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Properties

Label 186.2.a
Level $186$
Weight $2$
Character orbit 186.a
Rep. character $\chi_{186}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $4$
Sturm bound $64$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 186 = 2 \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 186.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(64\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(186))\).

Total New Old
Modular forms 36 5 31
Cusp forms 29 5 24
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(31\)FrickeDim
\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(-\)\(+\)$-$\(1\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(-\)\(-\)$-$\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(5\)

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(186))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 31
186.2.a.a 186.a 1.a $1$ $1.485$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
186.2.a.b 186.a 1.a $1$ $1.485$ \(\Q\) None \(-1\) \(1\) \(3\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-2q^{7}+\cdots\)
186.2.a.c 186.a 1.a $1$ $1.485$ \(\Q\) None \(1\) \(1\) \(1\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
186.2.a.d 186.a 1.a $2$ $1.485$ \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(3\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(1+\beta )q^{5}-q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(186))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(186)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 2}\)