - 15 \beta_{7} + 15 \beta_{8} + 17 \beta_{9} + 47 \beta_{10} + \beta_{11} + 5 \beta_{12} + 44 \beta_{13} - 2 \beta_{14} - 26 \beta_{15} ) q^{79} + ( 358 + 6 \beta_{1} - 86 \beta_{2} + 19 \beta_{3} + \beta_{4} + 22 \beta_{5} + 30 \beta_{6} + 19 \beta_{7} + 24 \beta_{8} + 7 \beta_{9} + 12 \beta_{10} + 22 \beta_{11} - 6 \beta_{12} + 7 \beta_{13} - 13 \beta_{14} - \beta_{15} ) q^{80} + ( -112 + 41 \beta_{1} + 3 \beta_{2} - 242 \beta_{3} - 3 \beta_{4} + 43 \beta_{5} + 2 \beta_{6} + 18 \beta_{7} + 16 \beta_{8} - \beta_{9} - 35 \beta_{10} - 47 \beta_{11} + 25 \beta_{12} - 4 \beta_{13} - 26 \beta_{14} - 10 \beta_{15} ) q^{81} + ( 15 - 24 \beta_{2} + 46 \beta_{3} + 20 \beta_{4} + 101 \beta_{5} + 41 \beta_{6} + 36 \beta_{7} + 41 \beta_{8} + 33 \beta_{9} + 13 \beta_{10} - 22 \beta_{11} + 33 \beta_{13} - 46 \beta_{14} - 20 \beta_{15} ) q^{82} + ( -373 - \beta_{1} - 41 \beta_{2} - 359 \beta_{3} + 11 \beta_{4} + \beta_{5} + 14 \beta_{6} + 51 \beta_{7} - 27 \beta_{8} - 43 \beta_{9} + 13 \beta_{10} + 7 \beta_{11} - 37 \beta_{12} + 4 \beta_{13} + 16 \beta_{14} + 22 \beta_{15} ) q^{83} + ( 140 - 14 \beta_{1} + 14 \beta_{2} + 105 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} - 14 \beta_{6} + 35 \beta_{7} - 14 \beta_{8} - 7 \beta_{10} - 28 \beta_{13} + 7 \beta_{15} ) q^{84} + ( 13 - 219 \beta_{1} + 180 \beta_{2} - 17 \beta_{3} + 52 \beta_{4} + 10 \beta_{6} - 36 \beta_{7} - 33 \beta_{8} + 16 \beta_{9} + 7 \beta_{10} + 13 \beta_{11} - 39 \beta_{12} - 23 \beta_{13} - 10 \beta_{14} + 26 \beta_{15} ) q^{85} + ( -57 - 24 \beta_{1} - 29 \beta_{2} + 50 \beta_{3} - 4 \beta_{4} + 15 \beta_{5} - 48 \beta_{6} + 194 \beta_{7} - 27 \beta_{8} - 9 \beta_{9} - 96 \beta_{10} - 57 \beta_{11} + 19 \beta_{12} - 20 \beta_{13} - 7 \beta_{14} - 2 \beta_{15} ) q^{86} + ( 16 + 22 \beta_{1} + 58 \beta_{2} + 131 \beta_{3} - 5 \beta_{4} - 34 \beta_{5} - 5 \beta_{6} + 25 \beta_{7} + 57 \beta_{8} + 31 \beta_{9} - 12 \beta_{10} - 19 \beta_{11} + 18 \beta_{12} + 42 \beta_{13} - 25 \beta_{14} - 3 \beta_{15} ) q^{87} + ( -213 + 64 \beta_{1} + \beta_{2} - 222 \beta_{3} + 4 \beta_{4} - 37 \beta_{5} - 9 \beta_{6} - 32 \beta_{7} - 8 \beta_{8} - 12 \beta_{9} + \beta_{10} + 9 \beta_{11} - 4 \beta_{12} + 45 \beta_{13} + 4 \beta_{14} + 8 \beta_{15} ) q^{88} + ( 567 + 30 \beta_{1} - 9 \beta_{2} + \beta_{3} + 23 \beta_{4} + 80 \beta_{5} - 4 \beta_{6} + 33 \beta_{7} - 33 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 8 \beta_{11} - 30 \beta_{12} - 3 \beta_{13} + 29 \beta_{14} - 23 \beta_{15} ) q^{89} + ( -430 + 53 \beta_{1} - 180 \beta_{2} - 516 \beta_{3} - 11 \beta_{4} - 27 \beta_{5} - 33 \beta_{6} - 26 \beta_{7} - 21 \beta_{8} - 22 \beta_{9} + 15 \beta_{10} + 51 \beta_{11} + 6 \beta_{12} + 21 \beta_{13} - 15 \beta_{14} + 27 \beta_{15} ) q^{90} + ( -42 - 7 \beta_{1} - 49 \beta_{2} + 21 \beta_{3} + 28 \beta_{5} - 7 \beta_{6} + 28 \beta_{7} + 7 \beta_{8} + 14 \beta_{9} + 7 \beta_{10} - 14 \beta_{11} + 7 \beta_{12} + 14 \beta_{13} - 28 \beta_{14} ) q^{91} + ( -467 - 74 \beta_{1} - 3 \beta_{2} - 439 \beta_{3} - 12 \beta_{4} + 246 \beta_{5} + 28 \beta_{6} + 122 \beta_{7} + 25 \beta_{8} + 38 \beta_{9} - 88 \beta_{10} - 51 \beta_{11} + 36 \beta_{12} + \beta_{13} - 13 \beta_{14} - 24 \beta_{15} ) q^{92} + ( 350 - 32 \beta_{1} - 79 \beta_{2} + 157 \beta_{3} + 11 \beta_{4} + 29 \beta_{5} - 32 \beta_{6} + 63 \beta_{7} + 31 \beta_{8} + 45 \beta_{9} - 52 \beta_{10} + 42 \beta_{12} + 2 \beta_{13} - 9 \beta_{14} - 5 \beta_{15} ) q^{93} + ( -5 + 55 \beta_{1} - 80 \beta_{2} - 409 \beta_{3} + 20 \beta_{4} + 40 \beta_{5} - 35 \beta_{6} + 61 \beta_{7} + 11 \beta_{8} + 3 \beta_{9} - 43 \beta_{10} - 5 \beta_{11} - 7 \beta_{12} + 18 \beta_{13} - 7 \beta_{14} + 10 \beta_{15} ) q^{94} + ( -9 + 75 \beta_{1} - 62 \beta_{2} + 108 \beta_{3} - 22 \beta_{4} + 36 \beta_{5} - 56 \beta_{6} + 31 \beta_{7} - 17 \beta_{8} - 16 \beta_{9} - 49 \beta_{10} - 9 \beta_{11} - 5 \beta_{12} - 12 \beta_{13} - 5 \beta_{14} - 11 \beta_{15} ) q^{95} + ( -489 - 84 \beta_{1} + 58 \beta_{2} - 232 \beta_{3} - 6 \beta_{4} + 109 \beta_{5} - 3 \beta_{6} + 110 \beta_{7} - 3 \beta_{8} + 11 \beta_{9} - 44 \beta_{10} + 16 \beta_{11} + 5 \beta_{12} - 9 \beta_{13} + 15 \beta_{14} + 13 \beta_{15} ) q^{96} + ( -45 + 173 \beta_{1} + 23 \beta_{2} - 48 \beta_{3} - 16 \beta_{4} + 7 \beta_{5} - 3 \beta_{6} - 37 \beta_{7} + 20 \beta_{8} + 24 \beta_{9} - 61 \beta_{10} + 24 \beta_{11} - 17 \beta_{12} + 3 \beta_{13} - 4 \beta_{14} - 32 \beta_{15} ) q^{97} + 49 \beta_{2} q^{98} + ( 51 + 24 \beta_{1} + 33 \beta_{2} + 8 \beta_{3} - 11 \beta_{4} + 20 \beta_{5} + 4 \beta_{6} + 46 \beta_{7} - 16 \beta_{8} + 15 \beta_{9} + 20 \beta_{10} - 46 \beta_{11} + 50 \beta_{12} + 25 \beta_{13} + 5 \beta_{14} - 41 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 3q^{2} + 2q^{3} - 43q^{4} - 30q^{5} + 19q^{6} + 56q^{7} + 12q^{8} - 124q^{9} + O(q^{10}) \) \( 16q - 3q^{2} + 2q^{3} - 43q^{4} - 30q^{5} + 19q^{6} + 56q^{7} + 12q^{8} - 124q^{9} - 28q^{10} - 24q^{11} + 268q^{12} - 68q^{13} + 21q^{14} + 56q^{15} - 103q^{16} + 336q^{17} - 479q^{18} + 352q^{19} - 330q^{20} + 70q^{21} - 151q^{22} - 228q^{23} - 195q^{24} - 244q^{25} + 1590q^{26} + 272q^{27} - 602q^{28} - 618q^{29} + 1030q^{30} - 72q^{31} - 786q^{32} - 700q^{33} + 261q^{34} - 420q^{35} + 727q^{36} + 420q^{37} - 1032q^{38} - 22q^{39} + 375q^{40} - 420q^{41} - 175q^{42} + 2q^{43} + 774q^{44} + 1406q^{45} + 804q^{46} - 570q^{47} + 1864q^{48} - 392q^{49} - 1110q^{50} - 2940q^{51} + 431q^{52} + 1056q^{53} + 2269q^{54} - 1676q^{55} + 42q^{56} + 122q^{57} - 37q^{58} + 150q^{59} - 6350q^{60} - 578q^{61} + 2340q^{62} - 350q^{63} - 224q^{64} + 366q^{65} + 5812q^{66} + 898q^{67} - 2526q^{68} - 2166q^{69} - 98q^{70} + 1764q^{71} + 1350q^{72} + 1944q^{73} + 222q^{74} - 2096q^{75} - 1423q^{76} + 168q^{77} - 5558q^{78} + 158q^{79} + 4950q^{80} + 476q^{81} - 422q^{82} - 2958q^{83} + 1715q^{84} + 774q^{85} + 114q^{86} + 44q^{87} - 1317q^{88} + 8760q^{89} - 3659q^{90} - 952q^{91} - 4629q^{92} + 3954q^{93} + 3234q^{94} - 930q^{95} - 5923q^{96} + 60q^{97} + 294q^{98} + 1214q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + 599392 x^{8} - 1089732 x^{7} + 4808401 x^{6} - 7939134 x^{5} + 26225236 x^{4} - 39450864 x^{3} + 62254768 x^{2} - 39660672 x + 21307456\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1081453415523969731 \nu^{15} - 10258576004084808911 \nu^{14} - 31433377624576942326 \nu^{13} - 563112173463188181077 \nu^{12} - 1308525829509857362353 \nu^{11} - 19607931782297760198339 \nu^{10} - 14926937550628598170271 \nu^{9} - 359065610897655236169716 \nu^{8} - 338828317369372158410328 \nu^{7} - 4446748427923459524058988 \nu^{6} - 805082242542684152633379 \nu^{5} - 24422079356124957269325774 \nu^{4} - 45221194666110986753154188 \nu^{3} - 98665615226027792616047816 \nu^{2} + 66635913671816055091959360 \nu + 202367040539048711739007360\)\()/ \)\(35\!\cdots\!16\)\( \)
\(\beta_{3}\)\(=\)\((\)\(5480043342153615460870 \nu^{15} - 15816131405703515847823 \nu^{14} + 323761712199266631472107 \nu^{13} - 688788532248435498730128 \nu^{12} + 11871367046005927356534519 \nu^{11} - 23658573685667169180098169 \nu^{10} + 246413116060118064521225473 \nu^{9} - 410084388547192134528065113 \nu^{8} + 3491874996428086949591717172 \nu^{7} - 5776274652209615948002027584 \nu^{6} + 28916019729366622881044804946 \nu^{5} - 43042265965218272972249226897 \nu^{4} + 157806969726691414058965486918 \nu^{3} - 190099815283061711298509725204 \nu^{2} + 398088886881135987217134517992 \nu - 255791123727576180195754455360\)\()/ \)\(20\!\cdots\!32\)\( \)
\(\beta_{4}\)\(=\)\((\)\(2012912600590333707615233 \nu^{15} + 17348590585241027539523665 \nu^{14} + 94621407753774784589725902 \nu^{13} + 993037120070939002537017207 \nu^{12} + 3796716706931644738614730395 \nu^{11} + 36625081764226040204233806189 \nu^{10} + 73402382013227503940205667901 \nu^{9} + 723727347815224198141460164624 \nu^{8} + 1135721563954343800778778826416 \nu^{7} + 9902840952227148167879741982936 \nu^{6} + 6858923188941733857349273824465 \nu^{5} + 65834791980623887700666878834770 \nu^{4} + 14608707215063202336236562518912 \nu^{3} + 297122594026016268556658910342592 \nu^{2} - 100709050684099700381174367837168 \nu + 231045738094646739733256452533408\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-2419979412074976894756057 \nu^{15} + 690978070507800231698429 \nu^{14} - 119581754227108303595820124 \nu^{13} - 55998582055525696780428259 \nu^{12} - 4200690232317939724117670625 \nu^{11} - 2418257592484333671692737539 \nu^{10} - 72246049429258322566246912299 \nu^{9} - 74308096401800431368886248118 \nu^{8} - 905517889815581466361886507104 \nu^{7} - 902875303474937966610420325984 \nu^{6} - 3847685508533632243901012702757 \nu^{5} - 7094929382538181471380784803912 \nu^{4} - 10989786345388241028658352425392 \nu^{3} - 47747185733973032384540726968792 \nu^{2} + 78275196242110151015388545820848 \nu - 124968951271698236619710625641344\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-3484841768378567142746415 \nu^{15} + 15441061097170197672163935 \nu^{14} - 209054059913060756453946760 \nu^{13} + 710941927764532698833803651 \nu^{12} - 7537113467269945077773324727 \nu^{11} + 24836620420560509104218027903 \nu^{10} - 156185615243464445899497136173 \nu^{9} + 439854028032648168864917823426 \nu^{8} - 2177314251406132652387582772784 \nu^{7} + 6119074886973611821750298063212 \nu^{6} - 18484405466972765634851884534251 \nu^{5} + 43049322538864499842014869850960 \nu^{4} - 105687321485134658455259760558300 \nu^{3} + 218372227058546907643945287277368 \nu^{2} - 271985052496919786199077239611520 \nu + 302987535457788659158644410668000\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-3967895809809560576198911 \nu^{15} + 16560462486271942740487663 \nu^{14} - 234189575775349980061021784 \nu^{13} + 745646160563660178634352387 \nu^{12} - 8410922277889247418122221383 \nu^{11} + 25962374376029156344194372543 \nu^{10} - 171986739826178435413384245133 \nu^{9} + 450922010247240876832183818802 \nu^{8} - 2378551688096364387138008448704 \nu^{7} + 6276300881535876310723612989260 \nu^{6} - 19665229471151252829278615349723 \nu^{5} + 42446179208647245906204839582832 \nu^{4} - 110743256857099048192023865897900 \nu^{3} + 210357418823247532968027990201016 \nu^{2} - 266279962675507008017343841720448 \nu + 137318945320257732621044058831872\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{8}\)\(=\)\((\)\(3539754752942150019523 \nu^{15} - 122684183678165230727 \nu^{14} + 174452770437151922132298 \nu^{13} + 144026909845419610167733 \nu^{12} + 6206184692543187159145479 \nu^{11} + 5970463195997448266509557 \nu^{10} + 109662495109947112821687583 \nu^{9} + 167763726599491300132839868 \nu^{8} + 1453858721703117529295747064 \nu^{7} + 2248924998488557613003892484 \nu^{6} + 7755005966247873356147868357 \nu^{5} + 19709283468356252697727458882 \nu^{4} + 32847078045789804696058658638 \nu^{3} + 113765355656410604286498506488 \nu^{2} - 78621694409674998087306087360 \nu + 246119870160029385338081735152\)\()/ \)\(11\!\cdots\!96\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-7280068900675734484690279 \nu^{15} + 11446153636711473335709037 \nu^{14} - 399874794936465044207502822 \nu^{13} + 360280730102682660021987843 \nu^{12} - 14492320041979894205147442201 \nu^{11} + 11390017912383838653016929477 \nu^{10} - 283004890614886032716404744351 \nu^{9} + 137833177935483126815879469364 \nu^{8} - 3884425764615758839447036925724 \nu^{7} + 2058220762092530560606261682136 \nu^{6} - 27253728189432891451132341965091 \nu^{5} + 12534091916150820900185835014826 \nu^{4} - 126509668657760850493065869739640 \nu^{3} + 38105179613335116901589366591128 \nu^{2} - 50783011643102454889071118893264 \nu - 205284968380959153271463859675360\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-7319948387281784389669773 \nu^{15} + 14558181372606579073367495 \nu^{14} - 405669210122006432169413390 \nu^{13} + 533869207735840794996603645 \nu^{12} - 14642820882018019513833389595 \nu^{11} + 17776306753632051927442731315 \nu^{10} - 286656145556427472084241752497 \nu^{9} + 267287902981829137028000228336 \nu^{8} - 3921206573626851282483236223656 \nu^{7} + 3910028060768276976452528832768 \nu^{6} - 28334821414581617480449856304705 \nu^{5} + 26993993655210492608442061806942 \nu^{4} - 143238139317065910496948640220900 \nu^{3} + 110265605700659445474951663784064 \nu^{2} - 236879356762672676499888477789632 \nu - 84469895684802761919532886457312\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{11}\)\(=\)\((\)\(7903516394153979760690095 \nu^{15} + 339918553831602736860403 \nu^{14} + 422384260504497774958418930 \nu^{13} + 287038659935656321202588421 \nu^{12} + 15463442556930391089376980609 \nu^{11} + 11762247029538296554737103899 \nu^{10} + 299921121748786699245029160867 \nu^{9} + 296141772705853916337422700160 \nu^{8} + 4160803665930533111087398266212 \nu^{7} + 3490590515274650502207947482752 \nu^{6} + 28552991936455017141471588973095 \nu^{5} + 18576889043716113263150032539234 \nu^{4} + 123463831593909081027748760161836 \nu^{3} + 67495284096965211046968788711944 \nu^{2} + 122014873901117059997121346268528 \nu + 141949203701560674914126334635904\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-2721304597174110961532076 \nu^{15} + 9048094403442331652662266 \nu^{14} - 151553379097130840326422727 \nu^{13} + 397617691162390390589943094 \nu^{12} - 5388447547802198588904990333 \nu^{11} + 13997126285582673342906832410 \nu^{10} - 104189163495539218710446065368 \nu^{9} + 246225657504241151342204736987 \nu^{8} - 1394686122021095991262430824120 \nu^{7} + 3540548246741204582005403861803 \nu^{6} - 9985232059753419424285906498029 \nu^{5} + 25336910673783162383732585102484 \nu^{4} - 53786723835166798388178093869730 \nu^{3} + 116569245380827873775947472689920 \nu^{2} - 98203978010517699355665744811228 \nu + 92513608124138838840741164947744\)\()/ \)\(32\!\cdots\!96\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-3758334808051921317076130 \nu^{15} + 9194422862022257270940262 \nu^{14} - 210007193213658225316023142 \nu^{13} + 357160353017772758790117497 \nu^{12} - 7566506626653361823078191221 \nu^{11} + 12077935686715285904931109629 \nu^{10} - 149110528403605821425308393217 \nu^{9} + 188268574467671702749056176032 \nu^{8} - 2044725032870816281370987842651 \nu^{7} + 2707456417709348954697102424061 \nu^{6} - 15156845979332218393959419182029 \nu^{5} + 18123776954292814802165438139294 \nu^{4} - 77915796984324620481437163570857 \nu^{3} + 89112906766388502173929455536686 \nu^{2} - 124751436905197424678574932355544 \nu + 21545017068179317477902313784072\)\()/ \)\(32\!\cdots\!96\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-21146356726947725192905591 \nu^{15} + 58767286733336790355500835 \nu^{14} - 1196330416887943045129382408 \nu^{13} + 2403009196834006542113868179 \nu^{12} - 42959088315298516018917495915 \nu^{11} + 81922122137665467762831879291 \nu^{10} - 849706798679780863898637199789 \nu^{9} + 1332777147748075249670995330234 \nu^{8} - 11540547990583665267978485758472 \nu^{7} + 18832230097674484669184056794620 \nu^{6} - 85641073438827558295167485810103 \nu^{5} + 127082005503905304042528831224040 \nu^{4} - 429738475328391092784191143486552 \nu^{3} + 593494422916143686190544599162424 \nu^{2} - 725567835929497996973806818530624 \nu + 148970662194419907159141497422496\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-10872325483943857459220828 \nu^{15} + 15566508988228390548574683 \nu^{14} - 585187983669420456361491803 \nu^{13} + 455203231410590555992780612 \nu^{12} - 21022720348551168765804749253 \nu^{11} + 14461547984148650089626023247 \nu^{10} - 400962187495723632085772482469 \nu^{9} + 170886788806443998108980017393 \nu^{8} - 5404792746292675942907645486018 \nu^{7} + 2949950815743123875587072783006 \nu^{6} - 36054572848513375121813745778140 \nu^{5} + 23841158775211753600977064940805 \nu^{4} - 167095585851342242693370543638498 \nu^{3} + 114533457622808501586770133476868 \nu^{2} - 137459777103900743839670307810344 \nu - 5347674804643293417862552358432\)\()/ \)\(64\!\cdots\!92\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} + 13 \beta_{3}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{14} + \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} - 20 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(-4 \beta_{15} + \beta_{14} + 3 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} - 29 \beta_{10} + \beta_{8} + 4 \beta_{7} + 35 \beta_{5} - 2 \beta_{4} - 256 \beta_{3} + \beta_{2} - 5 \beta_{1} - 256\)
\(\nu^{5}\)\(=\)\(-2 \beta_{15} + 33 \beta_{14} - 39 \beta_{13} + 4 \beta_{12} - 6 \beta_{11} - 71 \beta_{10} + 31 \beta_{9} - 6 \beta_{8} + 48 \beta_{7} - 34 \beta_{6} - 3 \beta_{5} - 4 \beta_{4} - 60 \beta_{3} + 449 \beta_{2} - 451 \beta_{1} - 6\)
\(\nu^{6}\)\(=\)\(90 \beta_{15} + 18 \beta_{14} - 54 \beta_{13} + 33 \beta_{12} + 102 \beta_{11} + 3 \beta_{10} - 54 \beta_{9} + 45 \beta_{8} + 681 \beta_{7} - 774 \beta_{6} - 306 \beta_{5} - 90 \beta_{4} - 51 \beta_{3} + 342 \beta_{2} - 33 \beta_{1} + 5692\)
\(\nu^{7}\)\(=\)\(216 \beta_{15} + 813 \beta_{14} + 327 \beta_{13} + 48 \beta_{12} + 243 \beta_{11} + 1080 \beta_{10} - 1734 \beta_{9} - 921 \beta_{8} - 2742 \beta_{7} - 1104 \beta_{6} - 3774 \beta_{5} + 108 \beta_{4} + 1119 \beta_{3} + 183 \beta_{2} + 10438 \beta_{1} + 2223\)
\(\nu^{8}\)\(=\)\(3042 \beta_{15} - 2088 \beta_{14} - 486 \beta_{13} - 4668 \beta_{12} + 954 \beta_{11} + 20536 \beta_{10} + 954 \beta_{9} - 2574 \beta_{8} - 22654 \beta_{7} + 20536 \beta_{6} - 19120 \beta_{5} + 6084 \beta_{4} + 137620 \beta_{3} - 13407 \beta_{2} + 8277 \beta_{1} + 954\)
\(\nu^{9}\)\(=\)\(-4464 \beta_{15} - 45584 \beta_{14} + 25024 \beta_{13} - 10260 \beta_{12} - 12 \beta_{11} + 30820 \beta_{10} + 25024 \beta_{9} + 36916 \beta_{8} + 21067 \beta_{7} + 64253 \beta_{6} + 111660 \beta_{5} + 4464 \beta_{4} + 55844 \beta_{3} - 267314 \beta_{2} + 10260 \beta_{1} - 76285\)
\(\nu^{10}\)\(=\)\(-185440 \beta_{15} + 22699 \beta_{14} + 61356 \beta_{13} + 117560 \beta_{12} - 145716 \beta_{11} - 542099 \beta_{10} + 47322 \beta_{9} + 70021 \beta_{8} + 184786 \beta_{7} + 5457 \beta_{6} + 844445 \beta_{5} - 92720 \beta_{4} - 3306307 \beta_{3} + 64564 \beta_{2} - 225536 \beta_{1} - 3311764\)
\(\nu^{11}\)\(=\)\(-163154 \beta_{15} + 679563 \beta_{14} - 1050552 \beta_{13} + 370531 \beta_{12} - 320025 \beta_{11} - 1851149 \beta_{10} + 516409 \beta_{9} - 370989 \beta_{8} + 1906365 \beta_{7} - 1014715 \beta_{6} + 222504 \beta_{5} - 326308 \beta_{4} - 3549363 \beta_{3} + 6568757 \beta_{2} - 6562474 \beta_{1} - 320025\)
\(\nu^{12}\)\(=\)\(2690190 \beta_{15} + 1674144 \beta_{14} - 2182167 \beta_{13} + 849000 \beta_{12} + 3469650 \beta_{11} - 340977 \beta_{10} - 2182167 \beta_{9} - 791199 \beta_{8} + 10754058 \beta_{7} - 14679909 \beta_{6} - 10721961 \beta_{5} - 2690190 \beta_{4} - 2523144 \beta_{3} + 10284309 \beta_{2} - 849000 \beta_{1} + 81748375\)
\(\nu^{13}\)\(=\)\(11040732 \beta_{15} + 12994026 \beta_{14} + 10754781 \beta_{13} - 2854938 \beta_{12} + 12258675 \beta_{11} + 30305421 \beta_{10} - 31508418 \beta_{9} - 18514392 \beta_{8} - 66457938 \beta_{7} - 22397763 \beta_{6} - 99618297 \beta_{5} + 5520366 \beta_{4} + 67291152 \beta_{3} + 3883371 \beta_{2} + 156701713 \beta_{1} + 89688915\)
\(\nu^{14}\)\(=\)\(75994776 \beta_{15} - 65254941 \beta_{14} + 34970778 \beta_{13} - 124260933 \beta_{12} + 25671753 \beta_{11} + 395283028 \beta_{10} + 10739835 \beta_{9} - 30284163 \beta_{8} - 452232154 \beta_{7} + 380351110 \beta_{6} - 337690573 \beta_{5} + 151989552 \beta_{4} + 2155785370 \beta_{3} - 351424308 \beta_{2} + 225106509 \beta_{1} + 25671753\)
\(\nu^{15}\)\(=\)\(-177240978 \beta_{15} - 834024074 \beta_{14} + 505632526 \beta_{13} - 267567516 \beta_{12} - 117171330 \beta_{11} + 595959064 \beta_{10} + 505632526 \beta_{9} + 805392274 \beta_{8} + 177969352 \beta_{7} + 1489324412 \beta_{6} + 2527097994 \beta_{5} + 177240978 \beta_{4} + 1101591590 \beta_{3} - 4427034893 \beta_{2} + 267567516 \beta_{1} - 2459310574\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
2.62188 + 4.54123i
2.28179 + 3.95218i
1.30789 + 2.26533i
0.797492 + 1.38130i
0.403686 + 0.699204i
−1.46974 2.54566i
−1.96709 3.40709i
−2.47591 4.28840i
2.62188 4.54123i
2.28179 3.95218i
1.30789 2.26533i
0.797492 1.38130i
0.403686 0.699204i
−1.46974 + 2.54566i
−1.96709 + 3.40709i
−2.47591 + 4.28840i
−2.62188 4.54123i −4.41799 + 2.73521i −9.74848 + 16.8849i 2.03009 3.51621i 24.0046 + 12.8917i 3.50000 + 6.06218i 60.2873 12.0372 24.1683i −21.2906
22.2 −2.28179 3.95218i 1.09409 5.07966i −6.41313 + 11.1079i −10.3955 + 18.0055i −22.5722 + 7.26670i 3.50000 + 6.06218i 22.0250 −24.6060 11.1152i 94.8813
22.3 −1.30789 2.26533i −3.13193 4.14620i 0.578868 1.00263i 6.77153 11.7286i −5.29628 + 12.5176i 3.50000 + 6.06218i −23.9546 −7.38198 + 25.9713i −35.4255
22.4 −0.797492 1.38130i 5.17737 0.441383i 2.72801 4.72505i 1.27816 2.21384i −4.73860 6.79949i 3.50000 + 6.06218i −21.4622 26.6104 4.57041i −4.07730
22.5 −0.403686 0.699204i −0.172376 + 5.19329i 3.67408 6.36369i −9.11444 + 15.7867i 3.70076 1.97593i 3.50000 + 6.06218i −12.3917 −26.9406 1.79040i 14.7175
22.6 1.46974 + 2.54566i 3.48146 3.85739i −0.320267 + 0.554718i 1.28443 2.22469i 14.9364 + 3.19326i 3.50000 + 6.06218i 21.6330 −2.75892 26.8587i 7.55109
22.7 1.96709 + 3.40709i 1.35403 + 5.01663i −3.73885 + 6.47588i 1.21571 2.10567i −14.4286 + 14.4814i 3.50000 + 6.06218i 2.05480 −23.3332 + 13.5853i 9.56561
22.8 2.47591 + 4.28840i −2.38464 4.61665i −8.26023 + 14.3071i −8.06998 + 13.9776i 13.8939 21.6567i 3.50000 + 6.06218i −42.1917 −15.6270 + 22.0181i −79.9221
43.1 −2.62188 + 4.54123i −4.41799 2.73521i −9.74848 16.8849i 2.03009 + 3.51621i 24.0046 12.8917i 3.50000 6.06218i 60.2873 12.0372 + 24.1683i −21.2906
43.2 −2.28179 + 3.95218i 1.09409 + 5.07966i −6.41313 11.1079i −10.3955 18.0055i −22.5722 7.26670i 3.50000 6.06218i 22.0250 −24.6060 + 11.1152i 94.8813
43.3 −1.30789 + 2.26533i −3.13193 + 4.14620i 0.578868 + 1.00263i 6.77153 + 11.7286i −5.29628 12.5176i 3.50000 6.06218i −23.9546 −7.38198 25.9713i −35.4255
43.4 −0.797492 + 1.38130i 5.17737 + 0.441383i 2.72801 + 4.72505i 1.27816 + 2.21384i −4.73860 + 6.79949i 3.50000 6.06218i −21.4622 26.6104 + 4.57041i −4.07730
43.5 −0.403686 + 0.699204i −0.172376 5.19329i 3.67408 + 6.36369i −9.11444 15.7867i 3.70076 + 1.97593i 3.50000 6.06218i −12.3917 −26.9406 + 1.79040i 14.7175
43.6 1.46974 2.54566i 3.48146 + 3.85739i −0.320267 0.554718i 1.28443 + 2.22469i 14.9364 3.19326i 3.50000 6.06218i 21.6330 −2.75892 + 26.8587i 7.55109
43.7 1.96709 3.40709i 1.35403 5.01663i −3.73885 6.47588i 1.21571 + 2.10567i −14.4286 14.4814i 3.50000 6.06218i 2.05480 −23.3332 13.5853i 9.56561
43.8 2.47591 4.28840i −2.38464 + 4.61665i −8.26023 14.3071i −8.06998 13.9776i 13.8939 + 21.6567i 3.50000 6.06218i −42.1917 −15.6270 22.0181i −79.9221
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.f.b 16
3.b odd 2 1 189.4.f.b 16
9.c even 3 1 inner 63.4.f.b 16
9.c even 3 1 567.4.a.i 8
9.d odd 6 1 189.4.f.b 16
9.d odd 6 1 567.4.a.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.b 16 1.a even 1 1 trivial
63.4.f.b 16 9.c even 3 1 inner
189.4.f.b 16 3.b odd 2 1
189.4.f.b 16 9.d odd 6 1
567.4.a.g 8 9.d odd 6 1
567.4.a.i 8 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 21307456 + 39660672 T + 62254768 T^{2} + 39450864 T^{3} + 26225236 T^{4} + 7939134 T^{5} + 4808401 T^{6} + 1089732 T^{7} + 599392 T^{8} + 76404 T^{9} + 42901 T^{10} + 4455 T^{11} + 2107 T^{12} + 129 T^{13} + 58 T^{14} + 3 T^{15} + T^{16} \)
$3$ \( 282429536481 - 20920706406 T + 24794911296 T^{2} - 3099363912 T^{3} + 1120809069 T^{4} - 258988914 T^{5} + 39858075 T^{6} - 12527136 T^{7} + 1372950 T^{8} - 463968 T^{9} + 54675 T^{10} - 13158 T^{11} + 2109 T^{12} - 216 T^{13} + 64 T^{14} - 2 T^{15} + T^{16} \)
$5$ \( 28841980243729 - 38849239074834 T + 34081386376861 T^{2} - 18503393913870 T^{3} + 7528104076330 T^{4} - 2078095242474 T^{5} + 431614721467 T^{6} - 51994843578 T^{7} + 4920908542 T^{8} - 20676930 T^{9} + 52776610 T^{10} + 1206468 T^{11} + 273883 T^{12} + 12060 T^{13} + 1072 T^{14} + 30 T^{15} + T^{16} \)
$7$ \( ( 49 - 7 T + T^{2} )^{8} \)
$11$ \( 19352068738871482384 - 35849465685845921328 T + 70022073886875510460 T^{2} + 6126106677903291300 T^{3} + 1180370289224013085 T^{4} + 5974651611808092 T^{5} + 6354194039535937 T^{6} + 123362981984280 T^{7} + 15045119406877 T^{8} + 147232944726 T^{9} + 17620195678 T^{10} + 209410236 T^{11} + 12555637 T^{12} + 79428 T^{13} + 4336 T^{14} + 24 T^{15} + T^{16} \)
$13$ \( \)\(64\!\cdots\!64\)\( + \)\(37\!\cdots\!48\)\( T + \)\(31\!\cdots\!56\)\( T^{2} + \)\(11\!\cdots\!68\)\( T^{3} + \)\(66\!\cdots\!89\)\( T^{4} + 20910710007378924876 T^{5} + 886830899435150467 T^{6} + 20768687397798344 T^{7} + 653974912972458 T^{8} + 12443182657628 T^{9} + 321160744711 T^{10} + 4528688904 T^{11} + 83171594 T^{12} + 699772 T^{13} + 11511 T^{14} + 68 T^{15} + T^{16} \)
$17$ \( ( 2073513126204 + 1554538753260 T + 250373208411 T^{2} - 8466840468 T^{3} - 32700159 T^{4} + 2480508 T^{5} - 8910 T^{6} - 168 T^{7} + T^{8} )^{2} \)
$19$ \( ( -604815137888177 + 18413307722920 T + 473899377403 T^{2} - 15172506716 T^{3} - 63882233 T^{4} + 3616846 T^{5} - 12980 T^{6} - 176 T^{7} + T^{8} )^{2} \)
$23$ \( \)\(21\!\cdots\!25\)\( + \)\(22\!\cdots\!70\)\( T + \)\(29\!\cdots\!74\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(52\!\cdots\!59\)\( T^{4} + \)\(79\!\cdots\!98\)\( T^{5} + \)\(47\!\cdots\!08\)\( T^{6} + 3517723859432587740 T^{7} + 152181350553640662 T^{8} + 1548626050062468 T^{9} + 32495897803311 T^{10} + 187084830924 T^{11} + 1900227492 T^{12} + 7913088 T^{13} + 71811 T^{14} + 228 T^{15} + T^{16} \)
$29$ \( \)\(24\!\cdots\!00\)\( + \)\(55\!\cdots\!40\)\( T + \)\(10\!\cdots\!56\)\( T^{2} + \)\(50\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!25\)\( T^{4} + \)\(12\!\cdots\!60\)\( T^{5} + \)\(74\!\cdots\!66\)\( T^{6} + 70465987056078523104 T^{7} + 1458128461714497559 T^{8} + 12124179768658290 T^{9} + 179453796065908 T^{10} + 1582604233902 T^{11} + 13305242827 T^{12} + 67672590 T^{13} + 273127 T^{14} + 618 T^{15} + T^{16} \)
$31$ \( \)\(45\!\cdots\!64\)\( - \)\(18\!\cdots\!28\)\( T + \)\(68\!\cdots\!40\)\( T^{2} - \)\(25\!\cdots\!60\)\( T^{3} + \)\(85\!\cdots\!73\)\( T^{4} - \)\(64\!\cdots\!46\)\( T^{5} + \)\(10\!\cdots\!38\)\( T^{6} - \)\(27\!\cdots\!32\)\( T^{7} + 8636930654645597013 T^{8} - 11092690384226676 T^{9} + 335166439923216 T^{10} - 26374414860 T^{11} + 9147853935 T^{12} + 110484 T^{13} + 115983 T^{14} + 72 T^{15} + T^{16} \)
$37$ \( ( 71324346320230788 - 1595546991527796 T - 52771182253767 T^{2} + 124534542204 T^{3} + 6138784098 T^{4} + 12199410 T^{5} - 131643 T^{6} - 210 T^{7} + T^{8} )^{2} \)
$41$ \( \)\(21\!\cdots\!96\)\( - \)\(49\!\cdots\!00\)\( T + \)\(11\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} - \)\(33\!\cdots\!36\)\( T^{5} + \)\(82\!\cdots\!29\)\( T^{6} + \)\(31\!\cdots\!84\)\( T^{7} + 49314133653098744023 T^{8} + 201906961420668972 T^{9} + 1916269491754600 T^{10} + 5615393885100 T^{11} + 28486127767 T^{12} + 62292312 T^{13} + 263530 T^{14} + 420 T^{15} + T^{16} \)
$43$ \( \)\(43\!\cdots\!56\)\( + \)\(39\!\cdots\!04\)\( T + \)\(65\!\cdots\!80\)\( T^{2} + \)\(19\!\cdots\!28\)\( T^{3} + \)\(81\!\cdots\!93\)\( T^{4} + \)\(34\!\cdots\!58\)\( T^{5} + \)\(19\!\cdots\!74\)\( T^{6} - \)\(71\!\cdots\!80\)\( T^{7} + \)\(30\!\cdots\!85\)\( T^{8} - 1017173615702195738 T^{9} + 24829751960735392 T^{10} - 6200975997630 T^{11} + 145631819591 T^{12} - 19080886 T^{13} + 461133 T^{14} - 2 T^{15} + T^{16} \)
$47$ \( \)\(70\!\cdots\!16\)\( + \)\(98\!\cdots\!84\)\( T + \)\(33\!\cdots\!28\)\( T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} + \)\(46\!\cdots\!58\)\( T^{5} + \)\(82\!\cdots\!16\)\( T^{6} + \)\(30\!\cdots\!48\)\( T^{7} + 46608343810770753639 T^{8} + 200539340662390434 T^{9} + 1543211389982358 T^{10} + 5116276763784 T^{11} + 28446777261 T^{12} + 85943106 T^{13} + 326565 T^{14} + 570 T^{15} + T^{16} \)
$53$ \( ( 86158340642449528308 - 2650393862685934092 T + 24282335794603149 T^{2} - 81691443067554 T^{3} + 17633836809 T^{4} + 437892894 T^{5} - 587934 T^{6} - 528 T^{7} + T^{8} )^{2} \)
$59$ \( \)\(12\!\cdots\!04\)\( - \)\(27\!\cdots\!44\)\( T + \)\(66\!\cdots\!48\)\( T^{2} + \)\(59\!\cdots\!08\)\( T^{3} + \)\(58\!\cdots\!85\)\( T^{4} + \)\(18\!\cdots\!36\)\( T^{5} + \)\(10\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!44\)\( T^{7} + \)\(13\!\cdots\!49\)\( T^{8} + 10902094912961533638 T^{9} + 77902055977792626 T^{10} + 14914159463124 T^{11} + 326995394229 T^{12} + 15347970 T^{13} + 696915 T^{14} - 150 T^{15} + T^{16} \)
$61$ \( \)\(63\!\cdots\!25\)\( - \)\(27\!\cdots\!70\)\( T + \)\(96\!\cdots\!64\)\( T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!23\)\( T^{4} - \)\(86\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!62\)\( T^{6} + \)\(42\!\cdots\!94\)\( T^{7} + \)\(16\!\cdots\!62\)\( T^{8} + 8730419005915000562 T^{9} + 47477314905361183 T^{10} + 113392302812838 T^{11} + 276580191338 T^{12} + 285723382 T^{13} + 670113 T^{14} + 578 T^{15} + T^{16} \)
$67$ \( \)\(26\!\cdots\!24\)\( + \)\(66\!\cdots\!88\)\( T + \)\(16\!\cdots\!76\)\( T^{2} - \)\(14\!\cdots\!32\)\( T^{3} + \)\(46\!\cdots\!29\)\( T^{4} - \)\(79\!\cdots\!94\)\( T^{5} + \)\(10\!\cdots\!25\)\( T^{6} - \)\(15\!\cdots\!18\)\( T^{7} + \)\(86\!\cdots\!67\)\( T^{8} - 67401351317987780362 T^{9} + 442845833216678812 T^{10} - 349068470579910 T^{11} + 1076951523329 T^{12} - 573468188 T^{13} + 1614654 T^{14} - 898 T^{15} + T^{16} \)
$71$ \( ( \)\(36\!\cdots\!25\)\( + 41762863933845347934 T - 84625618285280943 T^{2} - 370893752864550 T^{3} + 541546496631 T^{4} + 1037538936 T^{5} - 1297368 T^{6} - 882 T^{7} + T^{8} )^{2} \)
$73$ \( ( \)\(45\!\cdots\!76\)\( + 20102375258878654764 T + 7761474434502279 T^{2} - 334588931457660 T^{3} + 98139802230 T^{4} + 1566589032 T^{5} - 1393785 T^{6} - 972 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(18\!\cdots\!81\)\( + \)\(48\!\cdots\!24\)\( T + \)\(66\!\cdots\!71\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!64\)\( T^{4} - \)\(61\!\cdots\!12\)\( T^{5} + \)\(60\!\cdots\!11\)\( T^{6} - \)\(11\!\cdots\!92\)\( T^{7} + \)\(18\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!08\)\( T^{9} + 3035745212081161276 T^{10} - 188665356763566 T^{11} + 3616369662683 T^{12} - 140819368 T^{13} + 2312334 T^{14} - 158 T^{15} + T^{16} \)
$83$ \( \)\(19\!\cdots\!00\)\( + \)\(17\!\cdots\!80\)\( T + \)\(12\!\cdots\!24\)\( T^{2} + \)\(23\!\cdots\!52\)\( T^{3} + \)\(30\!\cdots\!17\)\( T^{4} + \)\(18\!\cdots\!20\)\( T^{5} + \)\(80\!\cdots\!95\)\( T^{6} + \)\(22\!\cdots\!88\)\( T^{7} + \)\(48\!\cdots\!11\)\( T^{8} + \)\(75\!\cdots\!68\)\( T^{9} + 10000025916310895868 T^{10} + 10530605267612676 T^{11} + 10919355505209 T^{12} + 8956255896 T^{13} + 6620490 T^{14} + 2958 T^{15} + T^{16} \)
$89$ \( ( \)\(44\!\cdots\!84\)\( - \)\(56\!\cdots\!32\)\( T + 1426979281925511527 T^{2} + 1461374513042748 T^{3} - 5942862637383 T^{4} + 1699594764 T^{5} + 5183942 T^{6} - 4380 T^{7} + T^{8} )^{2} \)
$97$ \( \)\(48\!\cdots\!00\)\( + \)\(42\!\cdots\!00\)\( T + \)\(34\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!85\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(57\!\cdots\!66\)\( T^{6} + \)\(10\!\cdots\!72\)\( T^{7} + \)\(21\!\cdots\!95\)\( T^{8} + \)\(14\!\cdots\!32\)\( T^{9} + 19460632456966806684 T^{10} + 7559097291028692 T^{11} + 13126774498371 T^{12} + 2499642396 T^{13} + 4191849 T^{14} - 60 T^{15} + T^{16} \)
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