# Properties

 Label 81.9.f Level $81$ Weight $9$ Character orbit 81.f Rep. character $\chi_{81}(8,\cdot)$ Character field $\Q(\zeta_{18})$ Dimension $138$ Newform subspaces $1$ Sturm bound $81$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 81.f (of order $$18$$ and degree $$6$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$27$$ Character field: $$\Q(\zeta_{18})$$ Newform subspaces: $$1$$ Sturm bound: $$81$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(81, [\chi])$$.

Total New Old
Modular forms 450 150 300
Cusp forms 414 138 276
Eisenstein series 36 12 24

## Trace form

 $$138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} + O(q^{10})$$ $$138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25} - 12 q^{28} + 943980 q^{29} + 920739 q^{31} + 3005136 q^{32} + 660474 q^{34} - 6225408 q^{35} - 3 q^{37} + 23716884 q^{38} - 975273 q^{40} - 16694382 q^{41} + 4412514 q^{43} - 17341119 q^{44} - 3 q^{46} + 11341869 q^{47} + 11347482 q^{49} + 40948977 q^{50} + 14465511 q^{52} - 12 q^{55} - 52215771 q^{56} - 19078611 q^{58} - 76116738 q^{59} + 34059450 q^{61} + 223709616 q^{62} + 100663293 q^{64} - 20396037 q^{65} - 103603884 q^{67} - 101921427 q^{68} + 135373629 q^{70} - 125718795 q^{71} - 7632642 q^{73} + 66643887 q^{74} - 203790342 q^{76} + 343269159 q^{77} - 68767890 q^{79} - 12 q^{82} - 383244663 q^{83} + 170435619 q^{85} - 71426730 q^{86} - 192774918 q^{88} - 135692730 q^{89} + 77546796 q^{91} + 1343159175 q^{92} - 44451609 q^{94} - 881099997 q^{95} + 31339344 q^{97} - 1293135102 q^{98} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(81, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.9.f.a $138$ $32.998$ None $$6$$ $$0$$ $$447$$ $$-6$$

## Decomposition of $$S_{9}^{\mathrm{old}}(81, [\chi])$$ into lower level spaces

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