# Properties

 Label 81.3.b Level $81$ Weight $3$ Character orbit 81.b Rep. character $\chi_{81}(80,\cdot)$ Character field $\Q$ Dimension $6$ Newform subspaces $2$ Sturm bound $27$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 81.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$27$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(81, [\chi])$$.

Total New Old
Modular forms 24 10 14
Cusp forms 12 6 6
Eisenstein series 12 4 8

## Trace form

 $$6 q - 6 q^{4} + O(q^{10})$$ $$6 q - 6 q^{4} - 24 q^{10} + 36 q^{13} + 6 q^{16} - 6 q^{19} - 66 q^{22} - 42 q^{25} + 120 q^{28} + 90 q^{34} - 60 q^{37} + 132 q^{40} + 18 q^{43} - 204 q^{46} - 174 q^{49} - 312 q^{52} + 120 q^{55} + 132 q^{58} + 288 q^{61} + 42 q^{64} + 54 q^{67} + 204 q^{70} + 66 q^{73} - 246 q^{76} - 288 q^{79} + 318 q^{82} - 36 q^{85} - 42 q^{88} - 276 q^{91} + 168 q^{94} - 198 q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(81, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.3.b.a $2$ $2.207$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\zeta_{6}q^{2}+q^{4}-2\zeta_{6}q^{5}+2q^{7}-5\zeta_{6}q^{8}+\cdots$$
81.3.b.b $4$ $2.207$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(81, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(81, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 2}$$