# Properties

 Label 320.6.a Level $320$ Weight $6$ Character orbit 320.a Rep. character $\chi_{320}(1,\cdot)$ Character field $\Q$ Dimension $40$ Newform subspaces $26$ Sturm bound $288$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.a (trivial) Character field: $$\Q$$ Newform subspaces: $$26$$ Sturm bound: $$288$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(320))$$.

Total New Old
Modular forms 252 40 212
Cusp forms 228 40 188
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.
$$+$$$$+$$$$+$$$$9$$
$$+$$$$-$$$$-$$$$11$$
$$-$$$$+$$$$-$$$$11$$
$$-$$$$-$$$$+$$$$9$$
Plus space$$+$$$$18$$
Minus space$$-$$$$22$$

## Trace form

 $$40 q + 3240 q^{9} + O(q^{10})$$ $$40 q + 3240 q^{9} + 464 q^{13} - 8496 q^{21} + 25000 q^{25} - 8144 q^{29} - 11344 q^{33} + 25600 q^{37} + 23216 q^{41} + 96040 q^{49} + 49456 q^{53} + 61616 q^{57} + 52224 q^{61} + 22320 q^{69} + 14896 q^{77} + 215176 q^{81} + 132400 q^{85} + 6320 q^{89} + 9312 q^{93} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
320.6.a.a $1$ $51.323$ $$\Q$$ None $$0$$ $$-26$$ $$25$$ $$22$$ $-$ $-$ $$q-26q^{3}+5^{2}q^{5}+22q^{7}+433q^{9}+\cdots$$
320.6.a.b $1$ $51.323$ $$\Q$$ None $$0$$ $$-24$$ $$-25$$ $$-172$$ $+$ $+$ $$q-24q^{3}-5^{2}q^{5}-172q^{7}+333q^{9}+\cdots$$
320.6.a.c $1$ $51.323$ $$\Q$$ None $$0$$ $$-22$$ $$25$$ $$218$$ $+$ $-$ $$q-22q^{3}+5^{2}q^{5}+218q^{7}+241q^{9}+\cdots$$
320.6.a.d $1$ $51.323$ $$\Q$$ None $$0$$ $$-18$$ $$25$$ $$-242$$ $-$ $-$ $$q-18q^{3}+5^{2}q^{5}-242q^{7}+3^{4}q^{9}+\cdots$$
320.6.a.e $1$ $51.323$ $$\Q$$ None $$0$$ $$-8$$ $$-25$$ $$108$$ $-$ $+$ $$q-8q^{3}-5^{2}q^{5}+108q^{7}-179q^{9}+\cdots$$
320.6.a.f $1$ $51.323$ $$\Q$$ None $$0$$ $$-6$$ $$25$$ $$-118$$ $+$ $-$ $$q-6q^{3}+5^{2}q^{5}-118q^{7}-207q^{9}+\cdots$$
320.6.a.g $1$ $51.323$ $$\Q$$ None $$0$$ $$-4$$ $$-25$$ $$-192$$ $-$ $+$ $$q-4q^{3}-5^{2}q^{5}-192q^{7}-227q^{9}+\cdots$$
320.6.a.h $1$ $51.323$ $$\Q$$ None $$0$$ $$-2$$ $$25$$ $$62$$ $-$ $-$ $$q-2q^{3}+5^{2}q^{5}+62q^{7}-239q^{9}+\cdots$$
320.6.a.i $1$ $51.323$ $$\Q$$ None $$0$$ $$2$$ $$25$$ $$-62$$ $+$ $-$ $$q+2q^{3}+5^{2}q^{5}-62q^{7}-239q^{9}+\cdots$$
320.6.a.j $1$ $51.323$ $$\Q$$ None $$0$$ $$4$$ $$-25$$ $$192$$ $+$ $+$ $$q+4q^{3}-5^{2}q^{5}+192q^{7}-227q^{9}+\cdots$$
320.6.a.k $1$ $51.323$ $$\Q$$ None $$0$$ $$6$$ $$25$$ $$118$$ $-$ $-$ $$q+6q^{3}+5^{2}q^{5}+118q^{7}-207q^{9}+\cdots$$
320.6.a.l $1$ $51.323$ $$\Q$$ None $$0$$ $$8$$ $$-25$$ $$-108$$ $+$ $+$ $$q+8q^{3}-5^{2}q^{5}-108q^{7}-179q^{9}+\cdots$$
320.6.a.m $1$ $51.323$ $$\Q$$ None $$0$$ $$18$$ $$25$$ $$242$$ $+$ $-$ $$q+18q^{3}+5^{2}q^{5}+242q^{7}+3^{4}q^{9}+\cdots$$
320.6.a.n $1$ $51.323$ $$\Q$$ None $$0$$ $$22$$ $$25$$ $$-218$$ $-$ $-$ $$q+22q^{3}+5^{2}q^{5}-218q^{7}+241q^{9}+\cdots$$
320.6.a.o $1$ $51.323$ $$\Q$$ None $$0$$ $$24$$ $$-25$$ $$172$$ $-$ $+$ $$q+24q^{3}-5^{2}q^{5}+172q^{7}+333q^{9}+\cdots$$
320.6.a.p $1$ $51.323$ $$\Q$$ None $$0$$ $$26$$ $$25$$ $$-22$$ $+$ $-$ $$q+26q^{3}+5^{2}q^{5}-22q^{7}+433q^{9}+\cdots$$
320.6.a.q $2$ $51.323$ $$\Q(\sqrt{129})$$ None $$0$$ $$-12$$ $$-50$$ $$-52$$ $-$ $+$ $$q+(-6-\beta )q^{3}-5^{2}q^{5}+(-26+3\beta )q^{7}+\cdots$$
320.6.a.r $2$ $51.323$ $$\Q(\sqrt{70})$$ None $$0$$ $$-8$$ $$-50$$ $$104$$ $+$ $+$ $$q+(-4+\beta )q^{3}-5^{2}q^{5}+(52+\beta )q^{7}+\cdots$$
320.6.a.s $2$ $51.323$ $$\Q(\sqrt{10})$$ None $$0$$ $$0$$ $$-50$$ $$0$$ $-$ $+$ $$q+\beta q^{3}-5^{2}q^{5}-7\beta q^{7}-203q^{9}+\cdots$$
320.6.a.t $2$ $51.323$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$50$$ $$0$$ $-$ $-$ $$q-3\beta q^{3}+5^{2}q^{5}+31\beta q^{7}-63q^{9}+\cdots$$
320.6.a.u $2$ $51.323$ $$\Q(\sqrt{85})$$ None $$0$$ $$0$$ $$50$$ $$0$$ $-$ $-$ $$q-\beta q^{3}+5^{2}q^{5}-3\beta q^{7}+97q^{9}+26\beta q^{11}+\cdots$$
320.6.a.v $2$ $51.323$ $$\Q(\sqrt{70})$$ None $$0$$ $$8$$ $$-50$$ $$-104$$ $+$ $+$ $$q+(4+\beta )q^{3}-5^{2}q^{5}+(-52+\beta )q^{7}+\cdots$$
320.6.a.w $2$ $51.323$ $$\Q(\sqrt{129})$$ None $$0$$ $$12$$ $$-50$$ $$52$$ $+$ $+$ $$q+(6+\beta )q^{3}-5^{2}q^{5}+(26-3\beta )q^{7}+\cdots$$
320.6.a.x $3$ $51.323$ 3.3.39180.1 None $$0$$ $$-10$$ $$75$$ $$-6$$ $+$ $-$ $$q+(-3-\beta _{1})q^{3}+5^{2}q^{5}+(-2-\beta _{1}+\cdots)q^{7}+\cdots$$
320.6.a.y $3$ $51.323$ 3.3.39180.1 None $$0$$ $$10$$ $$75$$ $$6$$ $+$ $-$ $$q+(3+\beta _{1})q^{3}+5^{2}q^{5}+(2+\beta _{1}-\beta _{2})q^{7}+\cdots$$
320.6.a.z $4$ $51.323$ 4.4.81998080.1 None $$0$$ $$0$$ $$-100$$ $$0$$ $-$ $+$ $$q-\beta _{1}q^{3}-5^{2}q^{5}+(6\beta _{1}-\beta _{2})q^{7}+(181+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(320))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(320)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 10}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 7}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(20))$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(80))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(160))$$$$^{\oplus 2}$$