# Properties

 Label 520.2.d Level $520$ Weight $2$ Character orbit 520.d Rep. character $\chi_{520}(209,\cdot)$ Character field $\Q$ Dimension $18$ Newform subspaces $3$ Sturm bound $168$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$168$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 92 18 74
Cusp forms 76 18 58
Eisenstein series 16 0 16

## Trace form

 $$18 q - 22 q^{9} + O(q^{10})$$ $$18 q - 22 q^{9} + 12 q^{11} - 8 q^{15} - 12 q^{19} - 16 q^{21} + 20 q^{25} + 12 q^{29} - 8 q^{31} + 10 q^{35} - 12 q^{41} + 28 q^{45} - 54 q^{49} + 12 q^{51} + 8 q^{55} - 44 q^{59} + 12 q^{61} - 24 q^{69} - 8 q^{71} + 30 q^{75} + 24 q^{79} + 26 q^{81} + 20 q^{85} + 28 q^{89} + 12 q^{91} + 52 q^{95} - 60 q^{99} + O(q^{100})$$

## 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.d.a $2$ $4.152$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(2+i)q^{5}-4iq^{7}+3q^{9}-2q^{11}+\cdots$$
520.2.d.b $6$ $4.152$ 6.0.350464.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{3}+\beta _{4})q^{3}+(\beta _{1}+\beta _{4})q^{5}+(-\beta _{3}+\cdots)q^{7}+\cdots$$
520.2.d.c $10$ $4.152$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-\beta _{9}q^{3}-\beta _{1}q^{5}-\beta _{4}q^{7}+(-4-\beta _{5}+\cdots)q^{9}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(520, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(260, [\chi])$$$$^{\oplus 2}$$