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Properties

Label 231.2.a
Level $231$
Weight $2$
Character orbit 231.a
Rep. character $\chi_{231}(1,\cdot)$
Character field $\Q$
Dimension $11$
Newform subspaces $5$
Sturm bound $64$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(231))\).

Total New Old
Modular forms 36 11 25
Cusp forms 29 11 18
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.

\(3\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(3\)
\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(11\)

Trace form

\( 11 q + q^{2} - q^{3} + 17 q^{4} + 10 q^{5} + 5 q^{6} - q^{7} - 3 q^{8} + 11 q^{9} + O(q^{10}) \) \( 11 q + q^{2} - q^{3} + 17 q^{4} + 10 q^{5} + 5 q^{6} - q^{7} - 3 q^{8} + 11 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + 2 q^{13} - 3 q^{14} + 2 q^{15} + 17 q^{16} + 22 q^{17} + q^{18} + 4 q^{19} - 2 q^{20} - q^{21} + q^{22} + 9 q^{24} + 29 q^{25} - 10 q^{26} - q^{27} - 7 q^{28} + 18 q^{29} - 18 q^{30} - 8 q^{31} - 35 q^{32} - q^{33} - 14 q^{34} + 2 q^{35} + 17 q^{36} - 6 q^{37} - 36 q^{38} - 14 q^{39} - 42 q^{40} + 30 q^{41} + q^{42} - 20 q^{43} - 7 q^{44} + 10 q^{45} - 32 q^{46} - 24 q^{47} - 31 q^{48} + 11 q^{49} - 65 q^{50} + 6 q^{51} - 58 q^{52} - 6 q^{53} + 5 q^{54} - 6 q^{55} - 15 q^{56} - 20 q^{57} - 46 q^{58} - 20 q^{59} - 6 q^{60} + 26 q^{61} - 16 q^{62} - q^{63} + 29 q^{64} + 36 q^{65} - 3 q^{66} - 20 q^{67} + 26 q^{68} + 16 q^{69} + 2 q^{70} + 8 q^{71} - 3 q^{72} + 38 q^{73} + 54 q^{74} - 15 q^{75} + 4 q^{76} - q^{77} - 14 q^{78} + 16 q^{79} + 6 q^{80} + 11 q^{81} - 14 q^{82} - 4 q^{83} - 7 q^{84} - 4 q^{85} + 12 q^{86} - 6 q^{87} + 9 q^{88} + 62 q^{89} - 2 q^{90} + 10 q^{91} + 8 q^{92} - 8 q^{93} + 24 q^{94} - 16 q^{95} + 33 q^{96} + 14 q^{97} + q^{98} - q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7 11
231.2.a.a 231.a 1.a $1$ $1.845$ \(\Q\) None \(-1\) \(-1\) \(-2\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+q^{7}+\cdots\)
231.2.a.b 231.a 1.a $2$ $1.845$ \(\Q(\sqrt{21}) \) None \(-1\) \(-2\) \(6\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(3+\beta )q^{4}+3q^{5}+\beta q^{6}+\cdots\)
231.2.a.c 231.a 1.a $2$ $1.845$ \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
231.2.a.d 231.a 1.a $3$ $1.845$ 3.3.837.1 None \(0\) \(-3\) \(0\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
231.2.a.e 231.a 1.a $3$ $1.845$ 3.3.229.1 None \(2\) \(3\) \(4\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}+q^{3}+(2+\beta _{1})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(231))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(231)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)