# Properties

 Label 63.6.a Level $63$ Weight $6$ Character orbit 63.a Rep. character $\chi_{63}(1,\cdot)$ Character field $\Q$ Dimension $13$ Newform subspaces $8$ Sturm bound $48$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 63.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$48$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(63))$$.

Total New Old
Modular forms 44 13 31
Cusp forms 36 13 23
Eisenstein series 8 0 8

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

$$3$$$$7$$FrickeDim
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$4$$
$$-$$$$+$$$$-$$$$4$$
$$-$$$$-$$$$+$$$$3$$
Plus space$$+$$$$5$$
Minus space$$-$$$$8$$

## Trace form

 $$13 q - 9 q^{2} + 193 q^{4} + 42 q^{5} + 49 q^{7} + 99 q^{8} + O(q^{10})$$ $$13 q - 9 q^{2} + 193 q^{4} + 42 q^{5} + 49 q^{7} + 99 q^{8} - 156 q^{10} - 876 q^{11} - 742 q^{13} + 147 q^{14} + 4645 q^{16} + 3090 q^{17} + 4868 q^{19} + 7788 q^{20} + 1788 q^{22} - 3168 q^{23} + 11083 q^{25} - 9372 q^{26} + 637 q^{28} - 3066 q^{29} - 9976 q^{31} - 741 q^{32} - 13230 q^{34} - 7938 q^{35} + 3902 q^{37} + 29862 q^{38} - 27144 q^{40} + 6138 q^{41} + 5276 q^{43} - 3168 q^{44} - 43020 q^{46} - 9840 q^{47} + 31213 q^{49} - 70131 q^{50} - 67084 q^{52} + 31110 q^{53} + 17856 q^{55} - 20433 q^{56} - 39414 q^{58} + 40092 q^{59} - 63190 q^{61} - 157404 q^{62} + 7201 q^{64} + 146412 q^{65} + 158996 q^{67} + 108798 q^{68} + 37044 q^{70} - 42864 q^{71} + 39146 q^{73} - 194946 q^{74} + 210194 q^{76} - 39396 q^{77} - 103504 q^{79} + 491748 q^{80} - 178806 q^{82} - 90108 q^{83} + 206004 q^{85} + 216840 q^{86} + 460572 q^{88} - 17070 q^{89} + 44198 q^{91} - 314760 q^{92} - 758436 q^{94} - 467832 q^{95} + 98282 q^{97} - 21609 q^{98} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(63))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
63.6.a.a $1$ $10.104$ $$\Q$$ None $$-10$$ $$0$$ $$106$$ $$-49$$ $-$ $+$ $$q-10q^{2}+68q^{4}+106q^{5}-7^{2}q^{7}+\cdots$$
63.6.a.b $1$ $10.104$ $$\Q$$ None $$-5$$ $$0$$ $$-94$$ $$-49$$ $-$ $+$ $$q-5q^{2}-7q^{4}-94q^{5}-7^{2}q^{7}+195q^{8}+\cdots$$
63.6.a.c $1$ $10.104$ $$\Q$$ None $$-1$$ $$0$$ $$34$$ $$-49$$ $-$ $+$ $$q-q^{2}-31q^{4}+34q^{5}-7^{2}q^{7}+63q^{8}+\cdots$$
63.6.a.d $1$ $10.104$ $$\Q$$ None $$6$$ $$0$$ $$-78$$ $$49$$ $-$ $-$ $$q+6q^{2}+4q^{4}-78q^{5}+7^{2}q^{7}-168q^{8}+\cdots$$
63.6.a.e $1$ $10.104$ $$\Q$$ None $$10$$ $$0$$ $$56$$ $$-49$$ $-$ $+$ $$q+10q^{2}+68q^{4}+56q^{5}-7^{2}q^{7}+\cdots$$
63.6.a.f $2$ $10.104$ $$\Q(\sqrt{57})$$ None $$-9$$ $$0$$ $$18$$ $$98$$ $-$ $-$ $$q+(-4-\beta )q^{2}+(-2+9\beta )q^{4}+(14+\cdots)q^{5}+\cdots$$
63.6.a.g $2$ $10.104$ $$\Q(\sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$-98$$ $+$ $+$ $$q+\beta q^{2}-4q^{4}-7\beta q^{5}-7^{2}q^{7}-6^{2}\beta q^{8}+\cdots$$
63.6.a.h $4$ $10.104$ 4.4.358541904.1 None $$0$$ $$0$$ $$0$$ $$196$$ $+$ $-$ $$q+\beta _{1}q^{2}+(24+\beta _{3})q^{4}+(3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(63))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(63)) \simeq$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(3))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(9))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(21))$$$$^{\oplus 2}$$