Properties

Label 13.4.c
Level $13$
Weight $4$
Character orbit 13.c
Rep. character $\chi_{13}(3,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $6$
Newform subspaces $2$
Sturm bound $4$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(13, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

Trace form

\( 6 q + q^{2} - 7 q^{3} - 13 q^{4} + 4 q^{5} + 30 q^{6} - 35 q^{7} - 30 q^{8} - 12 q^{9} - 63 q^{10} + 15 q^{11} + 312 q^{12} + 34 q^{13} + 68 q^{14} - 124 q^{15} + 7 q^{16} - 57 q^{17} - 614 q^{18} + 111 q^{19}+ \cdots + 5892 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.4.c.a 13.c 13.c $2$ $0.767$ \(\Q(\sqrt{-3}) \) None 13.4.c.a \(-4\) \(-2\) \(34\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\cdots\)
13.4.c.b 13.c 13.c $4$ $0.767$ \(\Q(\sqrt{-3}, \sqrt{17})\) None 13.4.c.b \(5\) \(-5\) \(-30\) \(-15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-\beta _{1}-2\beta _{2}-\beta _{3})q^{2}+(-1+3\beta _{1}+\cdots)q^{3}+\cdots\)