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Properties

Label 238.2.e
Level $238$
Weight $2$
Character orbit 238.e
Rep. character $\chi_{238}(137,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $24$
Newform subspaces $6$
Sturm bound $72$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 238.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(72\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 80 24 56
Cusp forms 64 24 40
Eisenstein series 16 0 16

Trace form

\( 24 q - 12 q^{4} + 4 q^{5} - 4 q^{7} - 20 q^{9} + O(q^{10}) \) \( 24 q - 12 q^{4} + 4 q^{5} - 4 q^{7} - 20 q^{9} - 4 q^{11} + 8 q^{13} - 4 q^{14} - 8 q^{15} - 12 q^{16} + 4 q^{17} + 4 q^{18} - 4 q^{19} - 8 q^{20} + 8 q^{21} + 8 q^{22} + 16 q^{23} - 16 q^{25} - 8 q^{26} + 8 q^{28} - 8 q^{29} + 4 q^{30} + 8 q^{31} - 28 q^{33} - 8 q^{34} + 4 q^{35} + 40 q^{36} - 4 q^{38} - 4 q^{39} + 8 q^{41} + 24 q^{42} - 32 q^{43} - 4 q^{44} + 32 q^{45} - 16 q^{46} + 20 q^{47} + 12 q^{49} + 24 q^{50} - 4 q^{52} - 20 q^{53} + 12 q^{54} + 40 q^{55} - 4 q^{56} + 24 q^{57} - 16 q^{58} - 8 q^{59} + 4 q^{60} - 24 q^{61} - 8 q^{62} + 36 q^{63} + 24 q^{64} - 12 q^{65} - 24 q^{66} - 4 q^{67} + 4 q^{68} - 80 q^{69} - 36 q^{70} + 56 q^{71} + 4 q^{72} + 36 q^{73} - 12 q^{74} - 52 q^{75} + 8 q^{76} + 36 q^{77} + 16 q^{78} - 4 q^{79} + 4 q^{80} - 72 q^{81} - 48 q^{83} - 16 q^{84} - 24 q^{86} + 24 q^{87} - 4 q^{88} - 28 q^{89} - 72 q^{90} - 24 q^{91} - 32 q^{92} + 28 q^{93} + 4 q^{94} + 24 q^{95} + 56 q^{97} + 24 q^{98} + 16 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.e.a 238.e 7.c $2$ $1.900$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
238.2.e.b 238.e 7.c $2$ $1.900$ \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
238.2.e.c 238.e 7.c $2$ $1.900$ \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
238.2.e.d 238.e 7.c $2$ $1.900$ \(\Q(\sqrt{-3}) \) None \(1\) \(2\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
238.2.e.e 238.e 7.c $6$ $1.900$ 6.0.4406832.1 None \(3\) \(0\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{4}q^{2}+(-\beta _{1}-\beta _{3}+\beta _{5})q^{3}+(-1+\cdots)q^{4}+\cdots\)
238.2.e.f 238.e 7.c $10$ $1.900$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(-5\) \(0\) \(1\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{4})q^{2}-\beta _{1}q^{3}+\beta _{4}q^{4}+\beta _{9}q^{5}+\cdots\)

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

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