Properties

Label 222.2.a
Level $222$
Weight $2$
Character orbit 222.a
Rep. character $\chi_{222}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $5$
Sturm bound $76$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 222.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(76\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(222))\).

Total New Old
Modular forms 42 5 37
Cusp forms 35 5 30
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.

\(2\)\(3\)\(37\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(5\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(222))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 37
222.2.a.a $1$ $1.773$ \(\Q\) None \(-1\) \(-1\) \(-4\) \(3\) $+$ $+$ $-$ \(q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}+3q^{7}+\cdots\)
222.2.a.b $1$ $1.773$ \(\Q\) None \(-1\) \(-1\) \(2\) \(0\) $+$ $+$ $-$ \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-q^{8}+\cdots\)
222.2.a.c $1$ $1.773$ \(\Q\) None \(-1\) \(1\) \(4\) \(-1\) $+$ $-$ $+$ \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}-q^{7}+\cdots\)
222.2.a.d $1$ $1.773$ \(\Q\) None \(1\) \(-1\) \(0\) \(3\) $-$ $+$ $+$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
222.2.a.e $1$ $1.773$ \(\Q\) None \(1\) \(1\) \(0\) \(-1\) $-$ $-$ $-$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(222))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(222)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 2}\)