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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7 11
231.2.a.a \(1\) \(1.845\) \(\Q\) None \(-1\) \(-1\) \(-2\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+q^{7}+\cdots\)
231.2.a.b \(2\) \(1.845\) \(\Q(\sqrt{21}) \) None \(-1\) \(-2\) \(6\) \(2\) \(+\) \(-\) \(+\) \(q-\beta q^{2}-q^{3}+(3+\beta )q^{4}+3q^{5}+\beta q^{6}+\cdots\)
231.2.a.c \(2\) \(1.845\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(2\) \(2\) \(-\) \(-\) \(-\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
231.2.a.d \(3\) \(1.845\) 3.3.837.1 None \(0\) \(-3\) \(0\) \(-3\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
231.2.a.e \(3\) \(1.845\) 3.3.229.1 None \(2\) \(3\) \(4\) \(-3\) \(-\) \(+\) \(+\) \(q+(1+\beta _{2})q^{2}+q^{3}+(2+\beta _{1})q^{4}+(1+\cdots)q^{5}+\cdots\)

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

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