# Properties

 Label 52.2.e Level $52$ Weight $2$ Character orbit 52.e Rep. character $\chi_{52}(9,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $4$ Newform subspaces $2$ Sturm bound $14$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$2$$ Sturm bound: $$14$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(52, [\chi])$$.

Total New Old
Modular forms 20 4 16
Cusp forms 8 4 4
Eisenstein series 12 0 12

## Trace form

 $$4 q - q^{3} - 2 q^{5} + 3 q^{7} - 7 q^{9} + O(q^{10})$$ $$4 q - q^{3} - 2 q^{5} + 3 q^{7} - 7 q^{9} + 5 q^{11} - 9 q^{13} - 12 q^{15} - 6 q^{17} + q^{19} + 22 q^{21} + 7 q^{23} + 6 q^{25} + 26 q^{27} - 8 q^{29} - 12 q^{31} + 15 q^{33} - 14 q^{35} + 4 q^{37} - 25 q^{39} - 6 q^{41} + 3 q^{43} - 9 q^{45} - 4 q^{47} - 3 q^{49} + 6 q^{51} + 6 q^{53} + 10 q^{55} - 26 q^{57} - 5 q^{59} - 2 q^{63} + 17 q^{65} - 9 q^{67} - 9 q^{69} + 17 q^{71} + 26 q^{73} + 11 q^{75} - 10 q^{77} - 16 q^{79} + 2 q^{81} + 48 q^{83} + 3 q^{85} + 21 q^{87} + 3 q^{89} - q^{91} + 28 q^{93} + 12 q^{95} - 13 q^{97} - 60 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(52, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
52.2.e.a $2$ $0.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$4$$ $$-1$$ $$q+(-3+3\zeta_{6})q^{3}+2q^{5}-\zeta_{6}q^{7}-6\zeta_{6}q^{9}+\cdots$$
52.2.e.b $2$ $0.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$-6$$ $$4$$ $$q+(2-2\zeta_{6})q^{3}-3q^{5}+4\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(52, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(52, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 2}$$