Properties

Label 222.2.e
Level $222$
Weight $2$
Character orbit 222.e
Rep. character $\chi_{222}(121,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $3$
Sturm bound $76$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 222.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 37 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(76\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(222, [\chi])\).

Total New Old
Modular forms 84 8 76
Cusp forms 68 8 60
Eisenstein series 16 0 16

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(222, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
222.2.e.a 222.e 37.c $2$ $1.773$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{3}-\zeta_{6}q^{4}-q^{6}+\cdots\)
222.2.e.b 222.e 37.c $2$ $1.773$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(-1\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{3}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots\)
222.2.e.c 222.e 37.c $4$ $1.773$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(2\) \(-1\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{1})q^{2}+\beta _{1}q^{3}-\beta _{1}q^{4}-\beta _{3}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(222, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(222, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(37, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(74, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(111, [\chi])\)\(^{\oplus 2}\)