Properties

Label 138.2.a
Level $138$
Weight $2$
Character orbit 138.a
Rep. character $\chi_{138}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $4$
Sturm bound $48$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(48\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(138))\).

Total New Old
Modular forms 28 5 23
Cusp forms 21 5 16
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.

\(2\)\(3\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

\( 5q + q^{2} + q^{3} + 5q^{4} - 2q^{5} + q^{6} + q^{8} + 5q^{9} + O(q^{10}) \) \( 5q + q^{2} + q^{3} + 5q^{4} - 2q^{5} + q^{6} + q^{8} + 5q^{9} + 2q^{10} - 12q^{11} + q^{12} - 2q^{13} - 2q^{15} + 5q^{16} - 6q^{17} + q^{18} - 8q^{19} - 2q^{20} + 4q^{21} - q^{23} + q^{24} - 5q^{25} - 2q^{26} + q^{27} - 2q^{29} - 6q^{30} + q^{32} - 6q^{34} - 16q^{35} + 5q^{36} + 10q^{37} - 12q^{38} + 6q^{39} + 2q^{40} + 10q^{41} - 4q^{42} - 16q^{43} - 12q^{44} - 2q^{45} + 3q^{46} + 8q^{47} + q^{48} + 13q^{49} + 7q^{50} - 10q^{51} - 2q^{52} + 22q^{53} + q^{54} + 8q^{55} + 8q^{57} - 2q^{58} - 4q^{59} - 2q^{60} + 2q^{61} - 8q^{62} + 5q^{64} + 20q^{65} - 12q^{66} + 16q^{67} - 6q^{68} + 3q^{69} - 24q^{70} + q^{72} - 14q^{73} + 30q^{74} - q^{75} - 8q^{76} + 32q^{77} - 2q^{78} - 8q^{79} - 2q^{80} + 5q^{81} + 2q^{82} + 20q^{83} + 4q^{84} + 12q^{85} + 4q^{86} - 10q^{87} + 18q^{89} + 2q^{90} - 32q^{91} - q^{92} + 24q^{94} + 16q^{95} + q^{96} - 14q^{97} + 25q^{98} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(138))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 23
138.2.a.a \(1\) \(1.102\) \(\Q\) None \(-1\) \(-1\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}-2q^{7}+\cdots\)
138.2.a.b \(1\) \(1.102\) \(\Q\) None \(-1\) \(1\) \(0\) \(2\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
138.2.a.c \(1\) \(1.102\) \(\Q\) None \(1\) \(-1\) \(2\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{8}+\cdots\)
138.2.a.d \(2\) \(1.102\) \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(-2\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+(-1+\beta )q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(138))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(138)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)