# Properties

 Label 81.7.d Level $81$ Weight $7$ Character orbit 81.d Rep. character $\chi_{81}(26,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $46$ Newform subspaces $6$ Sturm bound $63$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 81.d (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ Sturm bound: $$63$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(81, [\chi])$$.

Total New Old
Modular forms 120 50 70
Cusp forms 96 46 50
Eisenstein series 24 4 20

## Trace form

 $$46 q + 706 q^{4} + 602 q^{7} + O(q^{10})$$ $$46 q + 706 q^{4} + 602 q^{7} + 252 q^{10} + 4202 q^{13} - 20606 q^{16} - 2284 q^{19} - 7326 q^{22} + 48721 q^{25} + 137468 q^{28} + 104642 q^{31} + 45144 q^{34} + 29900 q^{37} - 236934 q^{40} + 317582 q^{43} - 157752 q^{46} - 321621 q^{49} - 671824 q^{52} + 1431144 q^{55} + 1238724 q^{58} - 45190 q^{61} - 3195188 q^{64} + 237374 q^{67} + 389682 q^{70} - 1011004 q^{73} - 1972504 q^{76} - 2012194 q^{79} + 2717856 q^{82} + 1192536 q^{85} + 1547838 q^{88} + 3005464 q^{91} - 4810572 q^{94} + 1460006 q^{97} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(81, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.7.d.a $2$ $18.634$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$286$$ $$q-2^{6}\zeta_{6}q^{4}+(286-286\zeta_{6})q^{7}-506\zeta_{6}q^{13}+\cdots$$
81.7.d.b $4$ $18.634$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-598$$ $$q+\zeta_{12}q^{2}-28\zeta_{12}^{2}q^{4}+(40\zeta_{12}-40\zeta_{12}^{3})q^{5}+\cdots$$
81.7.d.c $4$ $18.634$ $$\Q(\sqrt{-3}, \sqrt{-10})$$ None $$0$$ $$0$$ $$0$$ $$806$$ $$q+\beta _{1}q^{2}+26\beta _{2}q^{4}+(-14\beta _{1}+14\beta _{3})q^{5}+\cdots$$
81.7.d.d $4$ $18.634$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-1048$$ $$q+\beta _{1}q^{2}+98\beta _{2}q^{4}+(-5\beta _{1}+5\beta _{3})q^{5}+\cdots$$
81.7.d.e $8$ $18.634$ 8.0.$$\cdots$$.3 None $$0$$ $$0$$ $$0$$ $$676$$ $$q-\beta _{3}q^{2}+(7^{2}+7^{2}\beta _{1}+\beta _{4}+\beta _{5})q^{4}+\cdots$$
81.7.d.f $24$ $18.634$ None $$0$$ $$0$$ $$0$$ $$480$$

## Decomposition of $$S_{7}^{\mathrm{old}}(81, [\chi])$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(81, [\chi]) \cong$$ $$S_{7}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 2}$$