Properties

Label 13.3.f
Level $13$
Weight $3$
Character orbit 13.f
Rep. character $\chi_{13}(2,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $4$
Newform subspaces $1$
Sturm bound $3$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 13.f (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 1 \)
Sturm bound: \(3\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(13, [\chi])\).

Total New Old
Modular forms 12 12 0
Cusp forms 4 4 0
Eisenstein series 8 8 0

Trace form

\( 4 q - 2 q^{2} - 2 q^{3} - 6 q^{4} - 14 q^{5} + 10 q^{6} + 16 q^{7} - 6 q^{8} + 10 q^{9} + 24 q^{10} + 4 q^{11} - 26 q^{13} - 40 q^{14} - 14 q^{15} - 2 q^{16} - 12 q^{17} - 2 q^{18} + 10 q^{19} + 26 q^{20}+ \cdots + 208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{3}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.3.f.a 13.f 13.f $4$ $0.354$ \(\Q(\zeta_{12})\) None 13.3.f.a \(-2\) \(-2\) \(-14\) \(16\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)