Properties

Label 6026.2.a.f
Level 6026
Weight 2
Character orbit 6026.a
Self dual Yes
Analytic conductor 48.118
Analytic rank 1
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6026.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.117852258\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( -\beta_{1} q^{3} \) \(+ q^{4}\) \( + \beta_{3} q^{5} \) \( -\beta_{1} q^{6} \) \( + ( -1 - \beta_{13} ) q^{7} \) \(+ q^{8}\) \( + \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( -\beta_{1} q^{3} \) \(+ q^{4}\) \( + \beta_{3} q^{5} \) \( -\beta_{1} q^{6} \) \( + ( -1 - \beta_{13} ) q^{7} \) \(+ q^{8}\) \( + \beta_{2} q^{9} \) \( + \beta_{3} q^{10} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{12} + \beta_{13} - \beta_{17} ) q^{11} \) \( -\beta_{1} q^{12} \) \( + ( -1 + \beta_{1} - \beta_{7} + \beta_{19} ) q^{13} \) \( + ( -1 - \beta_{13} ) q^{14} \) \( + ( -1 - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{10} - \beta_{15} + \beta_{18} ) q^{15} \) \(+ q^{16}\) \( + ( -\beta_{3} + \beta_{5} - \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{17} \) \( + \beta_{2} q^{18} \) \( + ( -2 + \beta_{1} + \beta_{7} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{19} \) \( + \beta_{3} q^{20} \) \( + ( -1 + \beta_{1} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{21} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{12} + \beta_{13} - \beta_{17} ) q^{22} \) \(+ q^{23}\) \( -\beta_{1} q^{24} \) \( + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{14} + \beta_{18} ) q^{25} \) \( + ( -1 + \beta_{1} - \beta_{7} + \beta_{19} ) q^{26} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} ) q^{27} \) \( + ( -1 - \beta_{13} ) q^{28} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{29} \) \( + ( -1 - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{10} - \beta_{15} + \beta_{18} ) q^{30} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{14} - 2 \beta_{19} ) q^{31} \) \(+ q^{32}\) \( + ( -\beta_{1} + \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{33} \) \( + ( -\beta_{3} + \beta_{5} - \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{34} \) \( + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{35} \) \( + \beta_{2} q^{36} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{14} - \beta_{15} - \beta_{19} ) q^{37} \) \( + ( -2 + \beta_{1} + \beta_{7} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{38} \) \( + ( -3 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{39} \) \( + \beta_{3} q^{40} \) \( + ( -3 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{41} \) \( + ( -1 + \beta_{1} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{42} \) \( + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} + \beta_{19} ) q^{43} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{12} + \beta_{13} - \beta_{17} ) q^{44} \) \( + ( -1 + \beta_{1} - \beta_{4} + 2 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{45} \) \(+ q^{46}\) \( + ( -2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{47} \) \( -\beta_{1} q^{48} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{49} \) \( + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{14} + \beta_{18} ) q^{50} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{14} - 2 \beta_{18} + \beta_{19} ) q^{51} \) \( + ( -1 + \beta_{1} - \beta_{7} + \beta_{19} ) q^{52} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{11} + \beta_{13} + \beta_{15} + \beta_{16} + \beta_{18} - \beta_{19} ) q^{53} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} ) q^{54} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{55} \) \( + ( -1 - \beta_{13} ) q^{56} \) \( + ( -2 + 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{14} + \beta_{15} - 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{57} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{58} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{59} \) \( + ( -1 - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{10} - \beta_{15} + \beta_{18} ) q^{60} \) \( + ( 1 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{61} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{14} - 2 \beta_{19} ) q^{62} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{63} \) \(+ q^{64}\) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{13} - \beta_{14} + \beta_{17} ) q^{65} \) \( + ( -\beta_{1} + \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{66} \) \( + ( -1 + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{12} - \beta_{19} ) q^{67} \) \( + ( -\beta_{3} + \beta_{5} - \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{68} \) \( -\beta_{1} q^{69} \) \( + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{70} \) \( + ( -3 - \beta_{1} - 3 \beta_{3} + 4 \beta_{4} - 6 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - \beta_{13} - 2 \beta_{15} - 3 \beta_{17} + 2 \beta_{18} ) q^{71} \) \( + \beta_{2} q^{72} \) \( + ( -5 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} - 3 \beta_{13} - 2 \beta_{14} - \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{73} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{14} - \beta_{15} - \beta_{19} ) q^{74} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} + \beta_{15} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{75} \) \( + ( -2 + \beta_{1} + \beta_{7} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{76} \) \( + ( -2 - \beta_{2} - \beta_{3} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{77} \) \( + ( -3 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{78} \) \( + ( -1 - 3 \beta_{1} + 2 \beta_{4} - \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{14} - \beta_{15} - \beta_{16} - 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{79} \) \( + \beta_{3} q^{80} \) \( + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{14} - \beta_{17} ) q^{81} \) \( + ( -3 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{82} \) \( + ( -3 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{83} \) \( + ( -1 + \beta_{1} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{84} \) \( + ( -4 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{85} \) \( + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} + \beta_{19} ) q^{86} \) \( + ( 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{87} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{12} + \beta_{13} - \beta_{17} ) q^{88} \) \( + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{7} + \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{18} ) q^{89} \) \( + ( -1 + \beta_{1} - \beta_{4} + 2 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{90} \) \( + ( 1 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} + 2 \beta_{12} - \beta_{15} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{91} \) \(+ q^{92}\) \( + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{15} + \beta_{16} - 3 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{93} \) \( + ( -2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{94} \) \( + ( 5 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} + \beta_{12} + 2 \beta_{13} + 5 \beta_{14} + \beta_{16} - 2 \beta_{18} - \beta_{19} ) q^{95} \) \( -\beta_{1} q^{96} \) \( + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{12} - 2 \beta_{13} + \beta_{16} + 3 \beta_{17} - 2 \beta_{18} ) q^{97} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{98} \) \( + ( \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{14} - \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut +\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 5q^{6} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 20q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut +\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 5q^{6} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 20q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut -\mathstrut 6q^{10} \) \(\mathstrut -\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 5q^{12} \) \(\mathstrut -\mathstrut 13q^{13} \) \(\mathstrut -\mathstrut 12q^{14} \) \(\mathstrut -\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 20q^{16} \) \(\mathstrut -\mathstrut 14q^{17} \) \(\mathstrut -\mathstrut q^{18} \) \(\mathstrut -\mathstrut 21q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut -\mathstrut 8q^{21} \) \(\mathstrut -\mathstrut 3q^{22} \) \(\mathstrut +\mathstrut 20q^{23} \) \(\mathstrut -\mathstrut 5q^{24} \) \(\mathstrut -\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 13q^{26} \) \(\mathstrut -\mathstrut 5q^{27} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut -\mathstrut 27q^{29} \) \(\mathstrut -\mathstrut 10q^{30} \) \(\mathstrut -\mathstrut 27q^{31} \) \(\mathstrut +\mathstrut 20q^{32} \) \(\mathstrut -\mathstrut 12q^{33} \) \(\mathstrut -\mathstrut 14q^{34} \) \(\mathstrut -\mathstrut 23q^{35} \) \(\mathstrut -\mathstrut q^{36} \) \(\mathstrut -\mathstrut 19q^{37} \) \(\mathstrut -\mathstrut 21q^{38} \) \(\mathstrut -\mathstrut 35q^{39} \) \(\mathstrut -\mathstrut 6q^{40} \) \(\mathstrut -\mathstrut 17q^{41} \) \(\mathstrut -\mathstrut 8q^{42} \) \(\mathstrut -\mathstrut 27q^{43} \) \(\mathstrut -\mathstrut 3q^{44} \) \(\mathstrut +\mathstrut 4q^{45} \) \(\mathstrut +\mathstrut 20q^{46} \) \(\mathstrut -\mathstrut 28q^{47} \) \(\mathstrut -\mathstrut 5q^{48} \) \(\mathstrut -\mathstrut 10q^{49} \) \(\mathstrut -\mathstrut 14q^{50} \) \(\mathstrut +\mathstrut 6q^{51} \) \(\mathstrut -\mathstrut 13q^{52} \) \(\mathstrut -\mathstrut 47q^{53} \) \(\mathstrut -\mathstrut 5q^{54} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 12q^{56} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 27q^{58} \) \(\mathstrut -\mathstrut 16q^{59} \) \(\mathstrut -\mathstrut 10q^{60} \) \(\mathstrut -\mathstrut 9q^{61} \) \(\mathstrut -\mathstrut 27q^{62} \) \(\mathstrut -\mathstrut 9q^{63} \) \(\mathstrut +\mathstrut 20q^{64} \) \(\mathstrut +\mathstrut 9q^{65} \) \(\mathstrut -\mathstrut 12q^{66} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 14q^{68} \) \(\mathstrut -\mathstrut 5q^{69} \) \(\mathstrut -\mathstrut 23q^{70} \) \(\mathstrut -\mathstrut 30q^{71} \) \(\mathstrut -\mathstrut q^{72} \) \(\mathstrut -\mathstrut 26q^{73} \) \(\mathstrut -\mathstrut 19q^{74} \) \(\mathstrut -\mathstrut 18q^{75} \) \(\mathstrut -\mathstrut 21q^{76} \) \(\mathstrut -\mathstrut 50q^{77} \) \(\mathstrut -\mathstrut 35q^{78} \) \(\mathstrut -\mathstrut 35q^{79} \) \(\mathstrut -\mathstrut 6q^{80} \) \(\mathstrut -\mathstrut 60q^{81} \) \(\mathstrut -\mathstrut 17q^{82} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 8q^{84} \) \(\mathstrut -\mathstrut 62q^{85} \) \(\mathstrut -\mathstrut 27q^{86} \) \(\mathstrut +\mathstrut q^{87} \) \(\mathstrut -\mathstrut 3q^{88} \) \(\mathstrut -\mathstrut 25q^{89} \) \(\mathstrut +\mathstrut 4q^{90} \) \(\mathstrut +\mathstrut 22q^{91} \) \(\mathstrut +\mathstrut 20q^{92} \) \(\mathstrut -\mathstrut 21q^{93} \) \(\mathstrut -\mathstrut 28q^{94} \) \(\mathstrut -\mathstrut 14q^{95} \) \(\mathstrut -\mathstrut 5q^{96} \) \(\mathstrut +\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut 10q^{98} \) \(\mathstrut -\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(5\) \(x^{19}\mathstrut -\mathstrut \) \(17\) \(x^{18}\mathstrut +\mathstrut \) \(115\) \(x^{17}\mathstrut +\mathstrut \) \(78\) \(x^{16}\mathstrut -\mathstrut \) \(1083\) \(x^{15}\mathstrut +\mathstrut \) \(248\) \(x^{14}\mathstrut +\mathstrut \) \(5359\) \(x^{13}\mathstrut -\mathstrut \) \(3723\) \(x^{12}\mathstrut -\mathstrut \) \(14776\) \(x^{11}\mathstrut +\mathstrut \) \(14837\) \(x^{10}\mathstrut +\mathstrut \) \(21886\) \(x^{9}\mathstrut -\mathstrut \) \(28084\) \(x^{8}\mathstrut -\mathstrut \) \(14682\) \(x^{7}\mathstrut +\mathstrut \) \(25315\) \(x^{6}\mathstrut +\mathstrut \) \(2042\) \(x^{5}\mathstrut -\mathstrut \) \(9137\) \(x^{4}\mathstrut +\mathstrut \) \(576\) \(x^{3}\mathstrut +\mathstrut \) \(1159\) \(x^{2}\mathstrut -\mathstrut \) \(78\) \(x\mathstrut -\mathstrut \) \(18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(99716990\) \(\nu^{19}\mathstrut -\mathstrut \) \(165884442\) \(\nu^{18}\mathstrut -\mathstrut \) \(2663674000\) \(\nu^{17}\mathstrut +\mathstrut \) \(3599346675\) \(\nu^{16}\mathstrut +\mathstrut \) \(30194840133\) \(\nu^{15}\mathstrut -\mathstrut \) \(31119918252\) \(\nu^{14}\mathstrut -\mathstrut \) \(187493827325\) \(\nu^{13}\mathstrut +\mathstrut \) \(136061253901\) \(\nu^{12}\mathstrut +\mathstrut \) \(686826922760\) \(\nu^{11}\mathstrut -\mathstrut \) \(311582977642\) \(\nu^{10}\mathstrut -\mathstrut \) \(1492751196985\) \(\nu^{9}\mathstrut +\mathstrut \) \(330166097514\) \(\nu^{8}\mathstrut +\mathstrut \) \(1839263430391\) \(\nu^{7}\mathstrut -\mathstrut \) \(50250467098\) \(\nu^{6}\mathstrut -\mathstrut \) \(1142544646218\) \(\nu^{5}\mathstrut -\mathstrut \) \(154167082409\) \(\nu^{4}\mathstrut +\mathstrut \) \(266688463738\) \(\nu^{3}\mathstrut +\mathstrut \) \(80734359230\) \(\nu^{2}\mathstrut -\mathstrut \) \(6210157485\) \(\nu\mathstrut -\mathstrut \) \(5029558767\)\()/\)\(1711992393\)
\(\beta_{4}\)\(=\)\((\)\(152897921\) \(\nu^{19}\mathstrut -\mathstrut \) \(788021802\) \(\nu^{18}\mathstrut -\mathstrut \) \(2581287385\) \(\nu^{17}\mathstrut +\mathstrut \) \(17942676336\) \(\nu^{16}\mathstrut +\mathstrut \) \(12991637169\) \(\nu^{15}\mathstrut -\mathstrut \) \(168614176551\) \(\nu^{14}\mathstrut +\mathstrut \) \(12131055271\) \(\nu^{13}\mathstrut +\mathstrut \) \(845435323468\) \(\nu^{12}\mathstrut -\mathstrut \) \(353315039944\) \(\nu^{11}\mathstrut -\mathstrut \) \(2433681484336\) \(\nu^{10}\mathstrut +\mathstrut \) \(1387964422985\) \(\nu^{9}\mathstrut +\mathstrut \) \(4014750146469\) \(\nu^{8}\mathstrut -\mathstrut \) \(2433986096297\) \(\nu^{7}\mathstrut -\mathstrut \) \(3564239814652\) \(\nu^{6}\mathstrut +\mathstrut \) \(1927564243203\) \(\nu^{5}\mathstrut +\mathstrut \) \(1484823148105\) \(\nu^{4}\mathstrut -\mathstrut \) \(514352194880\) \(\nu^{3}\mathstrut -\mathstrut \) \(239014121539\) \(\nu^{2}\mathstrut +\mathstrut \) \(19091308350\) \(\nu\mathstrut -\mathstrut \) \(1903874094\)\()/\)\(1711992393\)
\(\beta_{5}\)\(=\)\((\)\(309851458\) \(\nu^{19}\mathstrut -\mathstrut \) \(1383955827\) \(\nu^{18}\mathstrut -\mathstrut \) \(6233624999\) \(\nu^{17}\mathstrut +\mathstrut \) \(33136669317\) \(\nu^{16}\mathstrut +\mathstrut \) \(46406228355\) \(\nu^{15}\mathstrut -\mathstrut \) \(328492457904\) \(\nu^{14}\mathstrut -\mathstrut \) \(134696224735\) \(\nu^{13}\mathstrut +\mathstrut \) \(1740985774982\) \(\nu^{12}\mathstrut -\mathstrut \) \(79154083532\) \(\nu^{11}\mathstrut -\mathstrut \) \(5302318876388\) \(\nu^{10}\mathstrut +\mathstrut \) \(1465883080024\) \(\nu^{9}\mathstrut +\mathstrut \) \(9252549939678\) \(\nu^{8}\mathstrut -\mathstrut \) \(3483690211072\) \(\nu^{7}\mathstrut -\mathstrut \) \(8681537826470\) \(\nu^{6}\mathstrut +\mathstrut \) \(3181938636582\) \(\nu^{5}\mathstrut +\mathstrut \) \(3828596607116\) \(\nu^{4}\mathstrut -\mathstrut \) \(868959764302\) \(\nu^{3}\mathstrut -\mathstrut \) \(655658120918\) \(\nu^{2}\mathstrut +\mathstrut \) \(12667611897\) \(\nu\mathstrut +\mathstrut \) \(10210385484\)\()/\)\(1711992393\)
\(\beta_{6}\)\(=\)\((\)\(322669970\) \(\nu^{19}\mathstrut -\mathstrut \) \(1360457148\) \(\nu^{18}\mathstrut -\mathstrut \) \(6450050938\) \(\nu^{17}\mathstrut +\mathstrut \) \(31439546223\) \(\nu^{16}\mathstrut +\mathstrut \) \(48925034667\) \(\nu^{15}\mathstrut -\mathstrut \) \(299751293436\) \(\nu^{14}\mathstrut -\mathstrut \) \(160904820680\) \(\nu^{13}\mathstrut +\mathstrut \) \(1522754372488\) \(\nu^{12}\mathstrut +\mathstrut \) \(94977561293\) \(\nu^{11}\mathstrut -\mathstrut \) \(4429190235145\) \(\nu^{10}\mathstrut +\mathstrut \) \(822808635896\) \(\nu^{9}\mathstrut +\mathstrut \) \(7345735584573\) \(\nu^{8}\mathstrut -\mathstrut \) \(2230153018763\) \(\nu^{7}\mathstrut -\mathstrut \) \(6497668918675\) \(\nu^{6}\mathstrut +\mathstrut \) \(2022451755138\) \(\nu^{5}\mathstrut +\mathstrut \) \(2669559331153\) \(\nu^{4}\mathstrut -\mathstrut \) \(479591138117\) \(\nu^{3}\mathstrut -\mathstrut \) \(444981502009\) \(\nu^{2}\mathstrut -\mathstrut \) \(13818490434\) \(\nu\mathstrut +\mathstrut \) \(11277017919\)\()/\)\(1711992393\)
\(\beta_{7}\)\(=\)\((\)\(322669970\) \(\nu^{19}\mathstrut -\mathstrut \) \(1360457148\) \(\nu^{18}\mathstrut -\mathstrut \) \(6450050938\) \(\nu^{17}\mathstrut +\mathstrut \) \(31439546223\) \(\nu^{16}\mathstrut +\mathstrut \) \(48925034667\) \(\nu^{15}\mathstrut -\mathstrut \) \(299751293436\) \(\nu^{14}\mathstrut -\mathstrut \) \(160904820680\) \(\nu^{13}\mathstrut +\mathstrut \) \(1522754372488\) \(\nu^{12}\mathstrut +\mathstrut \) \(94977561293\) \(\nu^{11}\mathstrut -\mathstrut \) \(4429190235145\) \(\nu^{10}\mathstrut +\mathstrut \) \(822808635896\) \(\nu^{9}\mathstrut +\mathstrut \) \(7345735584573\) \(\nu^{8}\mathstrut -\mathstrut \) \(2230153018763\) \(\nu^{7}\mathstrut -\mathstrut \) \(6497668918675\) \(\nu^{6}\mathstrut +\mathstrut \) \(2022451755138\) \(\nu^{5}\mathstrut +\mathstrut \) \(2669559331153\) \(\nu^{4}\mathstrut -\mathstrut \) \(481303130510\) \(\nu^{3}\mathstrut -\mathstrut \) \(443269509616\) \(\nu^{2}\mathstrut -\mathstrut \) \(6970520862\) \(\nu\mathstrut +\mathstrut \) \(7853033133\)\()/\)\(1711992393\)
\(\beta_{8}\)\(=\)\((\)\(146343330\) \(\nu^{19}\mathstrut -\mathstrut \) \(627942667\) \(\nu^{18}\mathstrut -\mathstrut \) \(2750855046\) \(\nu^{17}\mathstrut +\mathstrut \) \(13927919237\) \(\nu^{16}\mathstrut +\mathstrut \) \(19013317515\) \(\nu^{15}\mathstrut -\mathstrut \) \(126771603270\) \(\nu^{14}\mathstrut -\mathstrut \) \(50992377522\) \(\nu^{13}\mathstrut +\mathstrut \) \(611952043660\) \(\nu^{12}\mathstrut -\mathstrut \) \(27928299023\) \(\nu^{11}\mathstrut -\mathstrut \) \(1685927227003\) \(\nu^{10}\mathstrut +\mathstrut \) \(474253792532\) \(\nu^{9}\mathstrut +\mathstrut \) \(2647408333697\) \(\nu^{8}\mathstrut -\mathstrut \) \(1049940919284\) \(\nu^{7}\mathstrut -\mathstrut \) \(2228055002861\) \(\nu^{6}\mathstrut +\mathstrut \) \(917906878673\) \(\nu^{5}\mathstrut +\mathstrut \) \(880564478145\) \(\nu^{4}\mathstrut -\mathstrut \) \(263557449692\) \(\nu^{3}\mathstrut -\mathstrut \) \(141470821499\) \(\nu^{2}\mathstrut +\mathstrut \) \(13425493694\) \(\nu\mathstrut +\mathstrut \) \(3805064997\)\()/\)\(570664131\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(477486794\) \(\nu^{19}\mathstrut +\mathstrut \) \(2377631685\) \(\nu^{18}\mathstrut +\mathstrut \) \(8657638597\) \(\nu^{17}\mathstrut -\mathstrut \) \(56024761614\) \(\nu^{16}\mathstrut -\mathstrut \) \(50354083845\) \(\nu^{15}\mathstrut +\mathstrut \) \(545581112568\) \(\nu^{14}\mathstrut +\mathstrut \) \(14470325336\) \(\nu^{13}\mathstrut -\mathstrut \) \(2832716706445\) \(\nu^{12}\mathstrut +\mathstrut \) \(1050361806526\) \(\nu^{11}\mathstrut +\mathstrut \) \(8410243049992\) \(\nu^{10}\mathstrut -\mathstrut \) \(4771815292994\) \(\nu^{9}\mathstrut -\mathstrut \) \(14170725778413\) \(\nu^{8}\mathstrut +\mathstrut \) \(9136296553166\) \(\nu^{7}\mathstrut +\mathstrut \) \(12597173558332\) \(\nu^{6}\mathstrut -\mathstrut \) \(7746619103055\) \(\nu^{5}\mathstrut -\mathstrut \) \(5106166101469\) \(\nu^{4}\mathstrut +\mathstrut \) \(2213572376246\) \(\nu^{3}\mathstrut +\mathstrut \) \(862543526110\) \(\nu^{2}\mathstrut -\mathstrut \) \(123020445429\) \(\nu\mathstrut -\mathstrut \) \(21496095405\)\()/\)\(1711992393\)
\(\beta_{10}\)\(=\)\((\)\(550477613\) \(\nu^{19}\mathstrut -\mathstrut \) \(2275865718\) \(\nu^{18}\mathstrut -\mathstrut \) \(11409071752\) \(\nu^{17}\mathstrut +\mathstrut \) \(54163977555\) \(\nu^{16}\mathstrut +\mathstrut \) \(89385080193\) \(\nu^{15}\mathstrut -\mathstrut \) \(531636343350\) \(\nu^{14}\mathstrut -\mathstrut \) \(296476727960\) \(\nu^{13}\mathstrut +\mathstrut \) \(2775395086510\) \(\nu^{12}\mathstrut +\mathstrut \) \(96329231267\) \(\nu^{11}\mathstrut -\mathstrut \) \(8265873305065\) \(\nu^{10}\mathstrut +\mathstrut \) \(2096214081833\) \(\nu^{9}\mathstrut +\mathstrut \) \(13944905779887\) \(\nu^{8}\mathstrut -\mathstrut \) \(5708901648527\) \(\nu^{7}\mathstrut -\mathstrut \) \(12398618288461\) \(\nu^{6}\mathstrut +\mathstrut \) \(5657252453124\) \(\nu^{5}\mathstrut +\mathstrut \) \(5021707486708\) \(\nu^{4}\mathstrut -\mathstrut \) \(1822112773976\) \(\nu^{3}\mathstrut -\mathstrut \) \(827450554963\) \(\nu^{2}\mathstrut +\mathstrut \) \(129553986273\) \(\nu\mathstrut +\mathstrut \) \(18210231207\)\()/\)\(1711992393\)
\(\beta_{11}\)\(=\)\((\)\(562516600\) \(\nu^{19}\mathstrut -\mathstrut \) \(2270260821\) \(\nu^{18}\mathstrut -\mathstrut \) \(11766008126\) \(\nu^{17}\mathstrut +\mathstrut \) \(53784538836\) \(\nu^{16}\mathstrut +\mathstrut \) \(94239617940\) \(\nu^{15}\mathstrut -\mathstrut \) \(525192157785\) \(\nu^{14}\mathstrut -\mathstrut \) \(334836535351\) \(\nu^{13}\mathstrut +\mathstrut \) \(2725240933223\) \(\nu^{12}\mathstrut +\mathstrut \) \(284298537844\) \(\nu^{11}\mathstrut -\mathstrut \) \(8055235042100\) \(\nu^{10}\mathstrut +\mathstrut \) \(1516003317697\) \(\nu^{9}\mathstrut +\mathstrut \) \(13445110060605\) \(\nu^{8}\mathstrut -\mathstrut \) \(4598705287915\) \(\nu^{7}\mathstrut -\mathstrut \) \(11746626266969\) \(\nu^{6}\mathstrut +\mathstrut \) \(4417753791675\) \(\nu^{5}\mathstrut +\mathstrut \) \(4609864588088\) \(\nu^{4}\mathstrut -\mathstrut \) \(1130822268817\) \(\nu^{3}\mathstrut -\mathstrut \) \(741450867083\) \(\nu^{2}\mathstrut +\mathstrut \) \(7395908349\) \(\nu\mathstrut +\mathstrut \) \(12725619606\)\()/\)\(1711992393\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(840747436\) \(\nu^{19}\mathstrut +\mathstrut \) \(3451389795\) \(\nu^{18}\mathstrut +\mathstrut \) \(17340713522\) \(\nu^{17}\mathstrut -\mathstrut \) \(81372015369\) \(\nu^{16}\mathstrut -\mathstrut \) \(135968571153\) \(\nu^{15}\mathstrut +\mathstrut \) \(791153125932\) \(\nu^{14}\mathstrut +\mathstrut \) \(460823919703\) \(\nu^{13}\mathstrut -\mathstrut \) \(4091073927500\) \(\nu^{12}\mathstrut -\mathstrut \) \(252098034391\) \(\nu^{11}\mathstrut +\mathstrut \) \(12065066105960\) \(\nu^{10}\mathstrut -\mathstrut \) \(2722915606537\) \(\nu^{9}\mathstrut -\mathstrut \) \(20128396990494\) \(\nu^{8}\mathstrut +\mathstrut \) \(7662576443257\) \(\nu^{7}\mathstrut +\mathstrut \) \(17620952974694\) \(\nu^{6}\mathstrut -\mathstrut \) \(7487287629810\) \(\nu^{5}\mathstrut -\mathstrut \) \(6931059756389\) \(\nu^{4}\mathstrut +\mathstrut \) \(2217972391426\) \(\nu^{3}\mathstrut +\mathstrut \) \(1077516056294\) \(\nu^{2}\mathstrut -\mathstrut \) \(98272405995\) \(\nu\mathstrut -\mathstrut \) \(15043298778\)\()/\)\(1711992393\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(306455260\) \(\nu^{19}\mathstrut +\mathstrut \) \(1193388237\) \(\nu^{18}\mathstrut +\mathstrut \) \(6440353895\) \(\nu^{17}\mathstrut -\mathstrut \) \(27863463012\) \(\nu^{16}\mathstrut -\mathstrut \) \(52656595317\) \(\nu^{15}\mathstrut +\mathstrut \) \(267919145877\) \(\nu^{14}\mathstrut +\mathstrut \) \(200567866756\) \(\nu^{13}\mathstrut -\mathstrut \) \(1368544326728\) \(\nu^{12}\mathstrut -\mathstrut \) \(271585338031\) \(\nu^{11}\mathstrut +\mathstrut \) \(3983590226324\) \(\nu^{10}\mathstrut -\mathstrut \) \(438853391215\) \(\nu^{9}\mathstrut -\mathstrut \) \(6556627302303\) \(\nu^{8}\mathstrut +\mathstrut \) \(1860096940774\) \(\nu^{7}\mathstrut +\mathstrut \) \(5660449894751\) \(\nu^{6}\mathstrut -\mathstrut \) \(1963791412764\) \(\nu^{5}\mathstrut -\mathstrut \) \(2191104403268\) \(\nu^{4}\mathstrut +\mathstrut \) \(583750175422\) \(\nu^{3}\mathstrut +\mathstrut \) \(334887564545\) \(\nu^{2}\mathstrut -\mathstrut \) \(25266898338\) \(\nu\mathstrut -\mathstrut \) \(6583539255\)\()/\)\(570664131\)
\(\beta_{14}\)\(=\)\((\)\(306841820\) \(\nu^{19}\mathstrut -\mathstrut \) \(1142759006\) \(\nu^{18}\mathstrut -\mathstrut \) \(6603875713\) \(\nu^{17}\mathstrut +\mathstrut \) \(26675284060\) \(\nu^{16}\mathstrut +\mathstrut \) \(56303390241\) \(\nu^{15}\mathstrut -\mathstrut \) \(256415042700\) \(\nu^{14}\mathstrut -\mathstrut \) \(234380734790\) \(\nu^{13}\mathstrut +\mathstrut \) \(1309458273684\) \(\nu^{12}\mathstrut +\mathstrut \) \(436085177527\) \(\nu^{11}\mathstrut -\mathstrut \) \(3812158841505\) \(\nu^{10}\mathstrut -\mathstrut \) \(5754095250\) \(\nu^{9}\mathstrut +\mathstrut \) \(6281885236696\) \(\nu^{8}\mathstrut -\mathstrut \) \(1212682461482\) \(\nu^{7}\mathstrut -\mathstrut \) \(5441991275711\) \(\nu^{6}\mathstrut +\mathstrut \) \(1522471869796\) \(\nu^{5}\mathstrut +\mathstrut \) \(2119661708299\) \(\nu^{4}\mathstrut -\mathstrut \) \(494972035686\) \(\nu^{3}\mathstrut -\mathstrut \) \(316169845004\) \(\nu^{2}\mathstrut +\mathstrut \) \(28438633600\) \(\nu\mathstrut +\mathstrut \) \(3636815862\)\()/\)\(570664131\)
\(\beta_{15}\)\(=\)\((\)\(1084667243\) \(\nu^{19}\mathstrut -\mathstrut \) \(4546314924\) \(\nu^{18}\mathstrut -\mathstrut \) \(21817310038\) \(\nu^{17}\mathstrut +\mathstrut \) \(105828203667\) \(\nu^{16}\mathstrut +\mathstrut \) \(165046457061\) \(\nu^{15}\mathstrut -\mathstrut \) \(1015296823950\) \(\nu^{14}\mathstrut -\mathstrut \) \(521211788708\) \(\nu^{13}\mathstrut +\mathstrut \) \(5180055874786\) \(\nu^{12}\mathstrut +\mathstrut \) \(90802281113\) \(\nu^{11}\mathstrut -\mathstrut \) \(15082153681294\) \(\nu^{10}\mathstrut +\mathstrut \) \(3787701103535\) \(\nu^{9}\mathstrut +\mathstrut \) \(24888060042309\) \(\nu^{8}\mathstrut -\mathstrut \) \(9712589542436\) \(\nu^{7}\mathstrut -\mathstrut \) \(21643300706941\) \(\nu^{6}\mathstrut +\mathstrut \) \(9087254168172\) \(\nu^{5}\mathstrut +\mathstrut \) \(8535466217248\) \(\nu^{4}\mathstrut -\mathstrut \) \(2585383086992\) \(\nu^{3}\mathstrut -\mathstrut \) \(1351112921560\) \(\nu^{2}\mathstrut +\mathstrut \) \(110179494222\) \(\nu\mathstrut +\mathstrut \) \(21903959223\)\()/\)\(1711992393\)
\(\beta_{16}\)\(=\)\((\)\(1089425851\) \(\nu^{19}\mathstrut -\mathstrut \) \(4793379879\) \(\nu^{18}\mathstrut -\mathstrut \) \(21455653031\) \(\nu^{17}\mathstrut +\mathstrut \) \(112365319416\) \(\nu^{16}\mathstrut +\mathstrut \) \(154602262989\) \(\nu^{15}\mathstrut -\mathstrut \) \(1087065350931\) \(\nu^{14}\mathstrut -\mathstrut \) \(412087129414\) \(\nu^{13}\mathstrut +\mathstrut \) \(5600687059850\) \(\nu^{12}\mathstrut -\mathstrut \) \(493304121437\) \(\nu^{11}\mathstrut -\mathstrut \) \(16492312061762\) \(\nu^{10}\mathstrut +\mathstrut \) \(5529256958347\) \(\nu^{9}\mathstrut +\mathstrut \) \(27580460910462\) \(\nu^{8}\mathstrut -\mathstrut \) \(12573673222303\) \(\nu^{7}\mathstrut -\mathstrut \) \(24392018763977\) \(\nu^{6}\mathstrut +\mathstrut \) \(11437902014430\) \(\nu^{5}\mathstrut +\mathstrut \) \(9847996099748\) \(\nu^{4}\mathstrut -\mathstrut \) \(3344500909942\) \(\nu^{3}\mathstrut -\mathstrut \) \(1588589428406\) \(\nu^{2}\mathstrut +\mathstrut \) \(167597701761\) \(\nu\mathstrut +\mathstrut \) \(26342084538\)\()/\)\(1711992393\)
\(\beta_{17}\)\(=\)\((\)\(4456799\) \(\nu^{19}\mathstrut -\mathstrut \) \(18879387\) \(\nu^{18}\mathstrut -\mathstrut \) \(90366943\) \(\nu^{17}\mathstrut +\mathstrut \) \(445558719\) \(\nu^{16}\mathstrut +\mathstrut \) \(685127538\) \(\nu^{15}\mathstrut -\mathstrut \) \(4338879945\) \(\nu^{14}\mathstrut -\mathstrut \) \(2109921236\) \(\nu^{13}\mathstrut +\mathstrut \) \(22488696304\) \(\nu^{12}\mathstrut -\mathstrut \) \(294234073\) \(\nu^{11}\mathstrut -\mathstrut \) \(66542626897\) \(\nu^{10}\mathstrut +\mathstrut \) \(18907936892\) \(\nu^{9}\mathstrut +\mathstrut \) \(111562121574\) \(\nu^{8}\mathstrut -\mathstrut \) \(47391522458\) \(\nu^{7}\mathstrut -\mathstrut \) \(98477264995\) \(\nu^{6}\mathstrut +\mathstrut \) \(44859980994\) \(\nu^{5}\mathstrut +\mathstrut \) \(39427171861\) \(\nu^{4}\mathstrut -\mathstrut \) \(13420672217\) \(\nu^{3}\mathstrut -\mathstrut \) \(6391055815\) \(\nu^{2}\mathstrut +\mathstrut \) \(740541705\) \(\nu\mathstrut +\mathstrut \) \(133089489\)\()/6661449\)
\(\beta_{18}\)\(=\)\((\)\(1571933387\) \(\nu^{19}\mathstrut -\mathstrut \) \(6592015260\) \(\nu^{18}\mathstrut -\mathstrut \) \(31890712168\) \(\nu^{17}\mathstrut +\mathstrut \) \(154671482871\) \(\nu^{16}\mathstrut +\mathstrut \) \(243686192967\) \(\nu^{15}\mathstrut -\mathstrut \) \(1496966561592\) \(\nu^{14}\mathstrut -\mathstrut \) \(779896124435\) \(\nu^{13}\mathstrut +\mathstrut \) \(7710757987687\) \(\nu^{12}\mathstrut +\mathstrut \) \(160415769779\) \(\nu^{11}\mathstrut -\mathstrut \) \(22682866537195\) \(\nu^{10}\mathstrut +\mathstrut \) \(5678248146920\) \(\nu^{9}\mathstrut +\mathstrut \) \(37854401980263\) \(\nu^{8}\mathstrut -\mathstrut \) \(14792189063120\) \(\nu^{7}\mathstrut -\mathstrut \) \(33358718318455\) \(\nu^{6}\mathstrut +\mathstrut \) \(14054638663002\) \(\nu^{5}\mathstrut +\mathstrut \) \(13409940203227\) \(\nu^{4}\mathstrut -\mathstrut \) \(4131055705850\) \(\nu^{3}\mathstrut -\mathstrut \) \(2180302356475\) \(\nu^{2}\mathstrut +\mathstrut \) \(204713127345\) \(\nu\mathstrut +\mathstrut \) \(44758625463\)\()/\)\(1711992393\)
\(\beta_{19}\)\(=\)\((\)\(600429471\) \(\nu^{19}\mathstrut -\mathstrut \) \(2519966401\) \(\nu^{18}\mathstrut -\mathstrut \) \(12156366042\) \(\nu^{17}\mathstrut +\mathstrut \) \(59005753619\) \(\nu^{16}\mathstrut +\mathstrut \) \(92842904574\) \(\nu^{15}\mathstrut -\mathstrut \) \(570148648821\) \(\nu^{14}\mathstrut -\mathstrut \) \(298853019066\) \(\nu^{13}\mathstrut +\mathstrut \) \(2934376722508\) \(\nu^{12}\mathstrut +\mathstrut \) \(81393002008\) \(\nu^{11}\mathstrut -\mathstrut \) \(8637298145164\) \(\nu^{10}\mathstrut +\mathstrut \) \(2074169438390\) \(\nu^{9}\mathstrut +\mathstrut \) \(14459887772984\) \(\nu^{8}\mathstrut -\mathstrut \) \(5460601016910\) \(\nu^{7}\mathstrut -\mathstrut \) \(12843632042951\) \(\nu^{6}\mathstrut +\mathstrut \) \(5219898632999\) \(\nu^{5}\mathstrut +\mathstrut \) \(5245158178278\) \(\nu^{4}\mathstrut -\mathstrut \) \(1558745920940\) \(\nu^{3}\mathstrut -\mathstrut \) \(859091649656\) \(\nu^{2}\mathstrut +\mathstrut \) \(81393303662\) \(\nu\mathstrut +\mathstrut \) \(16459377834\)\()/\)\(570664131\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(\beta_{19}\mathstrut -\mathstrut \) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{6}\)\(=\)\(\beta_{19}\mathstrut +\mathstrut \) \(2\) \(\beta_{18}\mathstrut -\mathstrut \) \(12\) \(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(12\) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(47\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(100\)
\(\nu^{7}\)\(=\)\(14\) \(\beta_{19}\mathstrut -\mathstrut \) \(9\) \(\beta_{18}\mathstrut -\mathstrut \) \(17\) \(\beta_{17}\mathstrut +\mathstrut \) \(13\) \(\beta_{16}\mathstrut -\mathstrut \) \(4\) \(\beta_{15}\mathstrut +\mathstrut \) \(37\) \(\beta_{14}\mathstrut +\mathstrut \) \(22\) \(\beta_{13}\mathstrut +\mathstrut \) \(9\) \(\beta_{12}\mathstrut +\mathstrut \) \(25\) \(\beta_{11}\mathstrut -\mathstrut \) \(16\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(95\) \(\beta_{7}\mathstrut +\mathstrut \) \(66\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(13\) \(\beta_{4}\mathstrut -\mathstrut \) \(30\) \(\beta_{3}\mathstrut +\mathstrut \) \(70\) \(\beta_{2}\mathstrut +\mathstrut \) \(127\) \(\beta_{1}\mathstrut +\mathstrut \) \(145\)
\(\nu^{8}\)\(=\)\(22\) \(\beta_{19}\mathstrut +\mathstrut \) \(28\) \(\beta_{18}\mathstrut -\mathstrut \) \(114\) \(\beta_{17}\mathstrut +\mathstrut \) \(17\) \(\beta_{16}\mathstrut -\mathstrut \) \(31\) \(\beta_{15}\mathstrut +\mathstrut \) \(109\) \(\beta_{14}\mathstrut +\mathstrut \) \(15\) \(\beta_{13}\mathstrut -\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(31\) \(\beta_{11}\mathstrut -\mathstrut \) \(49\) \(\beta_{10}\mathstrut -\mathstrut \) \(78\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(155\) \(\beta_{7}\mathstrut +\mathstrut \) \(79\) \(\beta_{6}\mathstrut -\mathstrut \) \(22\) \(\beta_{5}\mathstrut -\mathstrut \) \(71\) \(\beta_{4}\mathstrut -\mathstrut \) \(117\) \(\beta_{3}\mathstrut +\mathstrut \) \(317\) \(\beta_{2}\mathstrut +\mathstrut \) \(131\) \(\beta_{1}\mathstrut +\mathstrut \) \(673\)
\(\nu^{9}\)\(=\)\(150\) \(\beta_{19}\mathstrut -\mathstrut \) \(51\) \(\beta_{18}\mathstrut -\mathstrut \) \(202\) \(\beta_{17}\mathstrut +\mathstrut \) \(129\) \(\beta_{16}\mathstrut -\mathstrut \) \(73\) \(\beta_{15}\mathstrut +\mathstrut \) \(343\) \(\beta_{14}\mathstrut +\mathstrut \) \(184\) \(\beta_{13}\mathstrut +\mathstrut \) \(58\) \(\beta_{12}\mathstrut +\mathstrut \) \(236\) \(\beta_{11}\mathstrut -\mathstrut \) \(186\) \(\beta_{10}\mathstrut -\mathstrut \) \(109\) \(\beta_{9}\mathstrut -\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(766\) \(\beta_{7}\mathstrut +\mathstrut \) \(454\) \(\beta_{6}\mathstrut -\mathstrut \) \(150\) \(\beta_{5}\mathstrut -\mathstrut \) \(127\) \(\beta_{4}\mathstrut -\mathstrut \) \(320\) \(\beta_{3}\mathstrut +\mathstrut \) \(524\) \(\beta_{2}\mathstrut +\mathstrut \) \(841\) \(\beta_{1}\mathstrut +\mathstrut \) \(1229\)
\(\nu^{10}\)\(=\)\(300\) \(\beta_{19}\mathstrut +\mathstrut \) \(281\) \(\beta_{18}\mathstrut -\mathstrut \) \(1009\) \(\beta_{17}\mathstrut +\mathstrut \) \(209\) \(\beta_{16}\mathstrut -\mathstrut \) \(349\) \(\beta_{15}\mathstrut +\mathstrut \) \(917\) \(\beta_{14}\mathstrut +\mathstrut \) \(164\) \(\beta_{13}\mathstrut -\mathstrut \) \(55\) \(\beta_{12}\mathstrut +\mathstrut \) \(353\) \(\beta_{11}\mathstrut -\mathstrut \) \(569\) \(\beta_{10}\mathstrut -\mathstrut \) \(568\) \(\beta_{9}\mathstrut -\mathstrut \) \(38\) \(\beta_{8}\mathstrut -\mathstrut \) \(1421\) \(\beta_{7}\mathstrut +\mathstrut \) \(578\) \(\beta_{6}\mathstrut -\mathstrut \) \(298\) \(\beta_{5}\mathstrut -\mathstrut \) \(553\) \(\beta_{4}\mathstrut -\mathstrut \) \(1064\) \(\beta_{3}\mathstrut +\mathstrut \) \(2153\) \(\beta_{2}\mathstrut +\mathstrut \) \(1190\) \(\beta_{1}\mathstrut +\mathstrut \) \(4730\)
\(\nu^{11}\)\(=\)\(1457\) \(\beta_{19}\mathstrut -\mathstrut \) \(161\) \(\beta_{18}\mathstrut -\mathstrut \) \(2077\) \(\beta_{17}\mathstrut +\mathstrut \) \(1166\) \(\beta_{16}\mathstrut -\mathstrut \) \(898\) \(\beta_{15}\mathstrut +\mathstrut \) \(2892\) \(\beta_{14}\mathstrut +\mathstrut \) \(1416\) \(\beta_{13}\mathstrut +\mathstrut \) \(295\) \(\beta_{12}\mathstrut +\mathstrut \) \(2033\) \(\beta_{11}\mathstrut -\mathstrut \) \(1898\) \(\beta_{10}\mathstrut -\mathstrut \) \(901\) \(\beta_{9}\mathstrut -\mathstrut \) \(84\) \(\beta_{8}\mathstrut -\mathstrut \) \(6038\) \(\beta_{7}\mathstrut +\mathstrut \) \(3034\) \(\beta_{6}\mathstrut -\mathstrut \) \(1453\) \(\beta_{5}\mathstrut -\mathstrut \) \(1131\) \(\beta_{4}\mathstrut -\mathstrut \) \(3015\) \(\beta_{3}\mathstrut +\mathstrut \) \(3856\) \(\beta_{2}\mathstrut +\mathstrut \) \(5903\) \(\beta_{1}\mathstrut +\mathstrut \) \(9882\)
\(\nu^{12}\)\(=\)\(3342\) \(\beta_{19}\mathstrut +\mathstrut \) \(2525\) \(\beta_{18}\mathstrut -\mathstrut \) \(8679\) \(\beta_{17}\mathstrut +\mathstrut \) \(2212\) \(\beta_{16}\mathstrut -\mathstrut \) \(3470\) \(\beta_{15}\mathstrut +\mathstrut \) \(7512\) \(\beta_{14}\mathstrut +\mathstrut \) \(1590\) \(\beta_{13}\mathstrut -\mathstrut \) \(694\) \(\beta_{12}\mathstrut +\mathstrut \) \(3546\) \(\beta_{11}\mathstrut -\mathstrut \) \(5751\) \(\beta_{10}\mathstrut -\mathstrut \) \(4058\) \(\beta_{9}\mathstrut -\mathstrut \) \(481\) \(\beta_{8}\mathstrut -\mathstrut \) \(12353\) \(\beta_{7}\mathstrut +\mathstrut \) \(4077\) \(\beta_{6}\mathstrut -\mathstrut \) \(3300\) \(\beta_{5}\mathstrut -\mathstrut \) \(4329\) \(\beta_{4}\mathstrut -\mathstrut \) \(9340\) \(\beta_{3}\mathstrut +\mathstrut \) \(14710\) \(\beta_{2}\mathstrut +\mathstrut \) \(10239\) \(\beta_{1}\mathstrut +\mathstrut \) \(34160\)
\(\nu^{13}\)\(=\)\(13444\) \(\beta_{19}\mathstrut +\mathstrut \) \(740\) \(\beta_{18}\mathstrut -\mathstrut \) \(19767\) \(\beta_{17}\mathstrut +\mathstrut \) \(10096\) \(\beta_{16}\mathstrut -\mathstrut \) \(9373\) \(\beta_{15}\mathstrut +\mathstrut \) \(23467\) \(\beta_{14}\mathstrut +\mathstrut \) \(10612\) \(\beta_{13}\mathstrut +\mathstrut \) \(935\) \(\beta_{12}\mathstrut +\mathstrut \) \(16881\) \(\beta_{11}\mathstrut -\mathstrut \) \(18032\) \(\beta_{10}\mathstrut -\mathstrut \) \(7152\) \(\beta_{9}\mathstrut -\mathstrut \) \(1147\) \(\beta_{8}\mathstrut -\mathstrut \) \(47285\) \(\beta_{7}\mathstrut +\mathstrut \) \(19942\) \(\beta_{6}\mathstrut -\mathstrut \) \(13352\) \(\beta_{5}\mathstrut -\mathstrut \) \(9674\) \(\beta_{4}\mathstrut -\mathstrut \) \(26801\) \(\beta_{3}\mathstrut +\mathstrut \) \(28087\) \(\beta_{2}\mathstrut +\mathstrut \) \(42988\) \(\beta_{1}\mathstrut +\mathstrut \) \(77594\)
\(\nu^{14}\)\(=\)\(33431\) \(\beta_{19}\mathstrut +\mathstrut \) \(21740\) \(\beta_{18}\mathstrut -\mathstrut \) \(73552\) \(\beta_{17}\mathstrut +\mathstrut \) \(21449\) \(\beta_{16}\mathstrut -\mathstrut \) \(32345\) \(\beta_{15}\mathstrut +\mathstrut \) \(60853\) \(\beta_{14}\mathstrut +\mathstrut \) \(14464\) \(\beta_{13}\mathstrut -\mathstrut \) \(7510\) \(\beta_{12}\mathstrut +\mathstrut \) \(33229\) \(\beta_{11}\mathstrut -\mathstrut \) \(54015\) \(\beta_{10}\mathstrut -\mathstrut \) \(28963\) \(\beta_{9}\mathstrut -\mathstrut \) \(5143\) \(\beta_{8}\mathstrut -\mathstrut \) \(104348\) \(\beta_{7}\mathstrut +\mathstrut \) \(28091\) \(\beta_{6}\mathstrut -\mathstrut \) \(32865\) \(\beta_{5}\mathstrut -\mathstrut \) \(34068\) \(\beta_{4}\mathstrut -\mathstrut \) \(80170\) \(\beta_{3}\mathstrut +\mathstrut \) \(100996\) \(\beta_{2}\mathstrut +\mathstrut \) \(85379\) \(\beta_{1}\mathstrut +\mathstrut \) \(251135\)
\(\nu^{15}\)\(=\)\(120065\) \(\beta_{19}\mathstrut +\mathstrut \) \(20697\) \(\beta_{18}\mathstrut -\mathstrut \) \(179378\) \(\beta_{17}\mathstrut +\mathstrut \) \(85390\) \(\beta_{16}\mathstrut -\mathstrut \) \(89704\) \(\beta_{15}\mathstrut +\mathstrut \) \(187312\) \(\beta_{14}\mathstrut +\mathstrut \) \(79155\) \(\beta_{13}\mathstrut -\mathstrut \) \(3253\) \(\beta_{12}\mathstrut +\mathstrut \) \(137936\) \(\beta_{11}\mathstrut -\mathstrut \) \(163762\) \(\beta_{10}\mathstrut -\mathstrut \) \(55697\) \(\beta_{9}\mathstrut -\mathstrut \) \(12964\) \(\beta_{8}\mathstrut -\mathstrut \) \(370131\) \(\beta_{7}\mathstrut +\mathstrut \) \(129516\) \(\beta_{6}\mathstrut -\mathstrut \) \(118726\) \(\beta_{5}\mathstrut -\mathstrut \) \(80930\) \(\beta_{4}\mathstrut -\mathstrut \) \(230736\) \(\beta_{3}\mathstrut +\mathstrut \) \(203180\) \(\beta_{2}\mathstrut +\mathstrut \) \(320473\) \(\beta_{1}\mathstrut +\mathstrut \) \(602574\)
\(\nu^{16}\)\(=\)\(313513\) \(\beta_{19}\mathstrut +\mathstrut \) \(183868\) \(\beta_{18}\mathstrut -\mathstrut \) \(617232\) \(\beta_{17}\mathstrut +\mathstrut \) \(196626\) \(\beta_{16}\mathstrut -\mathstrut \) \(289842\) \(\beta_{15}\mathstrut +\mathstrut \) \(490002\) \(\beta_{14}\mathstrut +\mathstrut \) \(126411\) \(\beta_{13}\mathstrut -\mathstrut \) \(74835\) \(\beta_{12}\mathstrut +\mathstrut \) \(297878\) \(\beta_{11}\mathstrut -\mathstrut \) \(485299\) \(\beta_{10}\mathstrut -\mathstrut \) \(208001\) \(\beta_{9}\mathstrut -\mathstrut \) \(50324\) \(\beta_{8}\mathstrut -\mathstrut \) \(865965\) \(\beta_{7}\mathstrut +\mathstrut \) \(189820\) \(\beta_{6}\mathstrut -\mathstrut \) \(307192\) \(\beta_{5}\mathstrut -\mathstrut \) \(268940\) \(\beta_{4}\mathstrut -\mathstrut \) \(677205\) \(\beta_{3}\mathstrut +\mathstrut \) \(696274\) \(\beta_{2}\mathstrut +\mathstrut \) \(698414\) \(\beta_{1}\mathstrut +\mathstrut \) \(1869005\)
\(\nu^{17}\)\(=\)\(1047731\) \(\beta_{19}\mathstrut +\mathstrut \) \(267647\) \(\beta_{18}\mathstrut -\mathstrut \) \(1576771\) \(\beta_{17}\mathstrut +\mathstrut \) \(711696\) \(\beta_{16}\mathstrut -\mathstrut \) \(815151\) \(\beta_{15}\mathstrut +\mathstrut \) \(1484509\) \(\beta_{14}\mathstrut +\mathstrut \) \(592816\) \(\beta_{13}\mathstrut -\mathstrut \) \(102869\) \(\beta_{12}\mathstrut +\mathstrut \) \(1118662\) \(\beta_{11}\mathstrut -\mathstrut \) \(1442100\) \(\beta_{10}\mathstrut -\mathstrut \) \(429821\) \(\beta_{9}\mathstrut -\mathstrut \) \(131962\) \(\beta_{8}\mathstrut -\mathstrut \) \(2902042\) \(\beta_{7}\mathstrut +\mathstrut \) \(832041\) \(\beta_{6}\mathstrut -\mathstrut \) \(1031854\) \(\beta_{5}\mathstrut -\mathstrut \) \(667258\) \(\beta_{4}\mathstrut -\mathstrut \) \(1947334\) \(\beta_{3}\mathstrut +\mathstrut \) \(1462671\) \(\beta_{2}\mathstrut +\mathstrut \) \(2425748\) \(\beta_{1}\mathstrut +\mathstrut \) \(4656292\)
\(\nu^{18}\)\(=\)\(2819553\) \(\beta_{19}\mathstrut +\mathstrut \) \(1541732\) \(\beta_{18}\mathstrut -\mathstrut \) \(5140078\) \(\beta_{17}\mathstrut +\mathstrut \) \(1735208\) \(\beta_{16}\mathstrut -\mathstrut \) \(2528798\) \(\beta_{15}\mathstrut +\mathstrut \) \(3929546\) \(\beta_{14}\mathstrut +\mathstrut \) \(1074956\) \(\beta_{13}\mathstrut -\mathstrut \) \(708412\) \(\beta_{12}\mathstrut +\mathstrut \) \(2590321\) \(\beta_{11}\mathstrut -\mathstrut \) \(4234197\) \(\beta_{10}\mathstrut -\mathstrut \) \(1507157\) \(\beta_{9}\mathstrut -\mathstrut \) \(466977\) \(\beta_{8}\mathstrut -\mathstrut \) \(7101037\) \(\beta_{7}\mathstrut +\mathstrut \) \(1257823\) \(\beta_{6}\mathstrut -\mathstrut \) \(2755579\) \(\beta_{5}\mathstrut -\mathstrut \) \(2126030\) \(\beta_{4}\mathstrut -\mathstrut \) \(5651200\) \(\beta_{3}\mathstrut +\mathstrut \) \(4817306\) \(\beta_{2}\mathstrut +\mathstrut \) \(5643656\) \(\beta_{1}\mathstrut +\mathstrut \) \(14033583\)
\(\nu^{19}\)\(=\)\(8984716\) \(\beta_{19}\mathstrut +\mathstrut \) \(2827347\) \(\beta_{18}\mathstrut -\mathstrut \) \(13552001\) \(\beta_{17}\mathstrut +\mathstrut \) \(5871457\) \(\beta_{16}\mathstrut -\mathstrut \) \(7162439\) \(\beta_{15}\mathstrut +\mathstrut \) \(11730170\) \(\beta_{14}\mathstrut +\mathstrut \) \(4471866\) \(\beta_{13}\mathstrut -\mathstrut \) \(1358968\) \(\beta_{12}\mathstrut +\mathstrut \) \(9037024\) \(\beta_{11}\mathstrut -\mathstrut \) \(12420347\) \(\beta_{10}\mathstrut -\mathstrut \) \(3304008\) \(\beta_{9}\mathstrut -\mathstrut \) \(1258618\) \(\beta_{8}\mathstrut -\mathstrut \) \(22801754\) \(\beta_{7}\mathstrut +\mathstrut \) \(5281303\) \(\beta_{6}\mathstrut -\mathstrut \) \(8816784\) \(\beta_{5}\mathstrut -\mathstrut \) \(5443768\) \(\beta_{4}\mathstrut -\mathstrut \) \(16213139\) \(\beta_{3}\mathstrut +\mathstrut \) \(10493112\) \(\beta_{2}\mathstrut +\mathstrut \) \(18549725\) \(\beta_{1}\mathstrut +\mathstrut \) \(35913688\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.80474
2.65160
2.60929
1.87740
1.76837
1.71702
1.51806
1.09509
0.700104
0.686421
0.178107
−0.100365
−0.435939
−0.590088
−1.29878
−1.82988
−1.91911
−1.95611
−2.02882
−2.44709
1.00000 −2.80474 1.00000 −0.947662 −2.80474 2.44948 1.00000 4.86656 −0.947662
1.2 1.00000 −2.65160 1.00000 0.347024 −2.65160 −2.28847 1.00000 4.03099 0.347024
1.3 1.00000 −2.60929 1.00000 1.96845 −2.60929 −1.34895 1.00000 3.80840 1.96845
1.4 1.00000 −1.87740 1.00000 3.35129 −1.87740 −2.95261 1.00000 0.524644 3.35129
1.5 1.00000 −1.76837 1.00000 1.49209 −1.76837 3.65436 1.00000 0.127121 1.49209
1.6 1.00000 −1.71702 1.00000 −4.22974 −1.71702 −1.91461 1.00000 −0.0518545 −4.22974
1.7 1.00000 −1.51806 1.00000 −3.34509 −1.51806 1.42565 1.00000 −0.695496 −3.34509
1.8 1.00000 −1.09509 1.00000 −0.379683 −1.09509 −3.84052 1.00000 −1.80078 −0.379683
1.9 1.00000 −0.700104 1.00000 1.01839 −0.700104 0.757569 1.00000 −2.50985 1.01839
1.10 1.00000 −0.686421 1.00000 1.74349 −0.686421 −1.83413 1.00000 −2.52883 1.74349
1.11 1.00000 −0.178107 1.00000 −1.41285 −0.178107 −1.81585 1.00000 −2.96828 −1.41285
1.12 1.00000 0.100365 1.00000 −2.25871 0.100365 1.56060 1.00000 −2.98993 −2.25871
1.13 1.00000 0.435939 1.00000 −0.796234 0.435939 3.50309 1.00000 −2.80996 −0.796234
1.14 1.00000 0.590088 1.00000 0.936829 0.590088 1.40799 1.00000 −2.65180 0.936829
1.15 1.00000 1.29878 1.00000 2.87965 1.29878 −3.57798 1.00000 −1.31316 2.87965
1.16 1.00000 1.82988 1.00000 0.255245 1.82988 −4.20483 1.00000 0.348474 0.255245
1.17 1.00000 1.91911 1.00000 −2.55109 1.91911 0.891228 1.00000 0.682993 −2.55109
1.18 1.00000 1.95611 1.00000 −3.19910 1.95611 0.785940 1.00000 0.826384 −3.19910
1.19 1.00000 2.02882 1.00000 −0.795313 2.02882 −4.29820 1.00000 1.11611 −0.795313
1.20 1.00000 2.44709 1.00000 −0.0770010 2.44709 −0.359767 1.00000 2.98826 −0.0770010
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(-1\)
\(131\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6026))\):

\(T_{3}^{20} + \cdots\)
\(T_{5}^{20} + \cdots\)