# Properties

 Label 84.9.p Level $84$ Weight $9$ Character orbit 84.p Rep. character $\chi_{84}(53,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $42$ Newform subspaces $2$ Sturm bound $144$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 84.p (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$144$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(84, [\chi])$$.

Total New Old
Modular forms 268 42 226
Cusp forms 244 42 202
Eisenstein series 24 0 24

## Trace form

 $$42 q - 4307 q^{7} - 1790 q^{9} + O(q^{10})$$ $$42 q - 4307 q^{7} - 1790 q^{9} + 57430 q^{13} + 68482 q^{15} + 153723 q^{19} + 154411 q^{21} + 1376167 q^{25} - 2389050 q^{27} - 551843 q^{31} + 1874885 q^{33} + 1614999 q^{37} + 6645319 q^{39} - 6682842 q^{43} - 527785 q^{45} - 5924535 q^{49} - 1103461 q^{51} + 71577224 q^{55} + 8395440 q^{57} + 2064556 q^{61} + 23513124 q^{63} + 5537483 q^{67} - 5588722 q^{69} + 2048483 q^{73} - 49612355 q^{75} - 64909811 q^{79} - 67596854 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 103876909 q^{91} - 21483884 q^{93} - 353400880 q^{97} - 94510994 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(84, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.9.p.a $2$ $34.220$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-81$$ $$0$$ $$-4273$$ $$q-3^{4}\zeta_{6}q^{3}+(-2769+1265\zeta_{6})q^{7}+\cdots$$
84.9.p.b $40$ $34.220$ None $$0$$ $$81$$ $$0$$ $$-34$$

## Decomposition of $$S_{9}^{\mathrm{old}}(84, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(84, [\chi]) \cong$$ $$S_{9}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$