# Properties

 Label 81.4.a Level $81$ Weight $4$ Character orbit 81.a Rep. character $\chi_{81}(1,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $5$ Sturm bound $36$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 81.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$36$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(81))$$.

Total New Old
Modular forms 33 14 19
Cusp forms 21 10 11
Eisenstein series 12 4 8

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.
$$+$$$$6$$
$$-$$$$4$$

## Trace form

 $$10 q + 34 q^{4} - 10 q^{7} + O(q^{10})$$ $$10 q + 34 q^{4} - 10 q^{7} + 48 q^{10} + 8 q^{13} + 262 q^{16} - 34 q^{19} - 102 q^{22} - 326 q^{25} - 304 q^{28} - 154 q^{31} - 594 q^{34} + 830 q^{37} + 600 q^{40} + 1016 q^{43} - 1356 q^{46} + 36 q^{49} - 988 q^{52} + 102 q^{55} - 3288 q^{58} - 64 q^{61} + 1738 q^{64} + 3032 q^{67} + 2976 q^{70} - 2680 q^{73} + 2930 q^{76} + 98 q^{79} + 3882 q^{82} - 2646 q^{85} - 4434 q^{88} + 3286 q^{91} + 2760 q^{94} + 3284 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(81))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
81.4.a.a $2$ $4.779$ $$\Q(\sqrt{33})$$ None $$-3$$ $$0$$ $$-15$$ $$7$$ $-$ $$q+(-1-\beta )q^{2}+(1+3\beta )q^{4}+(-8+\beta )q^{5}+\cdots$$
81.4.a.b $2$ $4.779$ $$\Q(\sqrt{57})$$ None $$-3$$ $$0$$ $$12$$ $$10$$ $+$ $$q+(-1-\beta )q^{2}+(7+3\beta )q^{4}+(7-2\beta )q^{5}+\cdots$$
81.4.a.c $2$ $4.779$ $$\Q(\sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$-44$$ $-$ $$q+\beta q^{2}-5q^{4}-7\beta q^{5}-22q^{7}-13\beta q^{8}+\cdots$$
81.4.a.d $2$ $4.779$ $$\Q(\sqrt{33})$$ None $$3$$ $$0$$ $$15$$ $$7$$ $+$ $$q+(1+\beta )q^{2}+(1+3\beta )q^{4}+(8-\beta )q^{5}+\cdots$$
81.4.a.e $2$ $4.779$ $$\Q(\sqrt{57})$$ None $$3$$ $$0$$ $$-12$$ $$10$$ $+$ $$q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(-7+2\beta )q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(81))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(81)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(27))$$$$^{\oplus 2}$$