# Properties

 Label 520.2.q Level $520$ Weight $2$ Character orbit 520.q Rep. character $\chi_{520}(81,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $28$ Newform subspaces $8$ Sturm bound $168$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$8$$ Sturm bound: $$168$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(520, [\chi])$$.

Total New Old
Modular forms 184 28 156
Cusp forms 152 28 124
Eisenstein series 32 0 32

## Trace form

 $$28 q + 4 q^{3} - 4 q^{7} - 14 q^{9} + O(q^{10})$$ $$28 q + 4 q^{3} - 4 q^{7} - 14 q^{9} + 2 q^{11} + 4 q^{13} - 12 q^{17} + 10 q^{19} + 40 q^{21} + 4 q^{23} + 28 q^{25} - 32 q^{27} - 4 q^{29} + 40 q^{31} + 8 q^{33} + 14 q^{35} + 8 q^{37} - 12 q^{39} + 12 q^{41} + 4 q^{43} - 80 q^{47} - 28 q^{49} + 8 q^{51} - 16 q^{53} + 4 q^{55} - 40 q^{57} - 16 q^{59} + 8 q^{61} - 24 q^{63} + 6 q^{65} + 4 q^{67} - 20 q^{69} + 8 q^{71} + 8 q^{73} + 4 q^{75} + 64 q^{77} + 40 q^{79} + 2 q^{81} - 16 q^{83} + 20 q^{87} - 6 q^{89} + 10 q^{91} - 12 q^{93} - 16 q^{97} + 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(520, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
520.2.q.a $2$ $4.152$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$-2$$ $$-3$$ $$q+(-2+2\zeta_{6})q^{3}-q^{5}-3\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots$$
520.2.q.b $2$ $4.152$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-2$$ $$3$$ $$q+(1-\zeta_{6})q^{3}-q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots$$
520.2.q.c $2$ $4.152$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-2$$ $$3$$ $$q+(1-\zeta_{6})q^{3}-q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots$$
520.2.q.d $2$ $4.152$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$2$$ $$3$$ $$q+(1-\zeta_{6})q^{3}+q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots$$
520.2.q.e $2$ $4.152$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-2$$ $$-3$$ $$q+(3-3\zeta_{6})q^{3}-q^{5}-3\zeta_{6}q^{7}-6\zeta_{6}q^{9}+\cdots$$
520.2.q.f $4$ $4.152$ $$\Q(\sqrt{-3}, \sqrt{17})$$ None $$0$$ $$1$$ $$4$$ $$-2$$ $$q+\beta _{1}q^{3}+q^{5}+(-1+\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots$$
520.2.q.g $6$ $4.152$ 6.0.27870912.1 None $$0$$ $$-1$$ $$-6$$ $$-9$$ $$q-\beta _{1}q^{3}-q^{5}+(-3+3\beta _{4})q^{7}+(-1+\cdots)q^{9}+\cdots$$
520.2.q.h $8$ $4.152$ 8.0.649638144.4 None $$0$$ $$0$$ $$8$$ $$4$$ $$q+\beta _{4}q^{3}+q^{5}+(1+\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(520, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(520, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(52, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(104, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(260, [\chi])$$$$^{\oplus 2}$$