# Properties

 Label 208.6.f Level $208$ Weight $6$ Character orbit 208.f Rep. character $\chi_{208}(129,\cdot)$ Character field $\Q$ Dimension $34$ Newform subspaces $5$ Sturm bound $168$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 208.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$168$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(208, [\chi])$$.

Total New Old
Modular forms 146 36 110
Cusp forms 134 34 100
Eisenstein series 12 2 10

## Trace form

 $$34 q + 20 q^{3} + 2590 q^{9} + O(q^{10})$$ $$34 q + 20 q^{3} + 2590 q^{9} - 62 q^{13} + 200 q^{17} + 3176 q^{23} - 17194 q^{25} + 10052 q^{27} - 4284 q^{29} + 8452 q^{35} + 1792 q^{39} + 25556 q^{43} - 69102 q^{49} - 105004 q^{51} + 28876 q^{53} - 73344 q^{55} - 52076 q^{61} + 6900 q^{65} - 63672 q^{69} - 36200 q^{75} + 44840 q^{77} + 99144 q^{79} + 293922 q^{81} - 237568 q^{87} - 151900 q^{91} + 92424 q^{95} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(208, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
208.6.f.a $2$ $33.360$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-8$$ $$0$$ $$0$$ $$q-4q^{3}+34iq^{5}-41iq^{7}-227q^{9}+\cdots$$
208.6.f.b $2$ $33.360$ $$\Q(\sqrt{-1})$$ None $$0$$ $$26$$ $$0$$ $$0$$ $$q+13q^{3}+17iq^{5}-35iq^{7}-74q^{9}+\cdots$$
208.6.f.c $6$ $33.360$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$-16$$ $$0$$ $$0$$ $$q+(-3-\beta _{2})q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-3\beta _{1}+\cdots)q^{7}+\cdots$$
208.6.f.d $6$ $33.360$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}-\beta _{1}q^{5}+\beta _{2}q^{7}+(118+\beta _{3}+\cdots)q^{9}+\cdots$$
208.6.f.e $18$ $33.360$ $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ None $$0$$ $$18$$ $$0$$ $$0$$ $$q+(1+\beta _{2})q^{3}+\beta _{8}q^{5}+\beta _{10}q^{7}+(90+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(208, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(208, [\chi]) \simeq$$ $$S_{6}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(52, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(104, [\chi])$$$$^{\oplus 2}$$