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Properties

Label 186.2.e
Level $186$
Weight $2$
Character orbit 186.e
Rep. character $\chi_{186}(25,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $4$
Sturm bound $64$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 186 = 2 \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 186.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 31 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(186, [\chi])\).

Total New Old
Modular forms 72 12 60
Cusp forms 56 12 44
Eisenstein series 16 0 16

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(186, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
186.2.e.a 186.e 31.c $2$ $1.485$ \(\Q(\sqrt{-3}) \) None 186.2.e.a \(-2\) \(1\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-q^{2}+(1-\zeta_{6})q^{3}+q^{4}-2\zeta_{6}q^{5}+\cdots\)
186.2.e.b 186.e 31.c $2$ $1.485$ \(\Q(\sqrt{-3}) \) None 186.2.e.b \(2\) \(-1\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+q^{2}+(-1+\zeta_{6})q^{3}+q^{4}+2\zeta_{6}q^{5}+\cdots\)
186.2.e.c 186.e 31.c $4$ $1.485$ \(\Q(\sqrt{-3}, \sqrt{19})\) None 186.2.e.c \(-4\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(1+\beta _{2}+\cdots)q^{6}+\cdots\)
186.2.e.d 186.e 31.c $4$ $1.485$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 186.2.e.d \(4\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+q^{2}-\beta _{2}q^{3}+q^{4}+2\beta _{1}q^{5}-\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(186, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(186, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(31, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(62, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(93, [\chi])\)\(^{\oplus 2}\)