Properties

Label 13.2.e
Level $13$
Weight $2$
Character orbit 13.e
Rep. character $\chi_{13}(4,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $2$
Newform subspaces $1$
Sturm bound $2$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(2\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(13, [\chi])\).

Total New Old
Modular forms 6 6 0
Cusp forms 2 2 0
Eisenstein series 4 4 0

Trace form

\( 2 q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} - q^{9} + O(q^{10}) \) \( 2 q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} - q^{9} - 3 q^{10} - 4 q^{12} - 5 q^{13} + 6 q^{15} + 5 q^{16} + 3 q^{17} - 6 q^{19} + 3 q^{20} + 6 q^{23} - 6 q^{24} + 4 q^{25} + 3 q^{26} - 8 q^{27} - 3 q^{29} - 6 q^{30} - 9 q^{32} + q^{36} + 15 q^{37} + 12 q^{38} + 14 q^{39} + 6 q^{40} - 9 q^{41} - 8 q^{43} - 3 q^{45} - 18 q^{46} + 10 q^{48} - 7 q^{49} - 6 q^{50} - 12 q^{51} + 2 q^{52} - 6 q^{53} + 12 q^{54} + 9 q^{58} + 12 q^{59} - q^{61} + 6 q^{62} - 2 q^{64} - 9 q^{65} + 6 q^{67} - 3 q^{68} + 12 q^{69} + 6 q^{71} + 3 q^{72} - 15 q^{74} - 4 q^{75} - 6 q^{76} - 24 q^{78} + 8 q^{79} - 15 q^{80} + 11 q^{81} + 9 q^{82} + 9 q^{85} - 6 q^{87} - 12 q^{89} + 6 q^{90} + 12 q^{92} - 12 q^{93} - 6 q^{94} + 6 q^{95} + 12 q^{97} + 21 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.2.e.a 13.e 13.e $2$ $0.104$ \(\Q(\sqrt{-3}) \) None 13.2.e.a \(-3\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)