# Properties

 Label 600.2.bc Level $600$ Weight $2$ Character orbit 600.bc Rep. character $\chi_{600}(169,\cdot)$ Character field $\Q(\zeta_{10})$ Dimension $56$ Newform subspaces $3$ Sturm bound $240$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.bc (of order $$10$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$25$$ Character field: $$\Q(\zeta_{10})$$ Newform subspaces: $$3$$ Sturm bound: $$240$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

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

Total New Old
Modular forms 512 56 456
Cusp forms 448 56 392
Eisenstein series 64 0 64

## Trace form

 $$56 q - 4 q^{5} + 14 q^{9} + O(q^{10})$$ $$56 q - 4 q^{5} + 14 q^{9} - 2 q^{15} + 12 q^{19} + 4 q^{21} + 60 q^{23} - 6 q^{25} + 16 q^{29} - 6 q^{31} - 4 q^{35} + 20 q^{37} - 16 q^{41} + 4 q^{45} + 40 q^{47} - 12 q^{49} + 48 q^{51} + 60 q^{53} + 32 q^{55} + 24 q^{59} - 4 q^{61} + 16 q^{65} - 8 q^{69} - 24 q^{71} - 40 q^{73} - 8 q^{75} - 4 q^{79} - 14 q^{81} - 60 q^{83} - 48 q^{85} - 60 q^{87} - 24 q^{89} - 8 q^{91} - 88 q^{95} - 30 q^{97} + O(q^{100})$$

## 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.bc.a $8$ $4.791$ $$\Q(\zeta_{20})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{20}q^{3}+(\zeta_{20}^{2}+\zeta_{20}^{3}+\zeta_{20}^{5}+\cdots)q^{5}+\cdots$$
600.2.bc.b $24$ $4.791$ None $$0$$ $$0$$ $$-4$$ $$0$$
600.2.bc.c $24$ $4.791$ None $$0$$ $$0$$ $$0$$ $$0$$

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

$$S_{2}^{\mathrm{old}}(600, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 2}$$