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Properties

Label 138.2.e
Level $138$
Weight $2$
Character orbit 138.e
Rep. character $\chi_{138}(13,\cdot)$
Character field $\Q(\zeta_{11})$
Dimension $40$
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.e (of order \(11\) and degree \(10\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 23 \)
Character field: \(\Q(\zeta_{11})\)
Newform subspaces: \( 4 \)
Sturm bound: \(48\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 280 40 240
Cusp forms 200 40 160
Eisenstein series 80 0 80

Trace form

\( 40 q - 4 q^{4} + 8 q^{5} + 8 q^{7} - 4 q^{9} + O(q^{10}) \) \( 40 q - 4 q^{4} + 8 q^{5} + 8 q^{7} - 4 q^{9} + 4 q^{10} + 24 q^{11} + 16 q^{13} + 8 q^{14} + 8 q^{15} - 4 q^{16} - 20 q^{17} - 16 q^{19} - 36 q^{20} + 4 q^{21} - 32 q^{22} - 64 q^{23} - 52 q^{25} - 28 q^{26} - 36 q^{28} - 12 q^{29} + 12 q^{30} - 12 q^{31} + 12 q^{33} + 24 q^{34} + 20 q^{35} - 4 q^{36} - 16 q^{37} + 32 q^{38} - 36 q^{39} + 4 q^{40} - 12 q^{41} + 12 q^{42} + 16 q^{43} + 24 q^{44} + 8 q^{45} + 20 q^{46} - 48 q^{47} + 44 q^{49} + 24 q^{50} - 16 q^{51} + 16 q^{52} - 12 q^{53} - 24 q^{55} + 8 q^{56} - 32 q^{57} - 12 q^{58} + 20 q^{59} + 8 q^{60} - 28 q^{61} + 40 q^{62} + 8 q^{63} - 4 q^{64} + 64 q^{65} + 24 q^{66} + 8 q^{67} + 24 q^{68} + 20 q^{69} - 16 q^{70} + 72 q^{71} + 44 q^{73} + 8 q^{74} + 32 q^{75} + 28 q^{76} + 64 q^{77} + 16 q^{78} - 44 q^{79} + 8 q^{80} - 4 q^{81} - 4 q^{82} + 20 q^{83} + 4 q^{84} + 8 q^{85} + 40 q^{86} + 40 q^{87} + 12 q^{88} + 28 q^{89} + 4 q^{90} + 56 q^{91} + 24 q^{92} + 32 q^{93} + 24 q^{94} - 28 q^{95} + 68 q^{97} - 56 q^{98} - 20 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.e.a 138.e 23.c $10$ $1.102$ \(\Q(\zeta_{22})\) None \(-1\) \(-1\) \(8\) \(8\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}q^{2}+\zeta_{22}^{8}q^{3}+\zeta_{22}^{2}q^{4}+(1+\cdots)q^{5}+\cdots\)
138.2.e.b 138.e 23.c $10$ $1.102$ \(\Q(\zeta_{22})\) None \(-1\) \(1\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}q^{2}-\zeta_{22}^{8}q^{3}+\zeta_{22}^{2}q^{4}+(-1+\cdots)q^{5}+\cdots\)
138.2.e.c 138.e 23.c $10$ $1.102$ \(\Q(\zeta_{22})\) None \(1\) \(-1\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}q^{2}+\zeta_{22}^{8}q^{3}+\zeta_{22}^{2}q^{4}+(1+\cdots)q^{5}+\cdots\)
138.2.e.d 138.e 23.c $10$ $1.102$ \(\Q(\zeta_{22})\) None \(1\) \(1\) \(2\) \(2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}q^{2}-\zeta_{22}^{8}q^{3}+\zeta_{22}^{2}q^{4}+(-1+\cdots)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(138, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(23, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(46, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(69, [\chi])\)\(^{\oplus 2}\)