# Properties

 Label 208.4.w Level $208$ Weight $4$ Character orbit 208.w Rep. character $\chi_{208}(17,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $40$ Newform subspaces $5$ Sturm bound $112$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 208.w (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$5$$ Sturm bound: $$112$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(208, [\chi])$$.

Total New Old
Modular forms 180 44 136
Cusp forms 156 40 116
Eisenstein series 24 4 20

## Trace form

 $$40 q + 7 q^{3} + 57 q^{7} - 163 q^{9} + O(q^{10})$$ $$40 q + 7 q^{3} + 57 q^{7} - 163 q^{9} + 3 q^{11} + 21 q^{13} - 78 q^{15} - 14 q^{17} + 3 q^{19} + 139 q^{23} - 782 q^{25} - 614 q^{27} - 72 q^{29} - 3 q^{33} + 296 q^{35} + 492 q^{37} - 1021 q^{39} - 180 q^{41} - 641 q^{43} + 237 q^{45} + 917 q^{49} + 862 q^{51} - 682 q^{53} - 168 q^{55} + 945 q^{59} + 294 q^{61} - 2022 q^{63} - 81 q^{65} - 1239 q^{67} + 573 q^{69} + 2631 q^{71} - 595 q^{75} - 218 q^{77} + 4128 q^{79} - 1152 q^{81} + 1089 q^{85} + 601 q^{87} + 2301 q^{89} + 2657 q^{91} + 246 q^{93} + 2034 q^{95} + 453 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(208, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
208.4.w.a $2$ $12.272$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-7$$ $$0$$ $$-39$$ $$q+(-7+7\zeta_{6})q^{3}+(-8+2^{4}\zeta_{6})q^{5}+\cdots$$
208.4.w.b $2$ $12.272$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$0$$ $$24$$ $$q+(2-2\zeta_{6})q^{3}+(1-2\zeta_{6})q^{5}+(2^{4}-8\zeta_{6})q^{7}+\cdots$$
208.4.w.c $8$ $12.272$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$36$$ $$q+\beta _{5}q^{3}+(2-3\beta _{1}+\beta _{2}-\beta _{4})q^{5}+\cdots$$
208.4.w.d $8$ $12.272$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$6$$ $$0$$ $$-18$$ $$q+(1+\beta _{1}-\beta _{6})q^{3}+(2+4\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
208.4.w.e $20$ $12.272$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$0$$ $$6$$ $$0$$ $$54$$ $$q+(1-\beta _{1}+\beta _{2})q^{3}+(-1+\beta _{1}-\beta _{7}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(208, [\chi])$$ into lower level spaces

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