Properties

Label 138.2.d
Level $138$
Weight $2$
Character orbit 138.d
Rep. character $\chi_{138}(137,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $1$
Sturm bound $48$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 69 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(48\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(138, [\chi])\).

Total New Old
Modular forms 28 8 20
Cusp forms 20 8 12
Eisenstein series 8 0 8

Trace form

\( 8 q - 4 q^{3} - 8 q^{4} - 4 q^{6} + O(q^{10}) \) \( 8 q - 4 q^{3} - 8 q^{4} - 4 q^{6} + 4 q^{12} - 8 q^{13} + 8 q^{16} - 16 q^{18} + 4 q^{24} + 48 q^{25} - 28 q^{27} + 16 q^{31} + 24 q^{39} + 16 q^{46} - 4 q^{48} - 56 q^{49} + 8 q^{52} + 28 q^{54} - 64 q^{55} + 8 q^{58} - 8 q^{64} + 28 q^{69} - 24 q^{70} + 16 q^{72} - 8 q^{73} - 4 q^{75} - 16 q^{78} + 8 q^{81} + 24 q^{85} - 56 q^{87} - 8 q^{93} + 56 q^{94} - 4 q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(138, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
138.2.d.a 138.d 69.c $8$ $1.102$ 8.0.\(\cdots\).3 None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{7})q^{3}-q^{4}+\beta _{3}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(138, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(138, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(69, [\chi])\)\(^{\oplus 2}\)