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

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

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
338.2.e.a \(4\) \(2.699\) \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(-1+\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
338.2.e.b \(4\) \(2.699\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+\zeta_{12}^{3}q^{5}+(4\zeta_{12}+\cdots)q^{7}+\cdots\)
338.2.e.c \(4\) \(2.699\) \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
338.2.e.d \(4\) \(2.699\) \(\Q(\zeta_{12})\) None \(0\) \(6\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(3-3\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
338.2.e.e \(12\) \(2.699\) 12.0.\(\cdots\).1 None \(0\) \(-6\) \(0\) \(0\) \(q+\beta _{6}q^{2}+(\beta _{4}+2\beta _{9})q^{3}+\beta _{7}q^{4}+(2\beta _{2}+\cdots)q^{5}+\cdots\)

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

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