# Properties

 Label 600.2.k Level $600$ Weight $2$ Character orbit 600.k Rep. character $\chi_{600}(301,\cdot)$ Character field $\Q$ Dimension $38$ Newform subspaces $6$ Sturm bound $240$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.k (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$240$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

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

Total New Old
Modular forms 132 38 94
Cusp forms 108 38 70
Eisenstein series 24 0 24

## Trace form

 $$38 q - 2 q^{2} + 4 q^{4} + 2 q^{6} - 4 q^{7} - 8 q^{8} - 38 q^{9} + O(q^{10})$$ $$38 q - 2 q^{2} + 4 q^{4} + 2 q^{6} - 4 q^{7} - 8 q^{8} - 38 q^{9} - 4 q^{12} + 12 q^{16} + 4 q^{17} + 2 q^{18} + 12 q^{22} + 8 q^{23} - 8 q^{24} - 12 q^{26} + 28 q^{28} - 4 q^{31} - 12 q^{32} + 4 q^{34} - 4 q^{36} - 8 q^{38} + 8 q^{39} - 4 q^{41} - 16 q^{42} - 56 q^{44} + 24 q^{47} - 16 q^{48} + 30 q^{49} - 8 q^{52} - 2 q^{54} + 56 q^{56} + 8 q^{57} + 24 q^{58} - 20 q^{62} + 4 q^{63} + 4 q^{64} - 16 q^{66} + 16 q^{68} - 40 q^{71} + 8 q^{72} + 28 q^{73} + 16 q^{74} - 16 q^{76} - 16 q^{78} - 36 q^{79} + 38 q^{81} - 44 q^{82} + 32 q^{84} - 28 q^{86} - 12 q^{87} - 28 q^{88} + 20 q^{89} + 24 q^{92} + 16 q^{94} - 28 q^{96} - 12 q^{97} + 6 q^{98} + O(q^{100})$$

## 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.k.a $2$ $4.791$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$0$$ $$0$$ $$-4$$ $$q+(-1+i)q^{2}-iq^{3}-2iq^{4}+(1+i)q^{6}+\cdots$$
600.2.k.b $2$ $4.791$ $$\Q(\sqrt{-1})$$ None $$2$$ $$0$$ $$0$$ $$4$$ $$q+(1+i)q^{2}+iq^{3}+2iq^{4}+(-1+i)q^{6}+\cdots$$
600.2.k.c $6$ $4.791$ 6.0.399424.1 None $$-2$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{2}-\beta _{3})q^{4}+\beta _{4}q^{6}+\cdots$$
600.2.k.d $8$ $4.791$ 8.0.214798336.3 None $$-2$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{3}q^{2}+\beta _{2}q^{3}+(-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots$$
600.2.k.e $8$ $4.791$ 8.0.214798336.3 None $$2$$ $$0$$ $$0$$ $$-8$$ $$q-\beta _{3}q^{2}-\beta _{2}q^{3}+(-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots$$
600.2.k.f $12$ $4.791$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-\beta _{5}q^{6}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(600, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$