# Properties

 Label 700.6.e Level $700$ Weight $6$ Character orbit 700.e Rep. character $\chi_{700}(449,\cdot)$ Character field $\Q$ Dimension $44$ Newform subspaces $10$ Sturm bound $720$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 700.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$720$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(700, [\chi])$$.

Total New Old
Modular forms 618 44 574
Cusp forms 582 44 538
Eisenstein series 36 0 36

## Trace form

 $$44 q - 3144 q^{9} + O(q^{10})$$ $$44 q - 3144 q^{9} - 104 q^{11} + 3512 q^{19} - 784 q^{21} + 15680 q^{29} + 812 q^{31} + 52148 q^{39} + 11708 q^{41} - 105644 q^{49} - 62908 q^{51} - 114716 q^{59} + 43836 q^{61} - 353980 q^{69} + 185412 q^{71} + 28360 q^{79} + 22452 q^{81} + 13500 q^{89} - 784 q^{91} - 171564 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(700, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
700.6.e.a $2$ $112.269$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+28iq^{3}+7^{2}iq^{7}-541q^{9}-577q^{11}+\cdots$$
700.6.e.b $2$ $112.269$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+26iq^{3}+7^{2}iq^{7}-433q^{9}+8q^{11}+\cdots$$
700.6.e.c $2$ $112.269$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+17iq^{3}-7^{2}iq^{7}-46q^{9}-317q^{11}+\cdots$$
700.6.e.d $2$ $112.269$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2iq^{3}+7^{2}iq^{7}+239q^{9}-720q^{11}+\cdots$$
700.6.e.e $4$ $112.269$ $$\Q(i, \sqrt{1009})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-11\beta _{2})q^{3}-7^{2}\beta _{2}q^{7}+(-153+\cdots)q^{9}+\cdots$$
700.6.e.f $4$ $112.269$ $$\Q(i, \sqrt{1009})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-8\beta _{2})q^{3}+7^{2}\beta _{2}q^{7}+(-90+\cdots)q^{9}+\cdots$$
700.6.e.g $6$ $112.269$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-2\beta _{2})q^{3}+7^{2}\beta _{2}q^{7}+(-94+\cdots)q^{9}+\cdots$$
700.6.e.h $6$ $112.269$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}+3\beta _{3})q^{3}-7^{2}\beta _{3}q^{7}+(23-\beta _{2}+\cdots)q^{9}+\cdots$$
700.6.e.i $8$ $112.269$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+7^{2}\beta _{3}q^{7}+(-52-11\beta _{2}+\cdots)q^{9}+\cdots$$
700.6.e.j $8$ $112.269$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-5\beta _{5})q^{3}+7^{2}\beta _{5}q^{7}+(19-6\beta _{2}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(700, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(700, [\chi]) \simeq$$ $$S_{6}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(140, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 2}$$