# Properties

 Label 1728.2.a Level $1728$ Weight $2$ Character orbit 1728.a Rep. character $\chi_{1728}(1,\cdot)$ Character field $\Q$ Dimension $32$ Newform subspaces $30$ Sturm bound $576$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.a (trivial) Character field: $$\Q$$ Newform subspaces: $$30$$ Sturm bound: $$576$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$5$$, $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(1728))$$.

Total New Old
Modular forms 324 32 292
Cusp forms 253 32 221
Eisenstein series 71 0 71

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

$$2$$$$3$$FrickeDim
$$+$$$$+$$$+$$$7$$
$$+$$$$-$$$-$$$9$$
$$-$$$$+$$$-$$$9$$
$$-$$$$-$$$+$$$7$$
Plus space$$+$$$$14$$
Minus space$$-$$$$18$$

## Trace form

 $$32 q + O(q^{10})$$ $$32 q - 8 q^{13} + 32 q^{25} - 8 q^{37} + 32 q^{49} + 8 q^{61} + 16 q^{85} + 16 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(1728))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
1728.2.a.a $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-4$$ $$-3$$ $+$ $-$ $$q-4q^{5}-3q^{7}-4q^{11}-q^{13}-4q^{17}+\cdots$$
1728.2.a.b $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-4$$ $$3$$ $-$ $+$ $$q-4q^{5}+3q^{7}+4q^{11}-q^{13}-4q^{17}+\cdots$$
1728.2.a.c $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-3$$ $$-1$$ $+$ $+$ $$q-3q^{5}-q^{7}+3q^{11}+4q^{13}-2q^{19}+\cdots$$
1728.2.a.d $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-3$$ $$1$$ $-$ $-$ $$q-3q^{5}+q^{7}-3q^{11}+4q^{13}+2q^{19}+\cdots$$
1728.2.a.e $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$-3$$ $-$ $-$ $$q-2q^{5}-3q^{7}+6q^{11}+3q^{13}-2q^{17}+\cdots$$
1728.2.a.f $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$-1$$ $+$ $+$ $$q-2q^{5}-q^{7}+2q^{11}-q^{13}+6q^{17}+\cdots$$
1728.2.a.g $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$1$$ $+$ $-$ $$q-2q^{5}+q^{7}-2q^{11}-q^{13}+6q^{17}+\cdots$$
1728.2.a.h $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-2$$ $$3$$ $-$ $+$ $$q-2q^{5}+3q^{7}-6q^{11}+3q^{13}-2q^{17}+\cdots$$
1728.2.a.i $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-1$$ $$-3$$ $-$ $+$ $$q-q^{5}-3q^{7}-5q^{11}-4q^{13}+8q^{17}+\cdots$$
1728.2.a.j $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-1$$ $$-3$$ $-$ $+$ $$q-q^{5}-3q^{7}+3q^{11}-4q^{17}+6q^{19}+\cdots$$
1728.2.a.k $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-1$$ $$3$$ $-$ $-$ $$q-q^{5}+3q^{7}-3q^{11}-4q^{17}-6q^{19}+\cdots$$
1728.2.a.l $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$-1$$ $$3$$ $+$ $-$ $$q-q^{5}+3q^{7}+5q^{11}-4q^{13}+8q^{17}+\cdots$$
1728.2.a.m $1$ $13.798$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-5$$ $-$ $+$ $$q-5q^{7}+7q^{13}-q^{19}-5q^{25}+4q^{31}+\cdots$$
1728.2.a.n $1$ $13.798$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-1$$ $+$ $+$ $$q-q^{7}-5q^{13}+7q^{19}-5q^{25}-4q^{31}+\cdots$$
1728.2.a.o $1$ $13.798$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$1$$ $-$ $-$ $$q+q^{7}-5q^{13}-7q^{19}-5q^{25}+4q^{31}+\cdots$$
1728.2.a.p $1$ $13.798$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$5$$ $+$ $-$ $$q+5q^{7}+7q^{13}+q^{19}-5q^{25}-4q^{31}+\cdots$$
1728.2.a.q $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$1$$ $$-3$$ $-$ $-$ $$q+q^{5}-3q^{7}-3q^{11}+4q^{17}+6q^{19}+\cdots$$
1728.2.a.r $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$1$$ $$-3$$ $-$ $-$ $$q+q^{5}-3q^{7}+5q^{11}-4q^{13}-8q^{17}+\cdots$$
1728.2.a.s $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$1$$ $$3$$ $+$ $+$ $$q+q^{5}+3q^{7}-5q^{11}-4q^{13}-8q^{17}+\cdots$$
1728.2.a.t $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$1$$ $$3$$ $-$ $+$ $$q+q^{5}+3q^{7}+3q^{11}+4q^{17}-6q^{19}+\cdots$$
1728.2.a.u $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$2$$ $$-3$$ $-$ $-$ $$q+2q^{5}-3q^{7}-6q^{11}+3q^{13}+2q^{17}+\cdots$$
1728.2.a.v $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$2$$ $$-1$$ $+$ $+$ $$q+2q^{5}-q^{7}-2q^{11}-q^{13}-6q^{17}+\cdots$$
1728.2.a.w $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$2$$ $$1$$ $+$ $-$ $$q+2q^{5}+q^{7}+2q^{11}-q^{13}-6q^{17}+\cdots$$
1728.2.a.x $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$2$$ $$3$$ $-$ $+$ $$q+2q^{5}+3q^{7}+6q^{11}+3q^{13}+2q^{17}+\cdots$$
1728.2.a.y $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$3$$ $$-1$$ $+$ $-$ $$q+3q^{5}-q^{7}-3q^{11}+4q^{13}-2q^{19}+\cdots$$
1728.2.a.z $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$3$$ $$1$$ $-$ $+$ $$q+3q^{5}+q^{7}+3q^{11}+4q^{13}+2q^{19}+\cdots$$
1728.2.a.ba $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$4$$ $$-3$$ $+$ $-$ $$q+4q^{5}-3q^{7}+4q^{11}-q^{13}+4q^{17}+\cdots$$
1728.2.a.bb $1$ $13.798$ $$\Q$$ None $$0$$ $$0$$ $$4$$ $$3$$ $-$ $+$ $$q+4q^{5}+3q^{7}-4q^{11}-q^{13}+4q^{17}+\cdots$$
1728.2.a.bc $2$ $13.798$ $$\Q(\sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $+$ $+$ $$q-\beta q^{5}+\beta q^{7}-q^{11}-4q^{13}+2\beta q^{19}+\cdots$$
1728.2.a.bd $2$ $13.798$ $$\Q(\sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $+$ $-$ $$q-\beta q^{5}-\beta q^{7}+q^{11}-4q^{13}-2\beta q^{19}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(1728))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(1728)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(24))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(27))$$$$^{\oplus 7}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(36))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(48))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(54))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(72))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(96))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(108))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(144))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(192))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(216))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(288))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(432))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(576))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(864))$$$$^{\oplus 2}$$