Properties

Label 338.2.e
Level $338$
Weight $2$
Character orbit 338.e
Rep. character $\chi_{338}(23,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $28$
Newform subspaces $5$
Sturm bound $91$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.e (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(91\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(338, [\chi])\).

Total New Old
Modular forms 120 28 92
Cusp forms 64 28 36
Eisenstein series 56 0 56

Trace form

\( 28 q + 14 q^{4} - 14 q^{9} + O(q^{10}) \) \( 28 q + 14 q^{4} - 14 q^{9} - 4 q^{14} - 14 q^{16} - 2 q^{17} + 2 q^{22} - 4 q^{23} - 20 q^{25} - 12 q^{27} + 6 q^{29} - 6 q^{30} + 10 q^{35} + 14 q^{36} + 4 q^{38} + 6 q^{42} - 4 q^{43} + 4 q^{49} + 28 q^{51} - 12 q^{53} + 8 q^{55} - 2 q^{56} + 10 q^{61} - 8 q^{62} - 28 q^{64} - 16 q^{66} + 2 q^{68} - 16 q^{69} + 8 q^{74} - 22 q^{75} + 40 q^{77} - 32 q^{79} + 10 q^{81} - 4 q^{82} + 20 q^{87} - 2 q^{88} + 28 q^{90} - 8 q^{92} + 6 q^{94} - 24 q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(338, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
338.2.e.a 338.e 13.e $4$ $2.699$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(-1+\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
338.2.e.b 338.e 13.e $4$ $2.699$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+\zeta_{12}^{3}q^{5}+(4\zeta_{12}+\cdots)q^{7}+\cdots\)
338.2.e.c 338.e 13.e $4$ $2.699$ \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
338.2.e.d 338.e 13.e $4$ $2.699$ \(\Q(\zeta_{12})\) None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(3-3\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
338.2.e.e 338.e 13.e $12$ $2.699$ 12.0.\(\cdots\).1 None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{6}q^{2}+(\beta _{4}+2\beta _{9})q^{3}+\beta _{7}q^{4}+(2\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(338, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(338, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)