# Properties

 Label 8.22.a Level $8$ Weight $22$ Character orbit 8.a Rep. character $\chi_{8}(1,\cdot)$ Character field $\Q$ Dimension $5$ Newform subspaces $2$ Sturm bound $22$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 8.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$22$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{22}(\Gamma_0(8))$$.

Total New Old
Modular forms 23 5 18
Cusp forms 19 5 14
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$Dim
$$+$$$$2$$
$$-$$$$3$$

## Trace form

 $$5 q - 8668 q^{3} - 22003634 q^{5} + 740760072 q^{7} + 29917044817 q^{9} + O(q^{10})$$ $$5 q - 8668 q^{3} - 22003634 q^{5} + 740760072 q^{7} + 29917044817 q^{9} + 13471294636 q^{11} - 356632250986 q^{13} - 1862971179880 q^{15} + 1203867602330 q^{17} + 55090597707316 q^{19} + 63483640028832 q^{21} - 215967104297192 q^{23} + 186686289431851 q^{25} + 3014062843435496 q^{27} + 3202555900321062 q^{29} + 2556850700943904 q^{31} - 6758392137812368 q^{33} + 7975251928887216 q^{35} - 26458703887352466 q^{37} + 86979314693434424 q^{39} - 229981070418310926 q^{41} + 231195720126249164 q^{43} - 859998337885344938 q^{45} + 305620128011018544 q^{47} - 1397370909615771011 q^{49} + 5114029246370682760 q^{51} - 4194423668594372354 q^{53} + 7292989855958396744 q^{55} - 10021125333234603248 q^{57} + 10009049496232751836 q^{59} - 13732181343142521274 q^{61} + 36014708061348485352 q^{63} - 23512551689317845212 q^{65} + 881069099479908964 q^{67} - 26426991957433856032 q^{69} - 31691319343619281336 q^{71} - 31008308284406859502 q^{73} - 26746902752653120100 q^{75} + 139184164190911890912 q^{77} - 217016023518521947312 q^{79} + 520915702457008912765 q^{81} - 317800628207406564748 q^{83} + 367067042098068889852 q^{85} - 384429083625446912136 q^{87} + 370148514466841267106 q^{89} - 887770236365977512720 q^{91} + 1575602859006642094720 q^{93} - 822320774641135950856 q^{95} + 459335324412563312746 q^{97} + 817207776598242539644 q^{99} + O(q^{100})$$

## Decomposition of $$S_{22}^{\mathrm{new}}(\Gamma_0(8))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
8.22.a.a $2$ $22.358$ $$\Q(\sqrt{358549})$$ None $$0$$ $$-105432$$ $$2108140$$ $$444771792$$ $+$ $$q+(-52716-\beta )q^{3}+(1054070+20\beta )q^{5}+\cdots$$
8.22.a.b $3$ $22.358$ $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ None $$0$$ $$96764$$ $$-24111774$$ $$295988280$$ $-$ $$q+(32255-\beta _{1})q^{3}+(-8037261+9\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{22}^{\mathrm{old}}(\Gamma_0(8))$$ into lower level spaces

$$S_{22}^{\mathrm{old}}(\Gamma_0(8)) \simeq$$ $$S_{22}^{\mathrm{new}}(\Gamma_0(1))$$$$^{\oplus 4}$$$$\oplus$$$$S_{22}^{\mathrm{new}}(\Gamma_0(2))$$$$^{\oplus 3}$$$$\oplus$$$$S_{22}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 2}$$