# Properties

 Label 600.2.f Level $600$ Weight $2$ Character orbit 600.f Rep. character $\chi_{600}(49,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $5$ Sturm bound $240$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$240$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(600, [\chi])$$.

Total New Old
Modular forms 144 10 134
Cusp forms 96 10 86
Eisenstein series 48 0 48

## Trace form

 $$10 q - 10 q^{9} + O(q^{10})$$ $$10 q - 10 q^{9} - 8 q^{11} + 20 q^{19} + 4 q^{21} - 4 q^{29} - 4 q^{31} + 8 q^{39} + 28 q^{41} - 30 q^{49} - 12 q^{51} - 8 q^{59} + 16 q^{61} - 8 q^{69} + 32 q^{79} + 10 q^{81} - 12 q^{89} - 36 q^{91} + 8 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(600, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
600.2.f.a $2$ $4.791$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+5iq^{7}-q^{9}-6q^{11}-3iq^{13}+\cdots$$
600.2.f.b $2$ $4.791$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-q^{9}-4q^{11}+6iq^{13}+6iq^{17}+\cdots$$
600.2.f.c $2$ $4.791$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-4iq^{7}-q^{9}-6iq^{13}+2iq^{17}+\cdots$$
600.2.f.d $2$ $4.791$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-3iq^{7}-q^{9}+2q^{11}-3iq^{13}+\cdots$$
600.2.f.e $2$ $4.791$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-q^{9}+4q^{11}+2iq^{13}+2iq^{17}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(600, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(600, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$