# Properties

 Label 400.2.c Level $400$ Weight $2$ Character orbit 400.c Rep. character $\chi_{400}(49,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $4$ Sturm bound $120$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$120$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(400, [\chi])$$.

Total New Old
Modular forms 78 10 68
Cusp forms 42 8 34
Eisenstein series 36 2 34

## Trace form

 $$8 q - 4 q^{9} + O(q^{10})$$ $$8 q - 4 q^{9} - 4 q^{11} + 12 q^{19} - 16 q^{21} + 8 q^{29} - 40 q^{39} - 12 q^{41} + 12 q^{51} + 32 q^{59} + 24 q^{61} + 24 q^{71} + 8 q^{79} + 16 q^{81} + 12 q^{89} - 24 q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(400, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.2.c.a $2$ $3.194$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+2iq^{7}-6q^{9}-q^{11}+4iq^{13}+\cdots$$
400.2.c.b $2$ $3.194$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+iq^{7}-q^{9}+iq^{13}+3iq^{17}+\cdots$$
400.2.c.c $2$ $3.194$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-2iq^{7}+2q^{9}+3q^{11}+4iq^{13}+\cdots$$
400.2.c.d $2$ $3.194$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{7}+3q^{9}-4q^{11}+iq^{13}+iq^{17}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(400, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(400, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$