Properties

Label 430.2.k.b
Level 430
Weight 2
Character orbit 430.k
Analytic conductor 3.434
Analytic rank 0
Dimension 18
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 5 x^{17} + 17 x^{16} - 43 x^{15} + 90 x^{14} - 114 x^{13} + 135 x^{12} + 2 x^{11} + 313 x^{10} - 632 x^{9} + 2382 x^{8} - 1755 x^{7} + 2203 x^{6} - 3817 x^{5} + 4882 x^{4} - 4217 x^{3} + 2612 x^{2} - 871 x + 169\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(10590185050212349241642389 \nu^{17} - 322207941785410405247122226 \nu^{16} + 1349453977643523664915922335 \nu^{15} - 4082679268463783607442741506 \nu^{14} + 9457476331730672322621963516 \nu^{13} - 17821969028073324909413190398 \nu^{12} + 16867368747268591527255091161 \nu^{11} - 18234194924664852374527770867 \nu^{10} - 13208813915998395021570022548 \nu^{9} - 95122259106268548109239994380 \nu^{8} + 154748876083706483404802355266 \nu^{7} - 502361011751002082761843249745 \nu^{6} + 122384045734723358025693748484 \nu^{5} - 276375838840025758269033949641 \nu^{4} + 1197939554289597365692341110239 \nu^{3} - 566851962221822965348422541776 \nu^{2} + 381975334912037766930584387756 \nu - 193990375754259281091475717071\)\()/ \)\(38\!\cdots\!84\)\( \)
\(\beta_{3}\)\(=\)\((\)\(136679740927497797476895111 \nu^{17} - 298655889357856214234058012 \nu^{16} + 545223620612472609677148799 \nu^{15} + 26714564701413009585410346 \nu^{14} - 2045286847801507297103118904 \nu^{13} + 13588676061869118162889294954 \nu^{12} - 13860468198203608476310425061 \nu^{11} + 38305137636735631494334067389 \nu^{10} + 63358554728879077013191207368 \nu^{9} + 37785874549937159252660354104 \nu^{8} + 138458639071502471234906245334 \nu^{7} + 631680410761680779804092618429 \nu^{6} - 31343544014941868774990422802 \nu^{5} + 227150940264044712240918268519 \nu^{4} - 303070392579553495659034185219 \nu^{3} + 946620228623494196092383846902 \nu^{2} - 704609662425666116302248436150 \nu + 376599157569722750767369137093\)\()/ \)\(19\!\cdots\!92\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(52\!\cdots\!21\)\( \nu^{17} + \)\(38\!\cdots\!46\)\( \nu^{16} - \)\(14\!\cdots\!63\)\( \nu^{15} + \)\(39\!\cdots\!54\)\( \nu^{14} - \)\(85\!\cdots\!20\)\( \nu^{13} + \)\(13\!\cdots\!86\)\( \nu^{12} - \)\(13\!\cdots\!77\)\( \nu^{11} + \)\(78\!\cdots\!99\)\( \nu^{10} - \)\(57\!\cdots\!36\)\( \nu^{9} + \)\(76\!\cdots\!28\)\( \nu^{8} - \)\(17\!\cdots\!54\)\( \nu^{7} + \)\(33\!\cdots\!01\)\( \nu^{6} - \)\(11\!\cdots\!48\)\( \nu^{5} + \)\(37\!\cdots\!89\)\( \nu^{4} - \)\(56\!\cdots\!35\)\( \nu^{3} + \)\(49\!\cdots\!68\)\( \nu^{2} - \)\(28\!\cdots\!76\)\( \nu + \)\(89\!\cdots\!47\)\(\)\()/ \)\(38\!\cdots\!84\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(38\!\cdots\!43\)\( \nu^{17} - \)\(17\!\cdots\!88\)\( \nu^{16} + \)\(59\!\cdots\!19\)\( \nu^{15} - \)\(14\!\cdots\!94\)\( \nu^{14} + \)\(29\!\cdots\!08\)\( \nu^{13} - \)\(32\!\cdots\!46\)\( \nu^{12} + \)\(38\!\cdots\!67\)\( \nu^{11} + \)\(20\!\cdots\!25\)\( \nu^{10} + \)\(12\!\cdots\!56\)\( \nu^{9} - \)\(18\!\cdots\!68\)\( \nu^{8} + \)\(87\!\cdots\!34\)\( \nu^{7} - \)\(33\!\cdots\!35\)\( \nu^{6} + \)\(74\!\cdots\!06\)\( \nu^{5} - \)\(97\!\cdots\!21\)\( \nu^{4} + \)\(15\!\cdots\!89\)\( \nu^{3} - \)\(10\!\cdots\!82\)\( \nu^{2} + \)\(49\!\cdots\!74\)\( \nu - \)\(23\!\cdots\!59\)\(\)\()/ \)\(19\!\cdots\!92\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(42\!\cdots\!03\)\( \nu^{17} + \)\(17\!\cdots\!98\)\( \nu^{16} - \)\(54\!\cdots\!87\)\( \nu^{15} + \)\(12\!\cdots\!80\)\( \nu^{14} - \)\(24\!\cdots\!14\)\( \nu^{13} + \)\(20\!\cdots\!58\)\( \nu^{12} - \)\(27\!\cdots\!57\)\( \nu^{11} - \)\(30\!\cdots\!21\)\( \nu^{10} - \)\(16\!\cdots\!22\)\( \nu^{9} + \)\(16\!\cdots\!74\)\( \nu^{8} - \)\(79\!\cdots\!58\)\( \nu^{7} - \)\(94\!\cdots\!07\)\( \nu^{6} - \)\(57\!\cdots\!22\)\( \nu^{5} + \)\(12\!\cdots\!41\)\( \nu^{4} - \)\(10\!\cdots\!65\)\( \nu^{3} + \)\(39\!\cdots\!52\)\( \nu^{2} - \)\(73\!\cdots\!70\)\( \nu + \)\(37\!\cdots\!03\)\(\)\()/ \)\(19\!\cdots\!92\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(95\!\cdots\!49\)\( \nu^{17} - \)\(48\!\cdots\!94\)\( \nu^{16} + \)\(15\!\cdots\!99\)\( \nu^{15} - \)\(39\!\cdots\!34\)\( \nu^{14} + \)\(80\!\cdots\!80\)\( \nu^{13} - \)\(95\!\cdots\!58\)\( \nu^{12} + \)\(10\!\cdots\!93\)\( \nu^{11} + \)\(24\!\cdots\!93\)\( \nu^{10} + \)\(26\!\cdots\!40\)\( \nu^{9} - \)\(66\!\cdots\!64\)\( \nu^{8} + \)\(21\!\cdots\!42\)\( \nu^{7} - \)\(15\!\cdots\!33\)\( \nu^{6} + \)\(10\!\cdots\!24\)\( \nu^{5} - \)\(37\!\cdots\!61\)\( \nu^{4} + \)\(40\!\cdots\!59\)\( \nu^{3} - \)\(29\!\cdots\!16\)\( \nu^{2} + \)\(14\!\cdots\!08\)\( \nu - \)\(29\!\cdots\!63\)\(\)\()/ \)\(38\!\cdots\!84\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(49\!\cdots\!51\)\( \nu^{17} - \)\(40\!\cdots\!42\)\( \nu^{16} + \)\(14\!\cdots\!13\)\( \nu^{15} - \)\(41\!\cdots\!90\)\( \nu^{14} + \)\(90\!\cdots\!00\)\( \nu^{13} - \)\(14\!\cdots\!70\)\( \nu^{12} + \)\(14\!\cdots\!63\)\( \nu^{11} - \)\(99\!\cdots\!85\)\( \nu^{10} + \)\(29\!\cdots\!96\)\( \nu^{9} - \)\(85\!\cdots\!28\)\( \nu^{8} + \)\(17\!\cdots\!34\)\( \nu^{7} - \)\(38\!\cdots\!59\)\( \nu^{6} + \)\(96\!\cdots\!72\)\( \nu^{5} - \)\(40\!\cdots\!19\)\( \nu^{4} + \)\(64\!\cdots\!09\)\( \nu^{3} - \)\(56\!\cdots\!40\)\( \nu^{2} + \)\(30\!\cdots\!80\)\( \nu - \)\(92\!\cdots\!73\)\(\)\()/ \)\(19\!\cdots\!92\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(25\!\cdots\!31\)\( \nu^{17} + \)\(12\!\cdots\!03\)\( \nu^{16} - \)\(43\!\cdots\!23\)\( \nu^{15} + \)\(10\!\cdots\!81\)\( \nu^{14} - \)\(22\!\cdots\!95\)\( \nu^{13} + \)\(28\!\cdots\!04\)\( \nu^{12} - \)\(32\!\cdots\!88\)\( \nu^{11} - \)\(14\!\cdots\!87\)\( \nu^{10} - \)\(76\!\cdots\!53\)\( \nu^{9} + \)\(16\!\cdots\!37\)\( \nu^{8} - \)\(60\!\cdots\!90\)\( \nu^{7} + \)\(46\!\cdots\!50\)\( \nu^{6} - \)\(48\!\cdots\!34\)\( \nu^{5} + \)\(96\!\cdots\!47\)\( \nu^{4} - \)\(12\!\cdots\!01\)\( \nu^{3} + \)\(11\!\cdots\!53\)\( \nu^{2} - \)\(55\!\cdots\!72\)\( \nu + \)\(13\!\cdots\!13\)\(\)\()/ \)\(97\!\cdots\!96\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(11\!\cdots\!59\)\( \nu^{17} - \)\(57\!\cdots\!06\)\( \nu^{16} + \)\(19\!\cdots\!77\)\( \nu^{15} - \)\(48\!\cdots\!02\)\( \nu^{14} + \)\(99\!\cdots\!04\)\( \nu^{13} - \)\(12\!\cdots\!10\)\( \nu^{12} + \)\(13\!\cdots\!67\)\( \nu^{11} + \)\(19\!\cdots\!79\)\( \nu^{10} + \)\(34\!\cdots\!00\)\( \nu^{9} - \)\(73\!\cdots\!36\)\( \nu^{8} + \)\(26\!\cdots\!58\)\( \nu^{7} - \)\(18\!\cdots\!79\)\( \nu^{6} + \)\(20\!\cdots\!32\)\( \nu^{5} - \)\(42\!\cdots\!19\)\( \nu^{4} + \)\(53\!\cdots\!97\)\( \nu^{3} - \)\(36\!\cdots\!64\)\( \nu^{2} + \)\(24\!\cdots\!32\)\( \nu - \)\(61\!\cdots\!33\)\(\)\()/ \)\(38\!\cdots\!84\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(12\!\cdots\!41\)\( \nu^{17} - \)\(52\!\cdots\!06\)\( \nu^{16} + \)\(16\!\cdots\!51\)\( \nu^{15} - \)\(38\!\cdots\!30\)\( \nu^{14} + \)\(74\!\cdots\!92\)\( \nu^{13} - \)\(63\!\cdots\!42\)\( \nu^{12} + \)\(79\!\cdots\!41\)\( \nu^{11} + \)\(10\!\cdots\!37\)\( \nu^{10} + \)\(43\!\cdots\!56\)\( \nu^{9} - \)\(47\!\cdots\!32\)\( \nu^{8} + \)\(24\!\cdots\!46\)\( \nu^{7} + \)\(37\!\cdots\!15\)\( \nu^{6} + \)\(17\!\cdots\!32\)\( \nu^{5} - \)\(30\!\cdots\!13\)\( \nu^{4} + \)\(27\!\cdots\!11\)\( \nu^{3} - \)\(14\!\cdots\!24\)\( \nu^{2} + \)\(43\!\cdots\!56\)\( \nu + \)\(88\!\cdots\!49\)\(\)\()/ \)\(38\!\cdots\!84\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(17\!\cdots\!27\)\( \nu^{17} - \)\(77\!\cdots\!86\)\( \nu^{16} + \)\(24\!\cdots\!65\)\( \nu^{15} - \)\(58\!\cdots\!62\)\( \nu^{14} + \)\(11\!\cdots\!96\)\( \nu^{13} - \)\(11\!\cdots\!98\)\( \nu^{12} + \)\(13\!\cdots\!87\)\( \nu^{11} + \)\(10\!\cdots\!47\)\( \nu^{10} + \)\(56\!\cdots\!44\)\( \nu^{9} - \)\(83\!\cdots\!24\)\( \nu^{8} + \)\(34\!\cdots\!50\)\( \nu^{7} - \)\(93\!\cdots\!43\)\( \nu^{6} + \)\(22\!\cdots\!48\)\( \nu^{5} - \)\(55\!\cdots\!35\)\( \nu^{4} + \)\(47\!\cdots\!53\)\( \nu^{3} - \)\(32\!\cdots\!00\)\( \nu^{2} + \)\(16\!\cdots\!08\)\( \nu - \)\(27\!\cdots\!09\)\(\)\()/ \)\(38\!\cdots\!84\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(19\!\cdots\!97\)\( \nu^{17} - \)\(90\!\cdots\!66\)\( \nu^{16} + \)\(30\!\cdots\!75\)\( \nu^{15} - \)\(75\!\cdots\!30\)\( \nu^{14} + \)\(15\!\cdots\!92\)\( \nu^{13} - \)\(18\!\cdots\!66\)\( \nu^{12} + \)\(22\!\cdots\!89\)\( \nu^{11} + \)\(55\!\cdots\!21\)\( \nu^{10} + \)\(62\!\cdots\!36\)\( \nu^{9} - \)\(99\!\cdots\!36\)\( \nu^{8} + \)\(43\!\cdots\!94\)\( \nu^{7} - \)\(22\!\cdots\!73\)\( \nu^{6} + \)\(41\!\cdots\!64\)\( \nu^{5} - \)\(61\!\cdots\!33\)\( \nu^{4} + \)\(78\!\cdots\!15\)\( \nu^{3} - \)\(64\!\cdots\!32\)\( \nu^{2} + \)\(36\!\cdots\!32\)\( \nu - \)\(74\!\cdots\!59\)\(\)\()/ \)\(38\!\cdots\!84\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(82\!\cdots\!81\)\( \nu^{17} - \)\(38\!\cdots\!74\)\( \nu^{16} + \)\(12\!\cdots\!23\)\( \nu^{15} - \)\(31\!\cdots\!18\)\( \nu^{14} + \)\(65\!\cdots\!88\)\( \nu^{13} - \)\(77\!\cdots\!84\)\( \nu^{12} + \)\(97\!\cdots\!03\)\( \nu^{11} + \)\(23\!\cdots\!35\)\( \nu^{10} + \)\(28\!\cdots\!90\)\( \nu^{9} - \)\(40\!\cdots\!76\)\( \nu^{8} + \)\(18\!\cdots\!82\)\( \nu^{7} - \)\(87\!\cdots\!61\)\( \nu^{6} + \)\(19\!\cdots\!42\)\( \nu^{5} - \)\(24\!\cdots\!37\)\( \nu^{4} + \)\(33\!\cdots\!09\)\( \nu^{3} - \)\(29\!\cdots\!32\)\( \nu^{2} + \)\(17\!\cdots\!94\)\( \nu - \)\(36\!\cdots\!01\)\(\)\()/ \)\(97\!\cdots\!96\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(11\!\cdots\!33\)\( \nu^{17} + \)\(56\!\cdots\!57\)\( \nu^{16} - \)\(18\!\cdots\!02\)\( \nu^{15} + \)\(46\!\cdots\!84\)\( \nu^{14} - \)\(95\!\cdots\!10\)\( \nu^{13} + \)\(11\!\cdots\!66\)\( \nu^{12} - \)\(12\!\cdots\!17\)\( \nu^{11} - \)\(38\!\cdots\!56\)\( \nu^{10} - \)\(34\!\cdots\!48\)\( \nu^{9} + \)\(70\!\cdots\!20\)\( \nu^{8} - \)\(26\!\cdots\!34\)\( \nu^{7} + \)\(15\!\cdots\!13\)\( \nu^{6} - \)\(18\!\cdots\!75\)\( \nu^{5} + \)\(39\!\cdots\!40\)\( \nu^{4} - \)\(49\!\cdots\!95\)\( \nu^{3} + \)\(34\!\cdots\!27\)\( \nu^{2} - \)\(18\!\cdots\!43\)\( \nu + \)\(23\!\cdots\!68\)\(\)\()/ \)\(97\!\cdots\!96\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(30\!\cdots\!61\)\( \nu^{17} - \)\(14\!\cdots\!20\)\( \nu^{16} + \)\(48\!\cdots\!95\)\( \nu^{15} - \)\(11\!\cdots\!34\)\( \nu^{14} + \)\(23\!\cdots\!18\)\( \nu^{13} - \)\(27\!\cdots\!60\)\( \nu^{12} + \)\(30\!\cdots\!17\)\( \nu^{11} + \)\(12\!\cdots\!09\)\( \nu^{10} + \)\(92\!\cdots\!48\)\( \nu^{9} - \)\(17\!\cdots\!66\)\( \nu^{8} + \)\(67\!\cdots\!72\)\( \nu^{7} - \)\(35\!\cdots\!45\)\( \nu^{6} + \)\(46\!\cdots\!40\)\( \nu^{5} - \)\(99\!\cdots\!67\)\( \nu^{4} + \)\(11\!\cdots\!43\)\( \nu^{3} - \)\(86\!\cdots\!62\)\( \nu^{2} + \)\(38\!\cdots\!60\)\( \nu - \)\(15\!\cdots\!73\)\(\)\()/ \)\(19\!\cdots\!92\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(76\!\cdots\!05\)\( \nu^{17} - \)\(35\!\cdots\!58\)\( \nu^{16} + \)\(11\!\cdots\!67\)\( \nu^{15} - \)\(27\!\cdots\!14\)\( \nu^{14} + \)\(56\!\cdots\!96\)\( \nu^{13} - \)\(61\!\cdots\!78\)\( \nu^{12} + \)\(73\!\cdots\!01\)\( \nu^{11} + \)\(36\!\cdots\!01\)\( \nu^{10} + \)\(25\!\cdots\!92\)\( \nu^{9} - \)\(37\!\cdots\!48\)\( \nu^{8} + \)\(16\!\cdots\!66\)\( \nu^{7} - \)\(61\!\cdots\!29\)\( \nu^{6} + \)\(13\!\cdots\!68\)\( \nu^{5} - \)\(23\!\cdots\!73\)\( \nu^{4} + \)\(25\!\cdots\!15\)\( \nu^{3} - \)\(19\!\cdots\!08\)\( \nu^{2} + \)\(10\!\cdots\!20\)\( \nu - \)\(16\!\cdots\!63\)\(\)\()/ \)\(38\!\cdots\!84\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{16} + \beta_{15} - \beta_{13} - \beta_{12} + 3 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 5 \beta_{2} + \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(6 \beta_{17} + \beta_{15} - 6 \beta_{14} - 14 \beta_{12} + 7 \beta_{11} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} - 6 \beta_{5} + 7 \beta_{2} + 7 \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(8 \beta_{17} + 2 \beta_{15} - 8 \beta_{14} - 6 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} - 9 \beta_{9} + 8 \beta_{8} - 30 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 9 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 9 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(2 \beta_{17} - 2 \beta_{16} - 11 \beta_{14} + 2 \beta_{13} + 22 \beta_{11} + 33 \beta_{10} + 3 \beta_{9} + 27 \beta_{8} - 22 \beta_{7} + 11 \beta_{6} + 24 \beta_{5} + 74 \beta_{4} - 33 \beta_{3} - 3 \beta_{2} - 24 \beta_{1} - 2\)
\(\nu^{7}\)\(=\)\(-57 \beta_{16} - 23 \beta_{15} - 23 \beta_{14} + 86 \beta_{13} + 65 \beta_{12} + 135 \beta_{11} - 86 \beta_{10} - 5 \beta_{9} + 34 \beta_{8} - 16 \beta_{7} + 58 \beta_{6} - 11 \beta_{5} + 8 \beta_{3} + 34 \beta_{2} - 34 \beta_{1} - 34\)
\(\nu^{8}\)\(=\)\(-28 \beta_{17} - 227 \beta_{16} - 96 \beta_{15} + 265 \beta_{13} + 492 \beta_{12} + 143 \beta_{11} - 473 \beta_{10} - 143 \beta_{9} - 73 \beta_{7} + 227 \beta_{6} - 73 \beta_{5} - 414 \beta_{4} + 414 \beta_{3} - 45 \beta_{1} - 473\)
\(\nu^{9}\)\(=\)\(-398 \beta_{17} - 415 \beta_{16} - 415 \beta_{15} + 197 \beta_{14} + 459 \beta_{13} + 1427 \beta_{12} - 318 \beta_{11} - 697 \beta_{10} - 383 \beta_{9} - 201 \beta_{8} + 415 \beta_{7} + 398 \beta_{6} + 335 \beta_{5} - 459 \beta_{4} + 398 \beta_{3} - 289 \beta_{2} - 687 \beta_{1} - 1012\)
\(\nu^{10}\)\(=\)\(-1485 \beta_{17} - 767 \beta_{16} - 1485 \beta_{15} + 767 \beta_{14} + 1485 \beta_{13} + 2856 \beta_{12} - 767 \beta_{11} - 2089 \beta_{10} - 995 \beta_{8} + 2725 \beta_{7} + 277 \beta_{6} + 718 \beta_{5} - 823 \beta_{4} - 643 \beta_{3} - 714 \beta_{2} - 2007 \beta_{1} - 823\)
\(\nu^{11}\)\(=\)\(-2966 \beta_{17} - 1517 \beta_{16} - 2774 \beta_{15} + 2966 \beta_{14} + 2463 \beta_{13} + 6416 \beta_{12} - 2581 \beta_{11} - 6722 \beta_{10} + 1709 \beta_{9} - 2774 \beta_{8} + 5854 \beta_{7} - 1137 \beta_{5} - 6722 \beta_{4} + 2463 \beta_{3} - 3080 \beta_{2} - 2581 \beta_{1} - 2966\)
\(\nu^{12}\)\(=\)\(-5854 \beta_{17} - 2389 \beta_{15} + 9957 \beta_{14} - 3598 \beta_{13} + 13729 \beta_{12} - 16990 \beta_{11} - 5854 \beta_{10} + 8252 \beta_{9} - 5854 \beta_{8} + 13645 \beta_{7} - 2389 \beta_{6} + 2389 \beta_{5} - 13729 \beta_{4} + 8406 \beta_{3} - 13525 \beta_{2} - 8252 \beta_{1} - 8406\)
\(\nu^{13}\)\(=\)\(-11127 \beta_{17} + 11127 \beta_{16} + 19379 \beta_{14} - 18169 \beta_{13} - 46663 \beta_{11} + 13891 \beta_{10} + 30614 \beta_{9} - 12961 \beta_{8} + 46663 \beta_{7} - 19379 \beta_{6} + 22525 \beta_{5} - 1007 \beta_{4} - 13891 \beta_{3} - 30614 \beta_{2} - 22525 \beta_{1} + 18169\)
\(\nu^{14}\)\(=\)\(43517 \beta_{16} + 19216 \beta_{15} + 19216 \beta_{14} - 43404 \beta_{13} - 93431 \beta_{12} - 59605 \beta_{11} + 43404 \beta_{10} + 67949 \beta_{9} - 24301 \beta_{8} + 73034 \beta_{7} - 67876 \beta_{6} + 12588 \beta_{5} - 49914 \beta_{3} - 31804 \beta_{2} + 24301 \beta_{1} + 112521\)
\(\nu^{15}\)\(=\)\(79662 \beta_{17} + 151782 \beta_{16} + 135767 \beta_{15} - 191427 \beta_{13} - 343209 \beta_{12} - 86921 \beta_{11} + 227417 \beta_{10} + 86921 \beta_{9} - 66956 \beta_{7} - 151782 \beta_{6} - 66956 \beta_{5} + 26046 \beta_{4} - 26046 \beta_{3} + 229243 \beta_{1} + 227417\)
\(\nu^{16}\)\(=\)\(318365 \beta_{17} + 467946 \beta_{16} + 467946 \beta_{15} - 148473 \beta_{14} - 639933 \beta_{13} - 1124641 \beta_{12} - 96570 \beta_{11} + 1062279 \beta_{10} + 79280 \beta_{9} + 169892 \beta_{8} - 467946 \beta_{7} - 318365 \beta_{6} - 53011 \beta_{5} + 639933 \beta_{4} - 318365 \beta_{3} + 220257 \beta_{2} + 538622 \beta_{1} + 656695\)
\(\nu^{17}\)\(=\)\(1085848 \beta_{17} + 953557 \beta_{16} + 1085848 \beta_{15} - 953557 \beta_{14} - 1085848 \beta_{13} - 3426670 \beta_{12} + 953557 \beta_{11} + 2473113 \beta_{10} - 156547 \beta_{9} + 695699 \beta_{8} - 1641232 \beta_{7} - 563408 \beta_{6} - 288838 \beta_{5} + 2178051 \beta_{4} - 1352707 \beta_{3} + 1368407 \beta_{2} + 1508941 \beta_{1} + 2178051\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3} - \beta_{4} - \beta_{10} + \beta_{12} + \beta_{13}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.532432 2.33273i
0.163144 + 0.714782i
0.468318 + 2.05184i
−0.767481 + 0.962391i
0.257241 0.322570i
1.28772 1.61475i
−0.767481 0.962391i
0.257241 + 0.322570i
1.28772 + 1.61475i
2.39969 1.15563i
0.695279 0.334829i
−1.47148 + 0.708626i
2.39969 + 1.15563i
0.695279 + 0.334829i
−1.47148 0.708626i
−0.532432 + 2.33273i
0.163144 0.714782i
0.468318 2.05184i
0.623490 + 0.781831i −1.49184 + 1.87071i −0.222521 + 0.974928i 0.900969 0.433884i −2.39273 4.52707 −0.900969 + 0.433884i −0.606400 2.65681i 0.900969 + 0.433884i
11.2 0.623490 + 0.781831i 0.457120 0.573211i −0.222521 + 0.974928i 0.900969 0.433884i 0.733164 −0.660533 −0.900969 + 0.433884i 0.547951 + 2.40073i 0.900969 + 0.433884i
11.3 0.623490 + 0.781831i 1.31220 1.64545i −0.222521 + 0.974928i 0.900969 0.433884i 2.10460 3.23129 −0.900969 + 0.433884i −0.318062 1.39352i 0.900969 + 0.433884i
21.1 −0.900969 0.433884i −1.10904 + 0.534087i 0.623490 + 0.781831i 0.222521 + 0.974928i 1.23094 −1.03973 −0.222521 0.974928i −0.925743 + 1.16084i 0.222521 0.974928i
21.2 −0.900969 0.433884i 0.371724 0.179013i 0.623490 + 0.781831i 0.222521 + 0.974928i −0.412583 −2.38473 −0.222521 0.974928i −1.76434 + 2.21241i 0.222521 0.974928i
21.3 −0.900969 0.433884i 1.86081 0.896118i 0.623490 + 0.781831i 0.222521 + 0.974928i −2.06534 1.71068 −0.222521 0.974928i 0.789110 0.989513i 0.222521 0.974928i
41.1 −0.900969 + 0.433884i −1.10904 0.534087i 0.623490 0.781831i 0.222521 0.974928i 1.23094 −1.03973 −0.222521 + 0.974928i −0.925743 1.16084i 0.222521 + 0.974928i
41.2 −0.900969 + 0.433884i 0.371724 + 0.179013i 0.623490 0.781831i 0.222521 0.974928i −0.412583 −2.38473 −0.222521 + 0.974928i −1.76434 2.21241i 0.222521 + 0.974928i
41.3 −0.900969 + 0.433884i 1.86081 + 0.896118i 0.623490 0.781831i 0.222521 0.974928i −2.06534 1.71068 −0.222521 + 0.974928i 0.789110 + 0.989513i 0.222521 + 0.974928i
121.1 −0.222521 + 0.974928i −0.592674 2.59667i −0.900969 0.433884i −0.623490 + 0.781831i 2.66345 2.84700 0.623490 0.781831i −3.68855 + 1.77631i −0.623490 0.781831i
121.2 −0.222521 + 0.974928i −0.171720 0.752353i −0.900969 0.433884i −0.623490 + 0.781831i 0.771701 −3.65146 0.623490 0.781831i 2.16636 1.04326i −0.623490 0.781831i
121.3 −0.222521 + 0.974928i 0.363425 + 1.59227i −0.900969 0.433884i −0.623490 + 0.781831i −1.63322 −1.57959 0.623490 0.781831i 0.299667 0.144312i −0.623490 0.781831i
231.1 −0.222521 0.974928i −0.592674 + 2.59667i −0.900969 + 0.433884i −0.623490 0.781831i 2.66345 2.84700 0.623490 + 0.781831i −3.68855 1.77631i −0.623490 + 0.781831i
231.2 −0.222521 0.974928i −0.171720 + 0.752353i −0.900969 + 0.433884i −0.623490 0.781831i 0.771701 −3.65146 0.623490 + 0.781831i 2.16636 + 1.04326i −0.623490 + 0.781831i
231.3 −0.222521 0.974928i 0.363425 1.59227i −0.900969 + 0.433884i −0.623490 0.781831i −1.63322 −1.57959 0.623490 + 0.781831i 0.299667 + 0.144312i −0.623490 + 0.781831i
391.1 0.623490 0.781831i −1.49184 1.87071i −0.222521 0.974928i 0.900969 + 0.433884i −2.39273 4.52707 −0.900969 0.433884i −0.606400 + 2.65681i 0.900969 0.433884i
391.2 0.623490 0.781831i 0.457120 + 0.573211i −0.222521 0.974928i 0.900969 + 0.433884i 0.733164 −0.660533 −0.900969 0.433884i 0.547951 2.40073i 0.900969 0.433884i
391.3 0.623490 0.781831i 1.31220 + 1.64545i −0.222521 0.974928i 0.900969 + 0.433884i 2.10460 3.23129 −0.900969 0.433884i −0.318062 + 1.39352i 0.900969 0.433884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.k.b 18
43.e even 7 1 inner 430.2.k.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.b 18 1.a even 1 1 trivial
430.2.k.b 18 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{18} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{3} \)
$3$ \( 1 - 2 T + T^{2} - 3 T^{3} + T^{4} + 18 T^{6} + 64 T^{7} - 171 T^{8} + 290 T^{9} - 781 T^{10} + 709 T^{11} - 1029 T^{12} + 953 T^{13} + 2528 T^{14} + 38 T^{15} + 12474 T^{16} - 47133 T^{17} + 61948 T^{18} - 141399 T^{19} + 112266 T^{20} + 1026 T^{21} + 204768 T^{22} + 231579 T^{23} - 750141 T^{24} + 1550583 T^{25} - 5124141 T^{26} + 5708070 T^{27} - 10097379 T^{28} + 11337408 T^{29} + 9565938 T^{30} + 4782969 T^{32} - 43046721 T^{33} + 43046721 T^{34} - 258280326 T^{35} + 387420489 T^{36} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{3} \)
$7$ \( ( 1 - 3 T + 35 T^{2} - 104 T^{3} + 659 T^{4} - 1784 T^{5} + 8247 T^{6} - 20223 T^{7} + 75730 T^{8} - 164555 T^{9} + 530110 T^{10} - 990927 T^{11} + 2828721 T^{12} - 4283384 T^{13} + 11075813 T^{14} - 12235496 T^{15} + 28824005 T^{16} - 17294403 T^{17} + 40353607 T^{18} )^{2} \)
$11$ \( 1 - 9 T + 25 T^{2} - 12 T^{3} - 24 T^{4} - 91 T^{5} - 540 T^{6} + 7247 T^{7} - 3173 T^{8} - 124295 T^{9} + 349798 T^{10} - 201170 T^{11} - 267079 T^{12} + 5753101 T^{13} - 42575665 T^{14} + 100911028 T^{15} + 151819518 T^{16} - 1253904551 T^{17} + 3739030263 T^{18} - 13792950061 T^{19} + 18370161678 T^{20} + 134312578268 T^{21} - 623350311265 T^{22} + 926542669151 T^{23} - 473146740319 T^{24} - 3920234190070 T^{25} + 74982307856038 T^{26} - 293081108252845 T^{27} - 82299448258973 T^{28} + 2067653676917917 T^{29} - 1694751323429340 T^{30} - 3141566805097721 T^{31} - 9113996005997784 T^{32} - 50126978032987812 T^{33} + 1148743246589304025 T^{34} - 4549023256493643939 T^{35} + 5559917313492231481 T^{36} \)
$13$ \( 1 - 4 T + 21 T^{2} + 56 T^{3} - 260 T^{4} + 1952 T^{5} + 3110 T^{6} - 2420 T^{7} + 66011 T^{8} + 439956 T^{9} - 292985 T^{10} + 2208328 T^{11} + 34844172 T^{12} - 36854292 T^{13} + 139942287 T^{14} + 1680146632 T^{15} - 929707432 T^{16} + 4297523544 T^{17} + 71524901083 T^{18} + 55867806072 T^{19} - 157120556008 T^{20} + 3691282150504 T^{21} + 3996891659007 T^{22} - 13683740639556 T^{23} + 168186163007148 T^{24} + 138569307049576 T^{25} - 238996865292185 T^{26} + 4665513126147588 T^{27} + 9100176905444339 T^{28} - 4337028153569540 T^{29} + 72457044730915910 T^{30} + 591212208068077856 T^{31} - 1023717860281815140 T^{32} + 2866410008789082392 T^{33} + 13973748792846776661 T^{34} - 34601663677525351732 T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \)
$17$ \( 1 + T - 23 T^{2} - 61 T^{3} + 797 T^{4} + 882 T^{5} - 9111 T^{6} - 55466 T^{7} + 169817 T^{8} + 861950 T^{9} + 2044903 T^{10} - 27646324 T^{11} - 18677652 T^{12} + 301514042 T^{13} + 2189324221 T^{14} - 6752066729 T^{15} - 26803962017 T^{16} - 10192305047 T^{17} + 842845786937 T^{18} - 173269185799 T^{19} - 7746345022913 T^{20} - 33172903839577 T^{21} + 182854548262141 T^{22} + 428106823131994 T^{23} - 450833113907988 T^{24} - 11344355903488052 T^{25} + 14264747318373223 T^{26} + 102216820146589150 T^{27} + 342350036192547833 T^{28} - 1900925000599171978 T^{29} - 5308271203400352471 T^{30} + 8735837825023036434 T^{31} + \)\(13\!\cdots\!13\)\( T^{32} - \)\(17\!\cdots\!73\)\( T^{33} - \)\(11\!\cdots\!63\)\( T^{34} + \)\(82\!\cdots\!77\)\( T^{35} + \)\(14\!\cdots\!09\)\( T^{36} \)
$19$ \( 1 + 2 T - 23 T^{2} - 24 T^{3} + 184 T^{4} + 1428 T^{5} + 12528 T^{6} + 12458 T^{7} - 294757 T^{8} - 220636 T^{9} + 6100277 T^{10} + 13014080 T^{11} + 26646360 T^{12} - 26340508 T^{13} - 628545959 T^{14} + 3968341152 T^{15} + 37315624628 T^{16} + 43754418476 T^{17} - 384806928499 T^{18} + 831333951044 T^{19} + 13470940490708 T^{20} + 27218851961568 T^{21} - 81912737922839 T^{22} - 65221705518292 T^{23} + 1253601481643160 T^{24} + 11632918321085120 T^{25} + 103604438997062357 T^{26} - 71196522887167444 T^{27} - 1807174696950649357 T^{28} + 1451235645354012302 T^{29} + 27728409306060865008 T^{30} + 60051660384103080252 T^{31} + \)\(14\!\cdots\!64\)\( T^{32} - \)\(36\!\cdots\!76\)\( T^{33} - \)\(66\!\cdots\!63\)\( T^{34} + \)\(10\!\cdots\!78\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \)
$23$ \( 1 + 19 T + 94 T^{2} - 364 T^{3} - 4934 T^{4} - 12944 T^{5} + 44670 T^{6} + 630217 T^{7} + 3232637 T^{8} + 217968 T^{9} - 91045708 T^{10} - 430615360 T^{11} + 84196748 T^{12} + 8365868928 T^{13} + 33581348290 T^{14} + 45796983038 T^{15} - 144611422188 T^{16} - 1864211343076 T^{17} - 11619108269332 T^{18} - 42876860890748 T^{19} - 76499442337452 T^{20} + 557211892623346 T^{21} + 9397438086821890 T^{22} + 53845601913650304 T^{23} + 12464140441088972 T^{24} - 1466170135597065920 T^{25} - 7129879099086223948 T^{26} + 392593643313767184 T^{27} + \)\(13\!\cdots\!13\)\( T^{28} + \)\(60\!\cdots\!59\)\( T^{29} + \)\(97\!\cdots\!70\)\( T^{30} - \)\(65\!\cdots\!52\)\( T^{31} - \)\(57\!\cdots\!06\)\( T^{32} - \)\(97\!\cdots\!48\)\( T^{33} + \)\(57\!\cdots\!34\)\( T^{34} + \)\(26\!\cdots\!57\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \)
$29$ \( 1 + 4 T - 51 T^{2} - 579 T^{3} - 1214 T^{4} + 26250 T^{5} + 148683 T^{6} - 209901 T^{7} - 4679822 T^{8} - 12680244 T^{9} + 95608714 T^{10} + 242480516 T^{11} - 2375176587 T^{12} - 7243939876 T^{13} + 48280620694 T^{14} + 562413939767 T^{15} + 693667484295 T^{16} - 10502405541311 T^{17} - 61444790128505 T^{18} - 304569760698019 T^{19} + 583374354292095 T^{20} + 13716713576977363 T^{21} + 34147965685073014 T^{22} - 148581530143677524 T^{23} - 1412810425440785427 T^{24} + 4182758908342495444 T^{25} + 47827916226314142154 T^{26} - \)\(18\!\cdots\!36\)\( T^{27} - \)\(19\!\cdots\!22\)\( T^{28} - \)\(25\!\cdots\!29\)\( T^{29} + \)\(52\!\cdots\!03\)\( T^{30} + \)\(26\!\cdots\!50\)\( T^{31} - \)\(36\!\cdots\!34\)\( T^{32} - \)\(49\!\cdots\!71\)\( T^{33} - \)\(12\!\cdots\!71\)\( T^{34} + \)\(29\!\cdots\!36\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$31$ \( 1 + 13 T + 82 T^{2} + 105 T^{3} - 1391 T^{4} - 5662 T^{5} + 25298 T^{6} + 405809 T^{7} + 2689426 T^{8} + 18321725 T^{9} + 105862757 T^{10} + 271821692 T^{11} - 314337076 T^{12} - 5345100555 T^{13} + 22049757012 T^{14} + 354024484218 T^{15} + 2321550409047 T^{16} + 12194450118645 T^{17} + 57150223625159 T^{18} + 378027953677995 T^{19} + 2231009943094167 T^{20} + 10546743409338438 T^{21} + 20363413645479252 T^{22} - 153025690899278805 T^{23} - 278975312024776756 T^{24} + 7478525318995095812 T^{25} + 90289396644094484837 T^{26} + \)\(48\!\cdots\!75\)\( T^{27} + \)\(22\!\cdots\!26\)\( T^{28} + \)\(10\!\cdots\!79\)\( T^{29} + \)\(19\!\cdots\!78\)\( T^{30} - \)\(13\!\cdots\!42\)\( T^{31} - \)\(10\!\cdots\!11\)\( T^{32} + \)\(24\!\cdots\!55\)\( T^{33} + \)\(59\!\cdots\!42\)\( T^{34} + \)\(29\!\cdots\!43\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \)
$37$ \( ( 1 - T + 154 T^{2} - 284 T^{3} + 12699 T^{4} - 29079 T^{5} + 767754 T^{6} - 1693550 T^{7} + 36154572 T^{8} - 70746236 T^{9} + 1337719164 T^{10} - 2318469950 T^{11} + 38889043362 T^{12} - 54498727719 T^{13} + 880598909943 T^{14} - 728666300156 T^{15} + 14619509078482 T^{16} - 3512479453921 T^{17} + 129961739795077 T^{18} )^{2} \)
$41$ \( 1 + 4 T - 76 T^{2} - 391 T^{3} + 1975 T^{4} + 9938 T^{5} - 26870 T^{6} + 433565 T^{7} + 4152628 T^{8} - 23937849 T^{9} - 280692273 T^{10} - 314064324 T^{11} + 5084566150 T^{12} + 15633564413 T^{13} + 29460875358 T^{14} + 1346278111323 T^{15} + 11524941557581 T^{16} - 48638796714869 T^{17} - 956776297373447 T^{18} - 1994190665309629 T^{19} + 19373426758293661 T^{20} + 92786833710492483 T^{21} + 83249392612497438 T^{22} + 1811245380978975013 T^{23} + 24152219232760042150 T^{24} - 61165369372547121444 T^{25} - \)\(22\!\cdots\!33\)\( T^{26} - \)\(78\!\cdots\!89\)\( T^{27} + \)\(55\!\cdots\!28\)\( T^{28} + \)\(23\!\cdots\!65\)\( T^{29} - \)\(60\!\cdots\!70\)\( T^{30} + \)\(91\!\cdots\!98\)\( T^{31} + \)\(74\!\cdots\!75\)\( T^{32} - \)\(60\!\cdots\!91\)\( T^{33} - \)\(48\!\cdots\!16\)\( T^{34} + \)\(10\!\cdots\!24\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \)
$43$ \( 1 + 13 T + 23 T^{2} + 341 T^{3} + 7465 T^{4} - 3559 T^{5} - 300996 T^{6} + 568061 T^{7} - 6089961 T^{8} - 178982555 T^{9} - 261868323 T^{10} + 1050344789 T^{11} - 23931288972 T^{12} - 12167512759 T^{13} + 1097418026995 T^{14} + 2155584799709 T^{15} + 6251828055461 T^{16} + 151946603608813 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 - 16 T + 58 T^{2} + 132 T^{3} + 2174 T^{4} - 59622 T^{5} + 583964 T^{6} - 2362132 T^{7} - 6973035 T^{8} - 13171090 T^{9} + 1946977070 T^{10} - 15462631128 T^{11} + 23862188930 T^{12} + 232936464174 T^{13} + 476845081058 T^{14} - 35831178696668 T^{15} + 291005595210458 T^{16} - 336150127799338 T^{17} - 5856096864612524 T^{18} - 15799056006568886 T^{19} + 642831359819901722 T^{20} - 3720100465824161764 T^{21} + 2326851881982182498 T^{22} + 53422815006541279218 T^{23} + \)\(25\!\cdots\!70\)\( T^{24} - \)\(78\!\cdots\!64\)\( T^{25} + \)\(46\!\cdots\!70\)\( T^{26} - \)\(14\!\cdots\!30\)\( T^{27} - \)\(36\!\cdots\!15\)\( T^{28} - \)\(58\!\cdots\!96\)\( T^{29} + \)\(67\!\cdots\!24\)\( T^{30} - \)\(32\!\cdots\!94\)\( T^{31} + \)\(55\!\cdots\!06\)\( T^{32} + \)\(15\!\cdots\!76\)\( T^{33} + \)\(32\!\cdots\!18\)\( T^{34} - \)\(42\!\cdots\!92\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \)
$53$ \( 1 - 14 T + 153 T^{2} - 1705 T^{3} + 14214 T^{4} - 166466 T^{5} + 1703087 T^{6} - 15265377 T^{7} + 137855194 T^{8} - 1013052036 T^{9} + 8625510462 T^{10} - 71977968032 T^{11} + 565915051273 T^{12} - 4612841434870 T^{13} + 31009525649816 T^{14} - 229284034127169 T^{15} + 1720930523999681 T^{16} - 12770176509314833 T^{17} + 101981507436118171 T^{18} - 676819354993686149 T^{19} + 4834093841915103929 T^{20} - 34135119148750539213 T^{21} + \)\(24\!\cdots\!96\)\( T^{22} - \)\(19\!\cdots\!10\)\( T^{23} + \)\(12\!\cdots\!17\)\( T^{24} - \)\(84\!\cdots\!84\)\( T^{25} + \)\(53\!\cdots\!82\)\( T^{26} - \)\(33\!\cdots\!88\)\( T^{27} + \)\(24\!\cdots\!06\)\( T^{28} - \)\(14\!\cdots\!69\)\( T^{29} + \)\(83\!\cdots\!67\)\( T^{30} - \)\(43\!\cdots\!18\)\( T^{31} + \)\(19\!\cdots\!66\)\( T^{32} - \)\(12\!\cdots\!85\)\( T^{33} + \)\(59\!\cdots\!13\)\( T^{34} - \)\(28\!\cdots\!82\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \)
$59$ \( 1 + 16 T + 110 T^{2} + 53 T^{3} - 9107 T^{4} - 138521 T^{5} - 1133458 T^{6} - 6642925 T^{7} - 15509483 T^{8} + 216413994 T^{9} + 3934924448 T^{10} + 39080271562 T^{11} + 325806014415 T^{12} + 2377170408214 T^{13} + 12590066317713 T^{14} - 2478684438573 T^{15} - 789252526438949 T^{16} - 11177472533604662 T^{17} - 102361406567292607 T^{18} - 659470879482675058 T^{19} - 2747388044533981469 T^{20} - 509069731309684167 T^{21} + \)\(15\!\cdots\!93\)\( T^{22} + \)\(16\!\cdots\!86\)\( T^{23} + \)\(13\!\cdots\!15\)\( T^{24} + \)\(97\!\cdots\!78\)\( T^{25} + \)\(57\!\cdots\!08\)\( T^{26} + \)\(18\!\cdots\!66\)\( T^{27} - \)\(79\!\cdots\!83\)\( T^{28} - \)\(20\!\cdots\!75\)\( T^{29} - \)\(20\!\cdots\!98\)\( T^{30} - \)\(14\!\cdots\!59\)\( T^{31} - \)\(56\!\cdots\!27\)\( T^{32} + \)\(19\!\cdots\!47\)\( T^{33} + \)\(23\!\cdots\!10\)\( T^{34} + \)\(20\!\cdots\!04\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \)
$61$ \( 1 - 24 T + 193 T^{2} - 1023 T^{3} + 12269 T^{4} - 101120 T^{5} + 137341 T^{6} + 6279714 T^{7} - 66107156 T^{8} - 91531315 T^{9} + 3450816819 T^{10} - 8829436552 T^{11} + 246170019400 T^{12} - 4618272414559 T^{13} + 35328002089660 T^{14} - 91573533385944 T^{15} - 1045706406860373 T^{16} + 6388716443127935 T^{17} + 8003958918844677 T^{18} + 389711703030804035 T^{19} - 3891073539927447933 T^{20} - 20785452181474955064 T^{21} + \)\(48\!\cdots\!60\)\( T^{22} - \)\(39\!\cdots\!59\)\( T^{23} + \)\(12\!\cdots\!00\)\( T^{24} - \)\(27\!\cdots\!92\)\( T^{25} + \)\(66\!\cdots\!39\)\( T^{26} - \)\(10\!\cdots\!15\)\( T^{27} - \)\(47\!\cdots\!56\)\( T^{28} + \)\(27\!\cdots\!54\)\( T^{29} + \)\(36\!\cdots\!61\)\( T^{30} - \)\(16\!\cdots\!20\)\( T^{31} + \)\(12\!\cdots\!29\)\( T^{32} - \)\(61\!\cdots\!23\)\( T^{33} + \)\(70\!\cdots\!73\)\( T^{34} - \)\(53\!\cdots\!04\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \)
$67$ \( 1 - 5 T - 243 T^{2} + 2637 T^{3} + 14043 T^{4} - 432442 T^{5} + 2109906 T^{6} + 27147279 T^{7} - 406935575 T^{8} + 400811603 T^{9} + 30519003442 T^{10} - 202859948048 T^{11} - 1206933969243 T^{12} + 21871443321101 T^{13} - 24381765716161 T^{14} - 1594432006027610 T^{15} + 11273924333523498 T^{16} + 50256293453176677 T^{17} - 1103332578696118791 T^{18} + 3367171661362837359 T^{19} + 50608646333186982522 T^{20} - \)\(47\!\cdots\!30\)\( T^{21} - \)\(49\!\cdots\!81\)\( T^{22} + \)\(29\!\cdots\!07\)\( T^{23} - \)\(10\!\cdots\!67\)\( T^{24} - \)\(12\!\cdots\!04\)\( T^{25} + \)\(12\!\cdots\!22\)\( T^{26} + \)\(10\!\cdots\!41\)\( T^{27} - \)\(74\!\cdots\!75\)\( T^{28} + \)\(33\!\cdots\!57\)\( T^{29} + \)\(17\!\cdots\!66\)\( T^{30} - \)\(23\!\cdots\!54\)\( T^{31} + \)\(51\!\cdots\!47\)\( T^{32} + \)\(64\!\cdots\!91\)\( T^{33} - \)\(40\!\cdots\!83\)\( T^{34} - \)\(55\!\cdots\!35\)\( T^{35} + \)\(74\!\cdots\!09\)\( T^{36} \)
$71$ \( 1 + 3 T + 202 T^{2} - 152 T^{3} + 27691 T^{4} - 92044 T^{5} + 2694570 T^{6} - 19553968 T^{7} + 262197087 T^{8} - 2018099301 T^{9} + 27616454078 T^{10} - 171467755434 T^{11} + 2659217155551 T^{12} - 13109941104883 T^{13} + 220898323319547 T^{14} - 1195438763913623 T^{15} + 14974987479690423 T^{16} - 102503631929816908 T^{17} + 1048608441396878449 T^{18} - 7277757867017000468 T^{19} + 75488911885119422343 T^{20} - \)\(42\!\cdots\!53\)\( T^{21} + \)\(56\!\cdots\!07\)\( T^{22} - \)\(23\!\cdots\!33\)\( T^{23} + \)\(34\!\cdots\!71\)\( T^{24} - \)\(15\!\cdots\!94\)\( T^{25} + \)\(17\!\cdots\!58\)\( T^{26} - \)\(92\!\cdots\!31\)\( T^{27} + \)\(85\!\cdots\!87\)\( T^{28} - \)\(45\!\cdots\!28\)\( T^{29} + \)\(44\!\cdots\!70\)\( T^{30} - \)\(10\!\cdots\!84\)\( T^{31} + \)\(22\!\cdots\!71\)\( T^{32} - \)\(89\!\cdots\!52\)\( T^{33} + \)\(84\!\cdots\!42\)\( T^{34} + \)\(88\!\cdots\!73\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$73$ \( 1 + 22 T + 272 T^{2} + 2669 T^{3} + 31125 T^{4} + 309208 T^{5} + 1961900 T^{6} + 14213181 T^{7} + 190802684 T^{8} + 2015970959 T^{9} + 10963042915 T^{10} + 81525488566 T^{11} + 933068681880 T^{12} + 3934730546441 T^{13} - 34153795489790 T^{14} - 225081571812169 T^{15} + 1756036748972213 T^{16} - 803992448271191 T^{17} - 175608418784800179 T^{18} - 58691448723796943 T^{19} + 9357919835272923077 T^{20} - 87560557821654547873 T^{21} - \)\(96\!\cdots\!90\)\( T^{22} + \)\(81\!\cdots\!13\)\( T^{23} + \)\(14\!\cdots\!20\)\( T^{24} + \)\(90\!\cdots\!02\)\( T^{25} + \)\(88\!\cdots\!15\)\( T^{26} + \)\(11\!\cdots\!67\)\( T^{27} + \)\(81\!\cdots\!16\)\( T^{28} + \)\(44\!\cdots\!37\)\( T^{29} + \)\(44\!\cdots\!00\)\( T^{30} + \)\(51\!\cdots\!64\)\( T^{31} + \)\(37\!\cdots\!25\)\( T^{32} + \)\(23\!\cdots\!33\)\( T^{33} + \)\(17\!\cdots\!92\)\( T^{34} + \)\(10\!\cdots\!66\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \)
$79$ \( ( 1 + 19 T + 531 T^{2} + 7400 T^{3} + 109592 T^{4} + 1180276 T^{5} + 11765861 T^{6} + 107015700 T^{7} + 850808559 T^{8} + 7993181266 T^{9} + 67213876161 T^{10} + 667884983700 T^{11} + 5801028341579 T^{12} + 45971845802356 T^{13} + 337220764879208 T^{14} + 1798847170855400 T^{15} + 10197275671650429 T^{16} + 28825067388224659 T^{17} + 119851595982618319 T^{18} )^{2} \)
$83$ \( 1 + 26 T + 41 T^{2} - 4688 T^{3} - 28457 T^{4} + 507415 T^{5} + 4946122 T^{6} - 38886347 T^{7} - 708143290 T^{8} - 839183842 T^{9} + 54673401155 T^{10} + 460215646841 T^{11} - 1917272402872 T^{12} - 58344829035493 T^{13} - 162190885833500 T^{14} + 4674551351114112 T^{15} + 42143643054278062 T^{16} - 154094583853300240 T^{17} - 4385792214629651212 T^{18} - 12789850459823919920 T^{19} + \)\(29\!\cdots\!18\)\( T^{20} + \)\(26\!\cdots\!44\)\( T^{21} - \)\(76\!\cdots\!00\)\( T^{22} - \)\(22\!\cdots\!99\)\( T^{23} - \)\(62\!\cdots\!68\)\( T^{24} + \)\(12\!\cdots\!07\)\( T^{25} + \)\(12\!\cdots\!55\)\( T^{26} - \)\(15\!\cdots\!26\)\( T^{27} - \)\(10\!\cdots\!10\)\( T^{28} - \)\(50\!\cdots\!49\)\( T^{29} + \)\(52\!\cdots\!42\)\( T^{30} + \)\(45\!\cdots\!45\)\( T^{31} - \)\(20\!\cdots\!53\)\( T^{32} - \)\(28\!\cdots\!16\)\( T^{33} + \)\(20\!\cdots\!21\)\( T^{34} + \)\(10\!\cdots\!98\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \)
$89$ \( 1 - 52 T + 1128 T^{2} - 13153 T^{3} + 81467 T^{4} + 68453 T^{5} - 9458026 T^{6} + 132595253 T^{7} - 801617131 T^{8} - 3309701708 T^{9} + 120931060118 T^{10} - 1201000189794 T^{11} + 3658479374927 T^{12} + 72912524472654 T^{13} - 1364925845819525 T^{14} + 10187321088410209 T^{15} + 12050739011899131 T^{16} - 1203687248326207182 T^{17} + 15537143357929513709 T^{18} - \)\(10\!\cdots\!98\)\( T^{19} + 95453903713253016651 T^{20} + \)\(71\!\cdots\!21\)\( T^{21} - \)\(85\!\cdots\!25\)\( T^{22} + \)\(40\!\cdots\!46\)\( T^{23} + \)\(18\!\cdots\!47\)\( T^{24} - \)\(53\!\cdots\!26\)\( T^{25} + \)\(47\!\cdots\!58\)\( T^{26} - \)\(11\!\cdots\!72\)\( T^{27} - \)\(24\!\cdots\!31\)\( T^{28} + \)\(36\!\cdots\!17\)\( T^{29} - \)\(23\!\cdots\!46\)\( T^{30} + \)\(15\!\cdots\!57\)\( T^{31} + \)\(15\!\cdots\!47\)\( T^{32} - \)\(22\!\cdots\!97\)\( T^{33} + \)\(17\!\cdots\!08\)\( T^{34} - \)\(71\!\cdots\!08\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \)
$97$ \( 1 - 27 T + 66 T^{2} + 6423 T^{3} - 102424 T^{4} + 202528 T^{5} + 13832941 T^{6} - 194404750 T^{7} + 369359831 T^{8} + 18779122634 T^{9} - 227205754104 T^{10} + 306742101966 T^{11} + 18629760587417 T^{12} - 187413939377042 T^{13} - 64509572163695 T^{14} + 16124698263546201 T^{15} - 117486606211482490 T^{16} - 549881906918518023 T^{17} + 14838596816271178503 T^{18} - 53338544971096248231 T^{19} - \)\(11\!\cdots\!10\)\( T^{20} + \)\(14\!\cdots\!73\)\( T^{21} - \)\(57\!\cdots\!95\)\( T^{22} - \)\(16\!\cdots\!94\)\( T^{23} + \)\(15\!\cdots\!93\)\( T^{24} + \)\(24\!\cdots\!58\)\( T^{25} - \)\(17\!\cdots\!44\)\( T^{26} + \)\(14\!\cdots\!78\)\( T^{27} + \)\(27\!\cdots\!19\)\( T^{28} - \)\(13\!\cdots\!50\)\( T^{29} + \)\(95\!\cdots\!81\)\( T^{30} + \)\(13\!\cdots\!56\)\( T^{31} - \)\(66\!\cdots\!56\)\( T^{32} + \)\(40\!\cdots\!39\)\( T^{33} + \)\(40\!\cdots\!86\)\( T^{34} - \)\(16\!\cdots\!99\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \)
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