# Properties

 Label 400.2.n Level $400$ Weight $2$ Character orbit 400.n Rep. character $\chi_{400}(143,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $18$ Newform subspaces $4$ Sturm bound $120$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.n (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$120$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 156 18 138
Cusp forms 84 18 66
Eisenstein series 72 0 72

## Trace form

 $$18 q + O(q^{10})$$ $$18 q + 6 q^{13} + 6 q^{17} + 48 q^{21} - 24 q^{33} - 30 q^{37} - 48 q^{41} + 18 q^{53} + 48 q^{57} + 18 q^{73} - 24 q^{77} - 66 q^{81} + 24 q^{93} + 18 q^{97} + O(q^{100})$$

## 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.n.a $2$ $3.194$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-3iq^{9}+(5-5i)q^{13}+(5+5i)q^{17}+\cdots$$
400.2.n.b $4$ $3.194$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{7}+3\zeta_{12}q^{9}+\cdots$$
400.2.n.c $4$ $3.194$ $$\Q(i, \sqrt{5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+\beta _{2}q^{7}-7\beta _{1}q^{9}+10q^{21}+\cdots$$
400.2.n.d $8$ $3.194$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}q^{3}+4\zeta_{24}^{5}q^{7}-2\zeta_{24}^{3}q^{9}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(400, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$