# Properties

 Label 400.6.c Level $400$ Weight $6$ Character orbit 400.c Rep. character $\chi_{400}(49,\cdot)$ Character field $\Q$ Dimension $44$ Newform subspaces $16$ Sturm bound $360$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$16$$ Sturm bound: $$360$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(400, [\chi])$$.

Total New Old
Modular forms 318 46 272
Cusp forms 282 44 238
Eisenstein series 36 2 34

## Trace form

 $$44 q - 3400 q^{9} + O(q^{10})$$ $$44 q - 3400 q^{9} - 724 q^{11} - 196 q^{19} - 2128 q^{21} - 12240 q^{29} + 12928 q^{31} - 10376 q^{39} - 8404 q^{41} - 85604 q^{49} - 115444 q^{51} + 77136 q^{59} + 2288 q^{61} + 43648 q^{69} + 25016 q^{71} + 58888 q^{79} + 156724 q^{81} - 132588 q^{89} + 160104 q^{91} + 244272 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(400, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.6.c.a $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+13iq^{3}-11iq^{7}-433q^{9}+768q^{11}+\cdots$$
400.6.c.b $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+12iq^{3}+86iq^{7}-333q^{9}-132q^{11}+\cdots$$
400.6.c.c $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+11iq^{3}-109iq^{7}-241q^{9}+480q^{11}+\cdots$$
400.6.c.d $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+10iq^{3}+12iq^{7}-157q^{9}-124q^{11}+\cdots$$
400.6.c.e $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+9iq^{3}+11^{2}iq^{7}-3^{4}q^{9}-656q^{11}+\cdots$$
400.6.c.f $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+6iq^{3}-44iq^{7}+99q^{9}-540q^{11}+\cdots$$
400.6.c.g $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+11iq^{3}-142iq^{7}+122q^{9}-777q^{11}+\cdots$$
400.6.c.h $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{3}-54iq^{7}+179q^{9}+604q^{11}+\cdots$$
400.6.c.i $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3iq^{3}+59iq^{7}+207q^{9}-192q^{11}+\cdots$$
400.6.c.j $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{3}+96iq^{7}+227q^{9}+148q^{11}+\cdots$$
400.6.c.k $2$ $64.154$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-31iq^{7}+239q^{9}+12^{2}q^{11}+\cdots$$
400.6.c.l $4$ $64.154$ $$\Q(i, \sqrt{129})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-3\beta _{1}+\beta _{2})q^{3}+(-13\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots$$
400.6.c.m $4$ $64.154$ $$\Q(i, \sqrt{409})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}+2\beta _{2})q^{3}+(-6\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots$$
400.6.c.n $4$ $64.154$ $$\Q(i, \sqrt{241})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-2\beta _{2})q^{3}+(-2\beta _{1}+20\beta _{2})q^{7}+\cdots$$
400.6.c.o $4$ $64.154$ $$\Q(i, \sqrt{241})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-4\beta _{1}-\beta _{2})q^{3}+(-4\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots$$
400.6.c.p $6$ $64.154$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+(-24\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(400, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(400, [\chi]) \cong$$ $$S_{6}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$