# Properties

 Label 81.4.c Level $81$ Weight $4$ Character orbit 81.c Rep. character $\chi_{81}(28,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $22$ Newform subspaces $7$ Sturm bound $36$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 81.c (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$7$$ Sturm bound: $$36$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(81, [\chi])$$.

Total New Old
Modular forms 66 26 40
Cusp forms 42 22 20
Eisenstein series 24 4 20

## Trace form

 $$22 q - 38 q^{4} + 32 q^{7} + O(q^{10})$$ $$22 q - 38 q^{4} + 32 q^{7} - 36 q^{10} - 58 q^{13} - 110 q^{16} + 476 q^{19} - 18 q^{22} - 317 q^{25} - 1252 q^{28} + 266 q^{31} + 1188 q^{34} + 692 q^{37} + 846 q^{40} - 76 q^{43} + 1152 q^{46} - 1239 q^{49} - 1732 q^{52} - 4356 q^{55} + 684 q^{58} - 22 q^{61} + 4180 q^{64} + 1346 q^{67} + 1494 q^{70} + 2528 q^{73} - 6124 q^{76} - 2614 q^{79} - 4464 q^{82} - 54 q^{85} + 5958 q^{88} + 6172 q^{91} + 6732 q^{94} + 86 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(81, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.4.c.a $2$ $4.779$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$0$$ $$-15$$ $$25$$ $$q+(-3+3\zeta_{6})q^{2}-\zeta_{6}q^{4}-15\zeta_{6}q^{5}+\cdots$$
81.4.c.b $2$ $4.779$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-20$$ $$q+8\zeta_{6}q^{4}+(-20+20\zeta_{6})q^{7}+70\zeta_{6}q^{13}+\cdots$$
81.4.c.c $2$ $4.779$ $$\Q(\sqrt{-3})$$ None $$3$$ $$0$$ $$15$$ $$25$$ $$q+(3-3\zeta_{6})q^{2}-\zeta_{6}q^{4}+15\zeta_{6}q^{5}+\cdots$$
81.4.c.d $4$ $4.779$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$-3$$ $$0$$ $$12$$ $$-10$$ $$q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots$$
81.4.c.e $4$ $4.779$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-22$$ $$q+\beta _{1}q^{2}+10\beta _{2}q^{4}+(4\beta _{1}+4\beta _{3})q^{5}+\cdots$$
81.4.c.f $4$ $4.779$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$44$$ $$q-\zeta_{12}^{2}q^{2}+(5-5\zeta_{12})q^{4}+(-7\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
81.4.c.g $4$ $4.779$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$3$$ $$0$$ $$-12$$ $$-10$$ $$q+(1+\beta _{1}+\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+3\beta _{3})q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(81, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(81, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 2}$$