# Properties

 Label 540.2.d Level $540$ Weight $2$ Character orbit 540.d Rep. character $\chi_{540}(109,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $3$ Sturm bound $216$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$540 = 2^{2} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 540.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$216$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(540, [\chi])$$.

Total New Old
Modular forms 126 8 118
Cusp forms 90 8 82
Eisenstein series 36 0 36

## Trace form

 $$8 q + O(q^{10})$$ $$8 q + 12 q^{25} - 4 q^{31} + 36 q^{49} + 4 q^{55} + 16 q^{61} - 48 q^{79} - 44 q^{85} - 36 q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(540, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
540.2.d.a $2$ $4.312$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+(-2+i)q^{5}-2iq^{7}+2q^{11}-6iq^{13}+\cdots$$
540.2.d.b $2$ $4.312$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(2+i)q^{5}+2iq^{7}-2q^{11}+6iq^{13}+\cdots$$
540.2.d.c $4$ $4.312$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+\beta _{2}q^{7}+(\beta _{1}-\beta _{3})q^{11}-3\beta _{2}q^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(540, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(540, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$