Properties

Label 238.2.b
Level $238$
Weight $2$
Character orbit 238.b
Rep. character $\chi_{238}(169,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $2$
Sturm bound $72$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 238.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(72\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(238, [\chi])\).

Total New Old
Modular forms 40 8 32
Cusp forms 32 8 24
Eisenstein series 8 0 8

Trace form

\( 8 q + 4 q^{2} + 8 q^{4} + 4 q^{8} + O(q^{10}) \) \( 8 q + 4 q^{2} + 8 q^{4} + 4 q^{8} - 4 q^{13} + 8 q^{16} - 12 q^{18} + 12 q^{19} + 4 q^{21} - 12 q^{26} + 4 q^{32} + 8 q^{33} - 4 q^{34} - 4 q^{35} - 4 q^{38} + 4 q^{42} - 24 q^{43} - 24 q^{47} - 8 q^{49} - 20 q^{50} + 4 q^{51} - 4 q^{52} - 16 q^{53} + 32 q^{55} - 60 q^{59} + 8 q^{64} + 8 q^{66} - 16 q^{67} - 16 q^{69} - 4 q^{70} - 12 q^{72} + 12 q^{76} - 8 q^{77} + 16 q^{81} + 4 q^{83} + 4 q^{84} + 28 q^{85} - 8 q^{86} + 40 q^{87} - 48 q^{89} - 16 q^{93} + 8 q^{94} - 4 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(238, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
238.2.b.a 238.b 17.b $2$ $1.900$ \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{2}+q^{4}+iq^{7}-q^{8}+3q^{9}+2q^{13}+\cdots\)
238.2.b.b 238.b 17.b $6$ $1.900$ 6.0.350464.1 None \(6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{2}+(\beta _{1}+\beta _{4})q^{3}+q^{4}+(\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(238, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(238, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(119, [\chi])\)\(^{\oplus 2}\)