# Properties

 Label 216.2.a Level $216$ Weight $2$ Character orbit 216.a Rep. character $\chi_{216}(1,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $4$ Sturm bound $72$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$72$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(216))$$.

Total New Old
Modular forms 48 4 44
Cusp forms 25 4 21
Eisenstein series 23 0 23

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.
$$+$$$$+$$$$+$$$$1$$
$$+$$$$-$$$$-$$$$1$$
$$-$$$$+$$$$-$$$$2$$
Plus space$$+$$$$1$$
Minus space$$-$$$$3$$

## Trace form

 $$4 q + O(q^{10})$$ $$4 q + 10 q^{13} + 2 q^{19} + 14 q^{25} - 22 q^{31} - 30 q^{37} - 20 q^{43} + 8 q^{49} + 22 q^{55} - 26 q^{61} + 2 q^{67} + 4 q^{73} + 22 q^{79} - 16 q^{85} + 18 q^{91} + 8 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(216))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
216.2.a.a $1$ $1.725$ $$\Q$$ None $$0$$ $$0$$ $$-4$$ $$-3$$ $+$ $+$ $$q-4q^{5}-3q^{7}-4q^{11}+q^{13}+4q^{17}+\cdots$$
216.2.a.b $1$ $1.725$ $$\Q$$ None $$0$$ $$0$$ $$-1$$ $$3$$ $+$ $-$ $$q-q^{5}+3q^{7}+5q^{11}+4q^{13}-8q^{17}+\cdots$$
216.2.a.c $1$ $1.725$ $$\Q$$ None $$0$$ $$0$$ $$1$$ $$3$$ $-$ $+$ $$q+q^{5}+3q^{7}-5q^{11}+4q^{13}+8q^{17}+\cdots$$
216.2.a.d $1$ $1.725$ $$\Q$$ None $$0$$ $$0$$ $$4$$ $$-3$$ $-$ $+$ $$q+4q^{5}-3q^{7}+4q^{11}+q^{13}-4q^{17}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(216))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(216)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(24))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(54))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(108))$$$$^{\oplus 2}$$