Properties

Label 98.2.e.a
Level $98$
Weight $2$
Character orbit 98.e
Analytic conductor $0.783$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 98.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.782533939809\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} + 15 x^{16} - 23 x^{15} + 72 x^{14} - 85 x^{13} + 432 x^{12} - 282 x^{11} + 1786 x^{10} - 1092 x^{9} + 7272 x^{8} - 10168 x^{7} + 25378 x^{6} - 43359 x^{5} + 57726 x^{4} - 56565 x^{3} + 38232 x^{2} - 6804 x + 729\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 7 \)
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.

\(f(q)\) \(=\) \( q -\beta_{13} q^{2} + ( \beta_{1} - \beta_{14} ) q^{3} + ( -1 + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{4} + ( -\beta_{11} - \beta_{14} + \beta_{17} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{6} + ( -1 + \beta_{1} + \beta_{10} + \beta_{12} ) q^{7} -\beta_{14} q^{8} + ( 2 - 2 \beta_{4} - \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{9} +O(q^{10})\) \( q -\beta_{13} q^{2} + ( \beta_{1} - \beta_{14} ) q^{3} + ( -1 + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{4} + ( -\beta_{11} - \beta_{14} + \beta_{17} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{6} + ( -1 + \beta_{1} + \beta_{10} + \beta_{12} ) q^{7} -\beta_{14} q^{8} + ( 2 - 2 \beta_{4} - \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{9} + ( \beta_{7} - \beta_{8} - \beta_{16} ) q^{10} + ( 2 \beta_{5} - \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{17} ) q^{11} + ( -\beta_{2} + \beta_{9} ) q^{12} + ( \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{13} + \beta_{17} ) q^{14} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{15} -\beta_{8} q^{16} + ( -1 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{17} + ( 2 + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{18} + ( 3 - 2 \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{10} - \beta_{13} ) q^{19} + ( -\beta_{7} + \beta_{9} + \beta_{11} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{20} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{15} - \beta_{17} ) q^{21} + ( -1 + 2 \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{22} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{23} + ( -\beta_{4} - \beta_{10} ) q^{24} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{13} + 2 \beta_{15} + 2 \beta_{17} ) q^{25} + ( -1 + \beta_{1} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{16} ) q^{26} + ( -3 + \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{7} + 3 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - \beta_{11} - \beta_{12} + 3 \beta_{13} + 3 \beta_{14} - \beta_{15} - \beta_{16} ) q^{27} + ( 1 - \beta_{2} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{14} ) q^{28} + ( -1 - \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{14} - \beta_{17} ) q^{29} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} - \beta_{14} - \beta_{16} + \beta_{17} ) q^{30} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{31} + \beta_{9} q^{32} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 8 \beta_{8} + 2 \beta_{9} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{17} ) q^{33} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{34} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{9} + \beta_{11} - \beta_{12} - 4 \beta_{13} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{35} + ( 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{36} + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{13} + \beta_{14} ) q^{38} + ( 1 - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{39} + ( -\beta_{10} + \beta_{15} + \beta_{17} ) q^{40} + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 5 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 5 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} ) q^{41} + ( 3 - \beta_{2} + \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{16} ) q^{42} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - \beta_{9} + 3 \beta_{13} + \beta_{14} - \beta_{15} ) q^{43} + ( 2 \beta_{1} - \beta_{11} - \beta_{14} + \beta_{17} ) q^{44} + ( 2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} - 4 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{45} + ( 2 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{46} + ( -\beta_{5} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{13} + 2 \beta_{14} + \beta_{17} ) q^{47} + ( 1 + \beta_{6} ) q^{48} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{49} + ( 3 + \beta_{2} - \beta_{3} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{11} - 3 \beta_{14} ) q^{50} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} - 6 \beta_{13} + \beta_{15} + \beta_{17} ) q^{51} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{16} ) q^{52} + ( 4 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - 4 \beta_{8} + \beta_{9} - \beta_{10} ) q^{53} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + 2 \beta_{11} - 3 \beta_{14} - \beta_{15} - 2 \beta_{17} ) q^{54} + ( 5 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 6 \beta_{8} - 2 \beta_{12} - 5 \beta_{13} - 2 \beta_{15} - 2 \beta_{17} ) q^{55} + ( 1 - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{56} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 4 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - 4 \beta_{13} - 5 \beta_{14} + 2 \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{57} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{58} + ( -3 + \beta_{5} + \beta_{6} + \beta_{8} - 4 \beta_{9} + 3 \beta_{13} + 4 \beta_{14} + \beta_{15} ) q^{59} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{15} - \beta_{17} ) q^{60} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{14} + \beta_{17} ) q^{61} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{14} + \beta_{17} ) q^{62} + ( -8 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + 10 \beta_{8} - 10 \beta_{9} + 11 \beta_{10} - \beta_{11} + 9 \beta_{13} + 6 \beta_{14} - \beta_{15} - \beta_{17} ) q^{63} -\beta_{10} q^{64} + ( -4 - \beta_{1} + \beta_{2} + \beta_{7} + 2 \beta_{8} + \beta_{12} + 4 \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{65} + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} - 3 \beta_{8} + 9 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{14} + \beta_{17} ) q^{66} + ( 2 - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{10} + \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{67} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} - \beta_{14} - \beta_{16} + \beta_{17} ) q^{68} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + 4 \beta_{8} - 5 \beta_{9} + 4 \beta_{10} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{69} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + \beta_{11} + 3 \beta_{13} + 4 \beta_{14} + \beta_{15} ) q^{70} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{71} + ( -2 + 2 \beta_{3} + 2 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{16} ) q^{72} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{9} - 6 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{16} ) q^{73} + ( -2 + \beta_{2} + \beta_{4} - \beta_{6} - \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{74} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + 8 \beta_{9} - 8 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{16} ) q^{75} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + 2 \beta_{13} + 2 \beta_{14} ) q^{76} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + \beta_{16} + \beta_{17} ) q^{77} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{78} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{6} - 2 \beta_{7} + 2 \beta_{11} - \beta_{12} + 2 \beta_{16} - 2 \beta_{17} ) q^{79} + ( 1 + \beta_{7} - \beta_{11} ) q^{80} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} + 7 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} - \beta_{14} + \beta_{17} ) q^{81} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 5 \beta_{9} + 3 \beta_{13} + 5 \beta_{14} + 2 \beta_{15} ) q^{82} + ( 4 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{8} + 4 \beta_{9} - 7 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{15} + 2 \beta_{17} ) q^{83} + ( 1 + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{10} - \beta_{11} - 3 \beta_{13} + \beta_{15} + \beta_{17} ) q^{84} + ( 3 \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 4 \beta_{10} + 3 \beta_{11} - \beta_{12} + 2 \beta_{13} + 4 \beta_{14} - 3 \beta_{15} - 4 \beta_{17} ) q^{85} + ( 3 - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - 3 \beta_{14} ) q^{86} + ( 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} + \beta_{11} - \beta_{12} + 5 \beta_{13} + 6 \beta_{14} - \beta_{15} - \beta_{17} ) q^{87} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{16} ) q^{88} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{6} + 3 \beta_{8} + 2 \beta_{10} + 3 \beta_{13} - 2 \beta_{15} - 2 \beta_{17} ) q^{89} + ( -5 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 5 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{90} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{8} + 6 \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{16} ) q^{91} + ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} ) q^{92} + ( 4 - \beta_{2} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{93} + ( 1 - \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{13} - 3 \beta_{14} ) q^{94} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 4 \beta_{7} - 3 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} + \beta_{14} + 4 \beta_{15} - 3 \beta_{16} + 3 \beta_{17} ) q^{95} + ( \beta_{5} - \beta_{13} ) q^{96} + ( 5 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{8} - 2 \beta_{10} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{16} - \beta_{17} ) q^{97} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 4 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{98} + ( 6 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{11} + 3 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 3q^{2} + 3q^{3} - 3q^{4} - 6q^{5} + 3q^{6} - 7q^{7} - 3q^{8} + 10q^{9} + O(q^{10}) \) \( 18q - 3q^{2} + 3q^{3} - 3q^{4} - 6q^{5} + 3q^{6} - 7q^{7} - 3q^{8} + 10q^{9} - 6q^{10} - q^{11} - 4q^{12} - 7q^{14} - 9q^{15} - 3q^{16} - 11q^{17} + 24q^{18} + 36q^{19} + q^{20} - 21q^{21} - q^{22} + 5q^{23} - 4q^{24} - 23q^{25} - 7q^{26} - 12q^{27} - 13q^{29} - 16q^{30} - 4q^{31} - 3q^{32} - 34q^{33} - 11q^{34} - 7q^{35} + 10q^{36} + 33q^{37} + 15q^{38} + 21q^{39} + q^{40} - 28q^{41} + 35q^{42} - 20q^{43} + 6q^{44} + 20q^{45} + 5q^{46} + 36q^{47} + 10q^{48} + 49q^{49} + 26q^{50} - 20q^{51} + 48q^{53} + 2q^{54} + 47q^{55} + 7q^{56} - 37q^{57} + 36q^{58} - 25q^{59} - 9q^{60} + q^{61} - 11q^{62} - 35q^{63} - 3q^{64} - 56q^{65} - 27q^{66} + 34q^{67} + 38q^{68} + 23q^{69} - 14q^{70} - 6q^{71} - 11q^{72} - 39q^{73} - 23q^{74} - 47q^{75} - 20q^{76} - 28q^{77} - 14q^{78} - 2q^{79} + 8q^{80} - 34q^{81} - 28q^{82} + 35q^{83} - 7q^{84} + 33q^{85} + 36q^{86} + 48q^{87} + 6q^{88} - 6q^{89} - 57q^{90} - 35q^{91} + 5q^{92} + 36q^{93} - 13q^{94} - 17q^{95} - 4q^{96} + 56q^{97} + 28q^{98} + 106q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 6 x^{17} + 15 x^{16} - 23 x^{15} + 72 x^{14} - 85 x^{13} + 432 x^{12} - 282 x^{11} + 1786 x^{10} - 1092 x^{9} + 7272 x^{8} - 10168 x^{7} + 25378 x^{6} - 43359 x^{5} + 57726 x^{4} - 56565 x^{3} + 38232 x^{2} - 6804 x + 729\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(10015658002884652210594435 \nu^{17} - 79513806052424886535080999 \nu^{16} + 325008551120043300765698265 \nu^{15} - 756991685900368306021245941 \nu^{14} + 1400721420122708642190436362 \nu^{13} - 2010945999033739260005960950 \nu^{12} + 7597976644312541423261696226 \nu^{11} - 8652663576611853540543729246 \nu^{10} + 44248918343919751452730754212 \nu^{9} - 14030907819417365887072158387 \nu^{8} + 187865311214703586839327158175 \nu^{7} - 105528207493893102592977311419 \nu^{6} + 864045388703495817784663010401 \nu^{5} - 719613121908098820715014186918 \nu^{4} + 2099865762766593631546061645529 \nu^{3} - 1722228805996632815732220879180 \nu^{2} + 967650595331333424467566781259 \nu + 71118173934409380488766390294\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{3}\)\(=\)\((\)\(15063944114204749831611655 \nu^{17} - 115631656173465529392831096 \nu^{16} + 353152647227254606333909665 \nu^{15} - 549699265959367850084592104 \nu^{14} + 1227462567742034605070636802 \nu^{13} - 2628749503893476320422432403 \nu^{12} + 7188519785503926976523617737 \nu^{11} - 11776823211587405847309643404 \nu^{10} + 26381419892223036176457799546 \nu^{9} - 44918138620408547170959066708 \nu^{8} + 122899553015186383044162308622 \nu^{7} - 247718221127857932097539167458 \nu^{6} + 579229036136375085942086148175 \nu^{5} - 806764686368952825190079739225 \nu^{4} + 1958830447042537083700638234147 \nu^{3} - 981876765226275758135851286679 \nu^{2} + 819519104662852361547822298254 \nu - 252969283425255760130639688702\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(62\!\cdots\!27\)\( \nu^{17} + \)\(38\!\cdots\!56\)\( \nu^{16} - \)\(10\!\cdots\!05\)\( \nu^{15} + \)\(18\!\cdots\!40\)\( \nu^{14} - \)\(50\!\cdots\!05\)\( \nu^{13} + \)\(59\!\cdots\!48\)\( \nu^{12} - \)\(29\!\cdots\!47\)\( \nu^{11} + \)\(22\!\cdots\!24\)\( \nu^{10} - \)\(13\!\cdots\!13\)\( \nu^{9} + \)\(80\!\cdots\!41\)\( \nu^{8} - \)\(50\!\cdots\!32\)\( \nu^{7} + \)\(71\!\cdots\!88\)\( \nu^{6} - \)\(18\!\cdots\!18\)\( \nu^{5} + \)\(32\!\cdots\!86\)\( \nu^{4} - \)\(41\!\cdots\!48\)\( \nu^{3} + \)\(49\!\cdots\!90\)\( \nu^{2} - \)\(23\!\cdots\!64\)\( \nu + \)\(29\!\cdots\!45\)\(\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(79\!\cdots\!31\)\( \nu^{17} + \)\(50\!\cdots\!85\)\( \nu^{16} - \)\(12\!\cdots\!55\)\( \nu^{15} + \)\(17\!\cdots\!08\)\( \nu^{14} - \)\(54\!\cdots\!62\)\( \nu^{13} + \)\(80\!\cdots\!07\)\( \nu^{12} - \)\(31\!\cdots\!35\)\( \nu^{11} + \)\(32\!\cdots\!75\)\( \nu^{10} - \)\(11\!\cdots\!72\)\( \nu^{9} + \)\(14\!\cdots\!00\)\( \nu^{8} - \)\(45\!\cdots\!45\)\( \nu^{7} + \)\(10\!\cdots\!77\)\( \nu^{6} - \)\(16\!\cdots\!72\)\( \nu^{5} + \)\(37\!\cdots\!18\)\( \nu^{4} - \)\(42\!\cdots\!32\)\( \nu^{3} + \)\(38\!\cdots\!16\)\( \nu^{2} - \)\(21\!\cdots\!87\)\( \nu + \)\(18\!\cdots\!12\)\(\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{6}\)\(=\)\((\)\(98969763333246802301305825 \nu^{17} - 513710134690333672589360613 \nu^{16} + 993991537625664669089712162 \nu^{15} - 1048806029795318531280661184 \nu^{14} + 5410056698869938196449955941 \nu^{13} - 3273923422382866649085772270 \nu^{12} + 36478660645677555573372972198 \nu^{11} + 5375198066276994932517168147 \nu^{10} + 154578251845407109438646931007 \nu^{9} + 22152728143589424115919705889 \nu^{8} + 657714726776550819254319835236 \nu^{7} - 461370730082250140024008381978 \nu^{6} + 1710812629708525923597234018715 \nu^{5} - 2373600084748988380011187627704 \nu^{4} + 2629307432802732653170282024980 \nu^{3} - 1701618818868706090540982163048 \nu^{2} + 301547760906705942114951426423 \nu + 158459249151164759917846330071\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(15\!\cdots\!90\)\( \nu^{17} + \)\(10\!\cdots\!08\)\( \nu^{16} - \)\(31\!\cdots\!80\)\( \nu^{15} + \)\(50\!\cdots\!10\)\( \nu^{14} - \)\(12\!\cdots\!47\)\( \nu^{13} + \)\(21\!\cdots\!75\)\( \nu^{12} - \)\(75\!\cdots\!15\)\( \nu^{11} + \)\(91\!\cdots\!87\)\( \nu^{10} - \)\(29\!\cdots\!31\)\( \nu^{9} + \)\(35\!\cdots\!47\)\( \nu^{8} - \)\(12\!\cdots\!10\)\( \nu^{7} + \)\(22\!\cdots\!49\)\( \nu^{6} - \)\(52\!\cdots\!74\)\( \nu^{5} + \)\(84\!\cdots\!02\)\( \nu^{4} - \)\(14\!\cdots\!69\)\( \nu^{3} + \)\(11\!\cdots\!23\)\( \nu^{2} - \)\(95\!\cdots\!38\)\( \nu + \)\(29\!\cdots\!71\)\(\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(65\!\cdots\!97\)\( \nu^{17} - \)\(42\!\cdots\!57\)\( \nu^{16} + \)\(11\!\cdots\!94\)\( \nu^{15} - \)\(17\!\cdots\!17\)\( \nu^{14} + \)\(50\!\cdots\!36\)\( \nu^{13} - \)\(71\!\cdots\!68\)\( \nu^{12} + \)\(29\!\cdots\!14\)\( \nu^{11} - \)\(29\!\cdots\!48\)\( \nu^{10} + \)\(11\!\cdots\!01\)\( \nu^{9} - \)\(11\!\cdots\!45\)\( \nu^{8} + \)\(46\!\cdots\!17\)\( \nu^{7} - \)\(86\!\cdots\!04\)\( \nu^{6} + \)\(17\!\cdots\!00\)\( \nu^{5} - \)\(33\!\cdots\!68\)\( \nu^{4} + \)\(44\!\cdots\!34\)\( \nu^{3} - \)\(44\!\cdots\!45\)\( \nu^{2} + \)\(30\!\cdots\!48\)\( \nu - \)\(53\!\cdots\!57\)\(\)\()/ \)\(35\!\cdots\!11\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(76\!\cdots\!84\)\( \nu^{17} + \)\(43\!\cdots\!11\)\( \nu^{16} - \)\(99\!\cdots\!05\)\( \nu^{15} + \)\(13\!\cdots\!67\)\( \nu^{14} - \)\(49\!\cdots\!24\)\( \nu^{13} + \)\(48\!\cdots\!54\)\( \nu^{12} - \)\(30\!\cdots\!67\)\( \nu^{11} + \)\(12\!\cdots\!83\)\( \nu^{10} - \)\(12\!\cdots\!99\)\( \nu^{9} + \)\(49\!\cdots\!12\)\( \nu^{8} - \)\(51\!\cdots\!48\)\( \nu^{7} + \)\(64\!\cdots\!77\)\( \nu^{6} - \)\(16\!\cdots\!21\)\( \nu^{5} + \)\(28\!\cdots\!40\)\( \nu^{4} - \)\(33\!\cdots\!30\)\( \nu^{3} + \)\(30\!\cdots\!64\)\( \nu^{2} - \)\(17\!\cdots\!40\)\( \nu - \)\(13\!\cdots\!25\)\(\)\()/ \)\(35\!\cdots\!11\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(34\!\cdots\!38\)\( \nu^{17} + \)\(20\!\cdots\!73\)\( \nu^{16} - \)\(50\!\cdots\!74\)\( \nu^{15} + \)\(76\!\cdots\!09\)\( \nu^{14} - \)\(24\!\cdots\!32\)\( \nu^{13} + \)\(28\!\cdots\!28\)\( \nu^{12} - \)\(14\!\cdots\!13\)\( \nu^{11} + \)\(90\!\cdots\!79\)\( \nu^{10} - \)\(60\!\cdots\!64\)\( \nu^{9} + \)\(35\!\cdots\!50\)\( \nu^{8} - \)\(24\!\cdots\!28\)\( \nu^{7} + \)\(34\!\cdots\!62\)\( \nu^{6} - \)\(85\!\cdots\!06\)\( \nu^{5} + \)\(14\!\cdots\!67\)\( \nu^{4} - \)\(19\!\cdots\!63\)\( \nu^{3} + \)\(17\!\cdots\!23\)\( \nu^{2} - \)\(12\!\cdots\!37\)\( \nu + \)\(15\!\cdots\!98\)\(\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(37\!\cdots\!36\)\( \nu^{17} + \)\(21\!\cdots\!45\)\( \nu^{16} - \)\(53\!\cdots\!05\)\( \nu^{15} + \)\(82\!\cdots\!55\)\( \nu^{14} - \)\(26\!\cdots\!42\)\( \nu^{13} + \)\(26\!\cdots\!26\)\( \nu^{12} - \)\(16\!\cdots\!54\)\( \nu^{11} + \)\(68\!\cdots\!25\)\( \nu^{10} - \)\(70\!\cdots\!41\)\( \nu^{9} + \)\(19\!\cdots\!40\)\( \nu^{8} - \)\(29\!\cdots\!38\)\( \nu^{7} + \)\(29\!\cdots\!95\)\( \nu^{6} - \)\(10\!\cdots\!33\)\( \nu^{5} + \)\(13\!\cdots\!21\)\( \nu^{4} - \)\(20\!\cdots\!48\)\( \nu^{3} + \)\(17\!\cdots\!84\)\( \nu^{2} - \)\(10\!\cdots\!28\)\( \nu - \)\(82\!\cdots\!95\)\(\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(11\!\cdots\!95\)\( \nu^{17} + \)\(59\!\cdots\!39\)\( \nu^{16} - \)\(11\!\cdots\!73\)\( \nu^{15} + \)\(13\!\cdots\!96\)\( \nu^{14} - \)\(63\!\cdots\!68\)\( \nu^{13} + \)\(37\!\cdots\!15\)\( \nu^{12} - \)\(42\!\cdots\!26\)\( \nu^{11} - \)\(60\!\cdots\!14\)\( \nu^{10} - \)\(18\!\cdots\!51\)\( \nu^{9} - \)\(29\!\cdots\!20\)\( \nu^{8} - \)\(75\!\cdots\!72\)\( \nu^{7} + \)\(51\!\cdots\!56\)\( \nu^{6} - \)\(20\!\cdots\!47\)\( \nu^{5} + \)\(27\!\cdots\!48\)\( \nu^{4} - \)\(31\!\cdots\!85\)\( \nu^{3} + \)\(21\!\cdots\!78\)\( \nu^{2} - \)\(38\!\cdots\!75\)\( \nu - \)\(60\!\cdots\!57\)\(\)\()/ \)\(35\!\cdots\!11\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(11\!\cdots\!06\)\( \nu^{17} - \)\(66\!\cdots\!13\)\( \nu^{16} + \)\(15\!\cdots\!47\)\( \nu^{15} - \)\(23\!\cdots\!02\)\( \nu^{14} + \)\(79\!\cdots\!95\)\( \nu^{13} - \)\(82\!\cdots\!38\)\( \nu^{12} + \)\(48\!\cdots\!00\)\( \nu^{11} - \)\(22\!\cdots\!29\)\( \nu^{10} + \)\(20\!\cdots\!36\)\( \nu^{9} - \)\(78\!\cdots\!06\)\( \nu^{8} + \)\(85\!\cdots\!13\)\( \nu^{7} - \)\(97\!\cdots\!98\)\( \nu^{6} + \)\(28\!\cdots\!84\)\( \nu^{5} - \)\(44\!\cdots\!49\)\( \nu^{4} + \)\(60\!\cdots\!19\)\( \nu^{3} - \)\(57\!\cdots\!48\)\( \nu^{2} + \)\(37\!\cdots\!29\)\( \nu - \)\(31\!\cdots\!28\)\(\)\()/ \)\(35\!\cdots\!11\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(41\!\cdots\!05\)\( \nu^{17} + \)\(23\!\cdots\!03\)\( \nu^{16} - \)\(57\!\cdots\!19\)\( \nu^{15} + \)\(83\!\cdots\!10\)\( \nu^{14} - \)\(27\!\cdots\!20\)\( \nu^{13} + \)\(29\!\cdots\!20\)\( \nu^{12} - \)\(17\!\cdots\!12\)\( \nu^{11} + \)\(86\!\cdots\!63\)\( \nu^{10} - \)\(70\!\cdots\!06\)\( \nu^{9} + \)\(31\!\cdots\!47\)\( \nu^{8} - \)\(29\!\cdots\!19\)\( \nu^{7} + \)\(36\!\cdots\!08\)\( \nu^{6} - \)\(96\!\cdots\!02\)\( \nu^{5} + \)\(15\!\cdots\!77\)\( \nu^{4} - \)\(20\!\cdots\!44\)\( \nu^{3} + \)\(19\!\cdots\!77\)\( \nu^{2} - \)\(10\!\cdots\!70\)\( \nu + \)\(39\!\cdots\!56\)\(\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(59\!\cdots\!48\)\( \nu^{17} - \)\(33\!\cdots\!72\)\( \nu^{16} + \)\(76\!\cdots\!05\)\( \nu^{15} - \)\(11\!\cdots\!52\)\( \nu^{14} + \)\(40\!\cdots\!40\)\( \nu^{13} - \)\(37\!\cdots\!18\)\( \nu^{12} + \)\(24\!\cdots\!93\)\( \nu^{11} - \)\(79\!\cdots\!94\)\( \nu^{10} + \)\(10\!\cdots\!32\)\( \nu^{9} - \)\(26\!\cdots\!80\)\( \nu^{8} + \)\(43\!\cdots\!51\)\( \nu^{7} - \)\(44\!\cdots\!90\)\( \nu^{6} + \)\(13\!\cdots\!98\)\( \nu^{5} - \)\(21\!\cdots\!19\)\( \nu^{4} + \)\(28\!\cdots\!07\)\( \nu^{3} - \)\(27\!\cdots\!97\)\( \nu^{2} + \)\(18\!\cdots\!82\)\( \nu - \)\(32\!\cdots\!74\)\(\)\()/ \)\(11\!\cdots\!37\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(21\!\cdots\!01\)\( \nu^{17} - \)\(12\!\cdots\!35\)\( \nu^{16} + \)\(31\!\cdots\!33\)\( \nu^{15} - \)\(49\!\cdots\!32\)\( \nu^{14} + \)\(15\!\cdots\!74\)\( \nu^{13} - \)\(17\!\cdots\!83\)\( \nu^{12} + \)\(92\!\cdots\!80\)\( \nu^{11} - \)\(56\!\cdots\!90\)\( \nu^{10} + \)\(38\!\cdots\!87\)\( \nu^{9} - \)\(21\!\cdots\!69\)\( \nu^{8} + \)\(15\!\cdots\!26\)\( \nu^{7} - \)\(21\!\cdots\!14\)\( \nu^{6} + \)\(53\!\cdots\!27\)\( \nu^{5} - \)\(92\!\cdots\!22\)\( \nu^{4} + \)\(11\!\cdots\!51\)\( \nu^{3} - \)\(12\!\cdots\!58\)\( \nu^{2} + \)\(72\!\cdots\!89\)\( \nu - \)\(90\!\cdots\!23\)\(\)\()/ \)\(35\!\cdots\!11\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(10\!\cdots\!83\)\( \nu^{17} - \)\(64\!\cdots\!06\)\( \nu^{16} + \)\(15\!\cdots\!35\)\( \nu^{15} - \)\(23\!\cdots\!09\)\( \nu^{14} + \)\(73\!\cdots\!57\)\( \nu^{13} - \)\(91\!\cdots\!32\)\( \nu^{12} + \)\(44\!\cdots\!72\)\( \nu^{11} - \)\(31\!\cdots\!41\)\( \nu^{10} + \)\(17\!\cdots\!64\)\( \nu^{9} - \)\(13\!\cdots\!84\)\( \nu^{8} + \)\(71\!\cdots\!25\)\( \nu^{7} - \)\(11\!\cdots\!09\)\( \nu^{6} + \)\(24\!\cdots\!45\)\( \nu^{5} - \)\(45\!\cdots\!98\)\( \nu^{4} + \)\(57\!\cdots\!43\)\( \nu^{3} - \)\(54\!\cdots\!68\)\( \nu^{2} + \)\(33\!\cdots\!92\)\( \nu - \)\(28\!\cdots\!02\)\(\)\()/ \)\(16\!\cdots\!91\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{17} + \beta_{16} - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - 5 \beta_{10} + 2 \beta_{9} - \beta_{8} + 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} - 4 \beta_{8} + \beta_{7} + \beta_{5} + 5 \beta_{3} + \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(8 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} - 12 \beta_{14} + 6 \beta_{11} - 29 \beta_{10} + 45 \beta_{9} - 29 \beta_{8} + 8 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 4 \beta_{3} + \beta_{2} + 2 \beta_{1} + 12\)
\(\nu^{5}\)\(=\)\(32 \beta_{17} - 17 \beta_{15} - 55 \beta_{14} + 23 \beta_{13} + 20 \beta_{12} + 17 \beta_{11} - 55 \beta_{10} + 98 \beta_{9} - 98 \beta_{8} + 20 \beta_{7} + 21 \beta_{6} + 41 \beta_{5} - 6 \beta_{4} + 21 \beta_{3} + 6 \beta_{1}\)
\(\nu^{6}\)\(=\)\(162 \beta_{17} - 78 \beta_{16} - 35 \beta_{15} - 242 \beta_{14} + 180 \beta_{13} + 162 \beta_{12} + 242 \beta_{9} - 380 \beta_{8} + 78 \beta_{7} + 101 \beta_{6} + 101 \beta_{5} - 63 \beta_{4} + 63 \beta_{3} - 20 \beta_{2} + 20 \beta_{1} - 180\)
\(\nu^{7}\)\(=\)\(581 \beta_{17} - 581 \beta_{16} - 695 \beta_{14} + 867 \beta_{13} + 738 \beta_{12} - 260 \beta_{11} + 867 \beta_{10} - 695 \beta_{8} + 260 \beta_{7} + 482 \beta_{6} + 338 \beta_{5} - 338 \beta_{4} + 131 \beta_{3} - 131 \beta_{2} - 1248\)
\(\nu^{8}\)\(=\)\(1788 \beta_{17} - 2877 \beta_{16} + 548 \beta_{15} - 1672 \beta_{14} + 3498 \beta_{13} + 2877 \beta_{12} - 1788 \beta_{11} + 6220 \beta_{10} - 3498 \beta_{9} + 548 \beta_{7} + 1302 \beta_{6} + 710 \beta_{5} - 1302 \beta_{4} - 710 \beta_{2} - 320 \beta_{1} - 6220\)
\(\nu^{9}\)\(=\)\(3907 \beta_{17} - 11007 \beta_{16} + 3907 \beta_{15} + 10554 \beta_{13} + 8778 \beta_{12} - 8778 \beta_{11} + 29522 \beta_{10} - 23691 \beta_{9} + 10554 \beta_{8} + 2950 \beta_{6} - 4471 \beta_{4} - 2181 \beta_{3} - 2950 \beta_{2} - 2181 \beta_{1} - 23691\)
\(\nu^{10}\)\(=\)\(-34258 \beta_{16} + 18780 \beta_{15} + 23343 \beta_{14} + 23343 \beta_{13} + 18780 \beta_{12} - 34258 \beta_{11} + 115569 \beta_{10} - 115569 \beta_{9} + 74768 \beta_{8} - 8317 \beta_{7} - 10908 \beta_{5} - 8876 \beta_{4} - 15775 \beta_{3} - 8876 \beta_{2} - 10908 \beta_{1} - 74768\)
\(\nu^{11}\)\(=\)\(-58350 \beta_{17} - 72822 \beta_{16} + 72822 \beta_{15} + 157800 \beta_{14} - 105493 \beta_{11} + 354367 \beta_{10} - 441790 \beta_{9} + 354367 \beta_{8} - 58350 \beta_{7} - 43108 \beta_{6} - 76748 \beta_{5} - 76748 \beta_{3} - 20041 \beta_{2} - 43108 \beta_{1} - 157800\)
\(\nu^{12}\)\(=\)\(-405403 \beta_{17} + 226992 \beta_{15} + 761719 \beta_{14} - 340865 \beta_{13} - 280742 \beta_{12} - 226992 \beta_{11} + 761719 \beta_{10} - 1370373 \beta_{9} + 1370373 \beta_{8} - 280742 \beta_{7} - 296017 \beta_{6} - 368968 \beta_{5} + 132121 \beta_{4} - 296017 \beta_{3} - 132121 \beta_{1}\)
\(\nu^{13}\)\(=\)\(-1953735 \beta_{17} + 1084493 \beta_{16} + 483774 \beta_{15} + 2929197 \beta_{14} - 2350098 \beta_{13} - 1953735 \beta_{12} - 2929197 \beta_{9} + 4232543 \beta_{8} - 1084493 \beta_{7} - 1430496 \beta_{6} - 1430496 \beta_{5} + 924173 \beta_{4} - 924173 \beta_{3} + 287115 \beta_{2} - 287115 \beta_{1} + 2350098\)
\(\nu^{14}\)\(=\)\(-7543291 \beta_{17} + 7543291 \beta_{16} + 9062260 \beta_{14} - 11313804 \beta_{13} - 9402630 \beta_{12} + 3361682 \beta_{11} - 11313804 \beta_{10} + 9062260 \beta_{8} - 3361682 \beta_{7} - 5498148 \beta_{6} - 4410053 \beta_{5} + 4410053 \beta_{4} - 1964630 \beta_{3} + 1964630 \beta_{2} + 16356758\)
\(\nu^{15}\)\(=\)\(-23320604 \beta_{17} + 36256752 \beta_{16} - 7185651 \beta_{15} + 19399238 \beta_{14} - 43593021 \beta_{13} - 36256752 \beta_{12} + 23320604 \beta_{11} - 78558012 \beta_{10} + 43593021 \beta_{9} - 7185651 \beta_{7} - 17014382 \beta_{6} - 9454465 \beta_{5} + 17014382 \beta_{4} + 9454465 \beta_{2} + 4223027 \beta_{1} + 78558012\)
\(\nu^{16}\)\(=\)\(-49934244 \beta_{17} + 139836396 \beta_{16} - 49934244 \beta_{15} - 134834431 \beta_{13} - 112154680 \beta_{12} + 112154680 \beta_{11} - 377923739 \beta_{10} + 303034222 \beta_{9} - 134834431 \beta_{8} - 36407370 \beta_{6} + 52607803 \beta_{4} + 29214494 \beta_{3} + 36407370 \beta_{2} + 29214494 \beta_{1} + 303034222\)
\(\nu^{17}\)\(=\)\(432441179 \beta_{16} - 239996980 \beta_{15} - 288598279 \beta_{14} - 288598279 \beta_{13} - 239996980 \beta_{12} + 432441179 \beta_{11} - 1457176683 \beta_{10} + 1457176683 \beta_{9} - 937092022 \beta_{8} + 106830454 \beta_{7} + 140518075 \beta_{5} + 112581903 \beta_{4} + 203121400 \beta_{3} + 112581903 \beta_{2} + 140518075 \beta_{1} + 937092022\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−1.30651 + 1.63831i
0.101116 0.126795i
1.35938 1.70461i
−0.392347 1.71899i
0.281648 + 1.23398i
0.432251 + 1.89382i
−1.88755 0.908997i
0.937338 + 0.451398i
3.47467 + 1.67331i
−1.88755 + 0.908997i
0.937338 0.451398i
3.47467 1.67331i
−0.392347 + 1.71899i
0.281648 1.23398i
0.432251 1.89382i
−1.30651 1.63831i
0.101116 + 0.126795i
1.35938 + 1.70461i
−0.222521 + 0.974928i −0.683017 + 0.856476i −0.900969 0.433884i −1.25326 + 1.57153i −0.683017 0.856476i −0.413525 + 2.61324i 0.623490 0.781831i 0.400524 + 1.75481i −1.25326 1.57153i
15.2 −0.222521 + 0.974928i 0.724605 0.908626i −0.900969 0.433884i 2.67921 3.35962i 0.724605 + 0.908626i −2.50613 + 0.848133i 0.623490 0.781831i 0.367014 + 1.60799i 2.67921 + 3.35962i
15.3 −0.222521 + 0.974928i 1.98287 2.48644i −0.900969 0.433884i −1.57994 + 1.98119i 1.98287 + 2.48644i 2.54314 0.729681i 0.623490 0.781831i −1.58305 6.93579i −1.57994 1.98119i
29.1 −0.900969 0.433884i −0.614868 2.69391i 0.623490 + 0.781831i 0.0678680 + 0.297349i −0.614868 + 2.69391i −1.53790 2.15287i −0.222521 0.974928i −4.17621 + 2.01115i 0.0678680 0.297349i
29.2 −0.900969 0.433884i 0.0591274 + 0.259054i 0.623490 + 0.781831i −0.871615 3.81880i 0.0591274 0.259054i 2.52187 + 0.800098i −0.222521 0.974928i 2.63929 1.27102i −0.871615 + 3.81880i
29.3 −0.900969 0.433884i 0.209730 + 0.918888i 0.623490 + 0.781831i 0.482195 + 2.11264i 0.209730 0.918888i −2.20649 + 1.45993i −0.222521 0.974928i 1.90254 0.916214i 0.482195 2.11264i
43.1 0.623490 0.781831i −2.78852 1.34288i −0.222521 0.974928i −0.309110 0.148859i −2.78852 + 1.34288i −2.03497 1.69083i −0.900969 0.433884i 4.10205 + 5.14381i −0.309110 + 0.148859i
43.2 0.623490 0.781831i 0.0363696 + 0.0175147i −0.222521 0.974928i 0.426714 + 0.205495i 0.0363696 0.0175147i 2.62504 0.330433i −0.900969 0.433884i −1.86945 2.34422i 0.426714 0.205495i
43.3 0.623490 0.781831i 2.57370 + 1.23943i −0.222521 0.974928i −2.64206 1.27235i 2.57370 1.23943i −2.49104 + 0.891482i −0.900969 0.433884i 3.21729 + 4.03436i −2.64206 + 1.27235i
57.1 0.623490 + 0.781831i −2.78852 + 1.34288i −0.222521 + 0.974928i −0.309110 + 0.148859i −2.78852 1.34288i −2.03497 + 1.69083i −0.900969 + 0.433884i 4.10205 5.14381i −0.309110 0.148859i
57.2 0.623490 + 0.781831i 0.0363696 0.0175147i −0.222521 + 0.974928i 0.426714 0.205495i 0.0363696 + 0.0175147i 2.62504 + 0.330433i −0.900969 + 0.433884i −1.86945 + 2.34422i 0.426714 + 0.205495i
57.3 0.623490 + 0.781831i 2.57370 1.23943i −0.222521 + 0.974928i −2.64206 + 1.27235i 2.57370 + 1.23943i −2.49104 0.891482i −0.900969 + 0.433884i 3.21729 4.03436i −2.64206 1.27235i
71.1 −0.900969 + 0.433884i −0.614868 + 2.69391i 0.623490 0.781831i 0.0678680 0.297349i −0.614868 2.69391i −1.53790 + 2.15287i −0.222521 + 0.974928i −4.17621 2.01115i 0.0678680 + 0.297349i
71.2 −0.900969 + 0.433884i 0.0591274 0.259054i 0.623490 0.781831i −0.871615 + 3.81880i 0.0591274 + 0.259054i 2.52187 0.800098i −0.222521 + 0.974928i 2.63929 + 1.27102i −0.871615 3.81880i
71.3 −0.900969 + 0.433884i 0.209730 0.918888i 0.623490 0.781831i 0.482195 2.11264i 0.209730 + 0.918888i −2.20649 1.45993i −0.222521 + 0.974928i 1.90254 + 0.916214i 0.482195 + 2.11264i
85.1 −0.222521 0.974928i −0.683017 0.856476i −0.900969 + 0.433884i −1.25326 1.57153i −0.683017 + 0.856476i −0.413525 2.61324i 0.623490 + 0.781831i 0.400524 1.75481i −1.25326 + 1.57153i
85.2 −0.222521 0.974928i 0.724605 + 0.908626i −0.900969 + 0.433884i 2.67921 + 3.35962i 0.724605 0.908626i −2.50613 0.848133i 0.623490 + 0.781831i 0.367014 1.60799i 2.67921 3.35962i
85.3 −0.222521 0.974928i 1.98287 + 2.48644i −0.900969 + 0.433884i −1.57994 1.98119i 1.98287 2.48644i 2.54314 + 0.729681i 0.623490 + 0.781831i −1.58305 + 6.93579i −1.57994 + 1.98119i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.2.e.a 18
3.b odd 2 1 882.2.u.j 18
4.b odd 2 1 784.2.u.c 18
7.b odd 2 1 686.2.e.a 18
7.c even 3 2 686.2.g.i 36
7.d odd 6 2 686.2.g.j 36
49.e even 7 1 inner 98.2.e.a 18
49.e even 7 1 4802.2.a.h 9
49.f odd 14 1 686.2.e.a 18
49.f odd 14 1 4802.2.a.e 9
49.g even 21 2 686.2.g.i 36
49.h odd 42 2 686.2.g.j 36
147.l odd 14 1 882.2.u.j 18
196.k odd 14 1 784.2.u.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.e.a 18 1.a even 1 1 trivial
98.2.e.a 18 49.e even 7 1 inner
686.2.e.a 18 7.b odd 2 1
686.2.e.a 18 49.f odd 14 1
686.2.g.i 36 7.c even 3 2
686.2.g.i 36 49.g even 21 2
686.2.g.j 36 7.d odd 6 2
686.2.g.j 36 49.h odd 42 2
784.2.u.c 18 4.b odd 2 1
784.2.u.c 18 196.k odd 14 1
882.2.u.j 18 3.b odd 2 1
882.2.u.j 18 147.l odd 14 1
4802.2.a.e 9 49.f odd 14 1
4802.2.a.h 9 49.e even 7 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{18} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{3} \)
$3$ \( 1 - 47 T + 736 T^{2} - 2217 T^{3} + 10982 T^{4} - 9074 T^{5} + 15670 T^{6} - 7337 T^{7} + 11566 T^{8} - 7200 T^{9} + 7631 T^{10} - 1885 T^{11} + 277 T^{12} - 29 T^{13} + 12 T^{14} + 9 T^{15} + 4 T^{16} - 3 T^{17} + T^{18} \)
$5$ \( 729 + 972 T + 2025 T^{2} + 14418 T^{3} + 234 T^{4} - 118758 T^{5} + 136735 T^{6} + 260614 T^{7} + 267705 T^{8} + 170517 T^{9} + 92243 T^{10} + 39036 T^{11} + 13353 T^{12} + 3448 T^{13} + 780 T^{14} + 166 T^{15} + 37 T^{16} + 6 T^{17} + T^{18} \)
$7$ \( 40353607 + 40353607 T - 18117946 T^{3} - 8235430 T^{4} + 1512630 T^{5} + 2489837 T^{6} + 638666 T^{7} - 203056 T^{8} - 172186 T^{9} - 29008 T^{10} + 13034 T^{11} + 7259 T^{12} + 630 T^{13} - 490 T^{14} - 154 T^{15} + 7 T^{17} + T^{18} \)
$11$ \( 6126662529 + 14117318280 T + 16194243168 T^{2} + 12046757454 T^{3} + 6448182129 T^{4} + 2436327558 T^{5} + 680531887 T^{6} + 144474632 T^{7} + 30425033 T^{8} + 6555946 T^{9} + 1307443 T^{10} + 182543 T^{11} + 24514 T^{12} + 364 T^{13} + 730 T^{14} + 46 T^{15} + 38 T^{16} + T^{17} + T^{18} \)
$13$ \( 19882681 + 71883539 T + 75934026 T^{2} - 2156098 T^{3} + 53604726 T^{4} - 11249028 T^{5} + 13551636 T^{6} - 2067065 T^{7} + 1483867 T^{8} + 41062 T^{9} + 34447 T^{10} - 9702 T^{11} + 3101 T^{12} + 224 T^{13} + 721 T^{14} + 14 T^{16} + T^{18} \)
$17$ \( 438986304 + 349982208 T + 3259471104 T^{2} - 1421224488 T^{3} + 663591312 T^{4} - 119862402 T^{5} + 3342763 T^{6} + 907451 T^{7} + 12656674 T^{8} + 2547177 T^{9} + 469218 T^{10} + 167585 T^{11} + 59742 T^{12} + 9834 T^{13} + 2175 T^{14} + 261 T^{15} + 53 T^{16} + 11 T^{17} + T^{18} \)
$19$ \( ( 500387 - 323830 T - 60732 T^{2} + 70778 T^{3} - 4789 T^{4} - 4437 T^{5} + 762 T^{6} + 49 T^{7} - 18 T^{8} + T^{9} )^{2} \)
$23$ \( 1225449 - 7422435 T + 20043936 T^{2} - 28602234 T^{3} + 19414530 T^{4} + 656250 T^{5} - 7271873 T^{6} + 838324 T^{7} + 3694646 T^{8} - 3627899 T^{9} + 2514647 T^{10} - 186577 T^{11} + 229502 T^{12} - 7259 T^{13} + 6353 T^{14} - 101 T^{15} + 89 T^{16} - 5 T^{17} + T^{18} \)
$29$ \( 1347921 + 23196780 T + 254290671 T^{2} + 283596498 T^{3} + 727402176 T^{4} + 148049349 T^{5} + 89139589 T^{6} - 32428621 T^{7} - 10910296 T^{8} + 1545147 T^{9} + 2177102 T^{10} + 498172 T^{11} + 40663 T^{12} - 9856 T^{13} - 1720 T^{14} - 18 T^{15} + 101 T^{16} + 13 T^{17} + T^{18} \)
$31$ \( ( -13 - 55 T + 973 T^{2} - 2134 T^{3} + 282 T^{4} + 515 T^{5} - 62 T^{6} - 42 T^{7} + 2 T^{8} + T^{9} )^{2} \)
$37$ \( 1703523546481 - 3913889303610 T + 4005241149640 T^{2} - 2414654901424 T^{3} + 1032144676369 T^{4} - 353102661625 T^{5} + 104555523254 T^{6} - 27768570742 T^{7} + 6714838141 T^{8} - 1475117058 T^{9} + 292835902 T^{10} - 51855229 T^{11} + 8065218 T^{12} - 1074633 T^{13} + 119060 T^{14} - 10496 T^{15} + 705 T^{16} - 33 T^{17} + T^{18} \)
$41$ \( 661949588881161 + 573823529046054 T + 522541700179947 T^{2} + 224691590458629 T^{3} + 60848696518911 T^{4} + 11313620160591 T^{5} + 1673024849230 T^{6} + 230242501293 T^{7} + 32867102555 T^{8} + 4290001989 T^{9} + 430054625 T^{10} + 25932319 T^{11} + 496237 T^{12} + 17969 T^{13} + 23891 T^{14} + 4585 T^{15} + 462 T^{16} + 28 T^{17} + T^{18} \)
$43$ \( 63645702961 + 36996504088 T + 91620147169 T^{2} + 8246715781 T^{3} + 6914508668 T^{4} - 393598233 T^{5} + 162855672 T^{6} - 131717720 T^{7} + 13573814 T^{8} + 8321924 T^{9} + 3051654 T^{10} + 218200 T^{11} - 56252 T^{12} - 17913 T^{13} + 800 T^{14} + 969 T^{15} + 206 T^{16} + 20 T^{17} + T^{18} \)
$47$ \( 9308601 - 49865544 T + 137054997 T^{2} - 244880631 T^{3} + 316017243 T^{4} - 312502578 T^{5} + 248123026 T^{6} - 163288450 T^{7} + 90813342 T^{8} - 42499887 T^{9} + 16799087 T^{10} - 5505573 T^{11} + 1475562 T^{12} - 318202 T^{13} + 53868 T^{14} - 6883 T^{15} + 625 T^{16} - 36 T^{17} + T^{18} \)
$53$ \( 97031627001 - 250155501930 T + 363412003302 T^{2} - 318061519128 T^{3} + 209866774842 T^{4} - 114516133515 T^{5} + 52573806922 T^{6} - 20432519512 T^{7} + 6815152143 T^{8} - 1957291065 T^{9} + 478409327 T^{10} - 97612140 T^{11} + 16323216 T^{12} - 2202277 T^{13} + 235776 T^{14} - 19498 T^{15} + 1186 T^{16} - 48 T^{17} + T^{18} \)
$59$ \( 83822409441 + 60194021589 T + 13727918394 T^{2} + 8766030870 T^{3} + 11144614347 T^{4} + 5161602423 T^{5} + 1039563109 T^{6} - 55625270 T^{7} - 39885921 T^{8} + 18212764 T^{9} + 17997078 T^{10} + 7087407 T^{11} + 1848536 T^{12} + 351252 T^{13} + 50902 T^{14} + 5553 T^{15} + 459 T^{16} + 25 T^{17} + T^{18} \)
$61$ \( 4441105256449 + 4894835398921 T + 568918787579 T^{2} - 871786428368 T^{3} + 377509347491 T^{4} - 132250224046 T^{5} + 46190613326 T^{6} - 6773297730 T^{7} + 1598762239 T^{8} - 52127225 T^{9} + 22211774 T^{10} + 801280 T^{11} + 516555 T^{12} + 11512 T^{13} + 6723 T^{14} + 425 T^{15} + 79 T^{16} - T^{17} + T^{18} \)
$67$ \( ( -149884139 - 5099891 T + 19662029 T^{2} - 574637 T^{3} - 557340 T^{4} + 26355 T^{5} + 5425 T^{6} - 297 T^{7} - 17 T^{8} + T^{9} )^{2} \)
$71$ \( 14085009 - 46848699 T + 1173382929 T^{2} + 221862861 T^{3} + 8210233665 T^{4} - 5180474523 T^{5} + 1589624470 T^{6} - 315771751 T^{7} + 48132912 T^{8} - 5942270 T^{9} + 3822025 T^{10} - 617747 T^{11} - 40502 T^{12} + 12845 T^{13} + 2312 T^{14} - 242 T^{15} + 3 T^{16} + 6 T^{17} + T^{18} \)
$73$ \( 351446147128609 + 802493508191986 T + 1115835500278486 T^{2} + 596512099485276 T^{3} + 266975232821443 T^{4} + 72777579463937 T^{5} + 14410271880603 T^{6} + 2297916849822 T^{7} + 316126001746 T^{8} + 41554709115 T^{9} + 5385235815 T^{10} + 590504220 T^{11} + 49881395 T^{12} + 3432274 T^{13} + 220281 T^{14} + 13673 T^{15} + 816 T^{16} + 39 T^{17} + T^{18} \)
$79$ \( ( 5529329 - 1532338 T - 2431936 T^{2} - 354690 T^{3} + 88277 T^{4} + 18382 T^{5} - 742 T^{6} - 243 T^{7} + T^{8} + T^{9} )^{2} \)
$83$ \( 2589575516688384 - 3482210671220736 T + 2515561263290880 T^{2} - 1010912235036480 T^{3} + 272171250033984 T^{4} - 58616229537012 T^{5} + 10902799991377 T^{6} - 1520431490312 T^{7} + 172583943161 T^{8} - 15203882043 T^{9} + 1037478127 T^{10} - 45652369 T^{11} + 2036601 T^{12} - 320691 T^{13} + 70420 T^{14} - 8638 T^{15} + 728 T^{16} - 35 T^{17} + T^{18} \)
$89$ \( 10124977628529 - 53963567963964 T + 176617362555720 T^{2} - 103730375826639 T^{3} + 42420666698406 T^{4} - 10970387846526 T^{5} + 1844764504678 T^{6} - 177534013002 T^{7} + 16977804372 T^{8} - 1657855643 T^{9} + 265358356 T^{10} + 26779092 T^{11} + 4649684 T^{12} + 97822 T^{13} - 1712 T^{14} - 1794 T^{15} + 30 T^{16} + 6 T^{17} + T^{18} \)
$97$ \( ( 8869 + 4263 T - 36603 T^{2} + 2310 T^{3} + 16170 T^{4} - 5159 T^{5} - 224 T^{6} + 238 T^{7} - 28 T^{8} + T^{9} )^{2} \)
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