# Properties

 Label 729.2.a Level $729$ Weight $2$ Character orbit 729.a Rep. character $\chi_{729}(1,\cdot)$ Character field $\Q$ Dimension $30$ Newform subspaces $5$ Sturm bound $162$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$162$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(729))$$.

Total New Old
Modular forms 99 42 57
Cusp forms 64 30 34
Eisenstein series 35 12 23

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

$$3$$Dim
$$+$$$$12$$
$$-$$$$18$$

## Trace form

 $$30 q + 24 q^{4} + O(q^{10})$$ $$30 q + 24 q^{4} + 6 q^{10} + 12 q^{16} + 6 q^{19} + 12 q^{22} + 6 q^{25} - 12 q^{28} + 6 q^{37} + 12 q^{40} + 6 q^{46} - 6 q^{49} + 24 q^{52} - 12 q^{55} - 24 q^{58} + 18 q^{61} - 6 q^{64} + 18 q^{67} + 30 q^{70} + 24 q^{73} + 12 q^{76} - 66 q^{82} - 54 q^{85} + 24 q^{88} + 24 q^{91} + 12 q^{94} - 54 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(729))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
729.2.a.a $6$ $5.821$ 6.6.1397493.1 None $$-3$$ $$0$$ $$-6$$ $$0$$ $+$ $$q+(-\beta _{1}-\beta _{5})q^{2}+(\beta _{1}-\beta _{2}-\beta _{3})q^{4}+\cdots$$
729.2.a.b $6$ $5.821$ 6.6.7459857.1 None $$-3$$ $$0$$ $$3$$ $$6$$ $-$ $$q+(-\beta _{1}-\beta _{3})q^{2}+(1+\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots$$
729.2.a.c $6$ $5.821$ $$\Q(\zeta_{36})^+$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $+$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots$$
729.2.a.d $6$ $5.821$ 6.6.1397493.1 None $$3$$ $$0$$ $$6$$ $$0$$ $-$ $$q+(\beta _{1}+\beta _{5})q^{2}+(\beta _{1}-\beta _{2}-\beta _{3})q^{4}+\cdots$$
729.2.a.e $6$ $5.821$ 6.6.7459857.1 None $$3$$ $$0$$ $$-3$$ $$6$$ $-$ $$q+(\beta _{1}+\beta _{3})q^{2}+(1+\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(729))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(729)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(81))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(243))$$$$^{\oplus 2}$$