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Properties

Label 198.2.e
Level $198$
Weight $2$
Character orbit 198.e
Rep. character $\chi_{198}(67,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $20$
Newform subspaces $5$
Sturm bound $72$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 198.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(72\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(198, [\chi])\).

Total New Old
Modular forms 80 20 60
Cusp forms 64 20 44
Eisenstein series 16 0 16

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(198, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
198.2.e.a 198.e 9.c $2$ $1.581$ \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
198.2.e.b 198.e 9.c $2$ $1.581$ \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(-3\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
198.2.e.c 198.e 9.c $4$ $1.581$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(-2\) \(2\) \(2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{2})q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+\cdots\)
198.2.e.d 198.e 9.c $6$ $1.581$ 6.0.954288.1 None \(-3\) \(1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{3})q^{2}+\beta _{4}q^{3}+\beta _{3}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
198.2.e.e 198.e 9.c $6$ $1.581$ 6.0.954288.1 None \(3\) \(-1\) \(5\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{3}q^{2}-\beta _{1}q^{3}+(-1-\beta _{3})q^{4}+(2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(198, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(198, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 2}\)