# Properties

 Label 200.6.a Level $200$ Weight $6$ Character orbit 200.a Rep. character $\chi_{200}(1,\cdot)$ Character field $\Q$ Dimension $24$ Newform subspaces $11$ Sturm bound $180$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 200.a (trivial) Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$180$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(200))$$.

Total New Old
Modular forms 162 24 138
Cusp forms 138 24 114
Eisenstein series 24 0 24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$5$$FrickeDim
$$+$$$$+$$$+$$$5$$
$$+$$$$-$$$-$$$7$$
$$-$$$$+$$$-$$$6$$
$$-$$$$-$$$+$$$6$$
Plus space$$+$$$$11$$
Minus space$$-$$$$13$$

## Trace form

 $$24 q + 20 q^{3} - 100 q^{7} + 1802 q^{9} + O(q^{10})$$ $$24 q + 20 q^{3} - 100 q^{7} + 1802 q^{9} - 582 q^{11} - 700 q^{13} - 380 q^{17} - 702 q^{19} + 844 q^{21} - 1260 q^{23} + 7640 q^{27} + 8756 q^{29} + 5756 q^{31} + 13760 q^{33} - 10780 q^{37} - 752 q^{39} + 21254 q^{41} - 21180 q^{43} + 5260 q^{47} + 98984 q^{49} - 25310 q^{51} + 5620 q^{53} - 63760 q^{57} - 41144 q^{59} + 35264 q^{61} - 28900 q^{63} + 146380 q^{67} + 178276 q^{69} + 69088 q^{71} + 96340 q^{73} - 225920 q^{77} - 64644 q^{79} + 78656 q^{81} - 80380 q^{83} + 399800 q^{87} + 275778 q^{89} - 72968 q^{91} + 118960 q^{93} - 338700 q^{97} - 270180 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(200))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
200.6.a.a $1$ $32.077$ $$\Q$$ None $$0$$ $$-20$$ $$0$$ $$24$$ $+$ $+$ $$q-20q^{3}+24q^{7}+157q^{9}+124q^{11}+\cdots$$
200.6.a.b $1$ $32.077$ $$\Q$$ None $$0$$ $$2$$ $$0$$ $$62$$ $-$ $+$ $$q+2q^{3}+62q^{7}-239q^{9}-12^{2}q^{11}+\cdots$$
200.6.a.c $1$ $32.077$ $$\Q$$ None $$0$$ $$8$$ $$0$$ $$108$$ $+$ $+$ $$q+8q^{3}+108q^{7}-179q^{9}-604q^{11}+\cdots$$
200.6.a.d $1$ $32.077$ $$\Q$$ None $$0$$ $$18$$ $$0$$ $$-242$$ $+$ $+$ $$q+18q^{3}-242q^{7}+3^{4}q^{9}+656q^{11}+\cdots$$
200.6.a.e $2$ $32.077$ $$\Q(\sqrt{241})$$ None $$0$$ $$-8$$ $$0$$ $$8$$ $-$ $-$ $$q+(-4-\beta )q^{3}+(4+2\beta )q^{7}+(14+8\beta )q^{9}+\cdots$$
200.6.a.f $2$ $32.077$ $$\Q(\sqrt{241})$$ None $$0$$ $$8$$ $$0$$ $$-8$$ $+$ $+$ $$q+(4+\beta )q^{3}+(-4-2\beta )q^{7}+(14+8\beta )q^{9}+\cdots$$
200.6.a.g $2$ $32.077$ $$\Q(\sqrt{129})$$ None $$0$$ $$12$$ $$0$$ $$-52$$ $-$ $+$ $$q+(6+\beta )q^{3}+(-26+3\beta )q^{7}+(309+\cdots)q^{9}+\cdots$$
200.6.a.h $3$ $32.077$ 3.3.47217.1 None $$0$$ $$-1$$ $$0$$ $$70$$ $-$ $+$ $$q-\beta _{1}q^{3}+(24-\beta _{1}-\beta _{2})q^{7}+(50+5\beta _{1}+\cdots)q^{9}+\cdots$$
200.6.a.i $3$ $32.077$ 3.3.47217.1 None $$0$$ $$1$$ $$0$$ $$-70$$ $+$ $-$ $$q+\beta _{1}q^{3}+(-24+\beta _{1}+\beta _{2})q^{7}+(50+\cdots)q^{9}+\cdots$$
200.6.a.j $4$ $32.077$ 4.4.1595208.1 None $$0$$ $$-4$$ $$0$$ $$-148$$ $-$ $-$ $$q+(-1-\beta _{1})q^{3}+(-37-\beta _{1}+\beta _{3})q^{7}+\cdots$$
200.6.a.k $4$ $32.077$ 4.4.1595208.1 None $$0$$ $$4$$ $$0$$ $$148$$ $+$ $-$ $$q+(1+\beta _{1})q^{3}+(37+\beta _{1}-\beta _{3})q^{7}+(5^{3}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(200))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(200)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(40))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(50))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(100))$$$$^{\oplus 2}$$