# Properties

 Label 63.4.a Level $63$ Weight $4$ Character orbit 63.a Rep. character $\chi_{63}(1,\cdot)$ Character field $\Q$ Dimension $7$ Newform subspaces $5$ Sturm bound $32$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$32$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(63))$$.

Total New Old
Modular forms 28 7 21
Cusp forms 20 7 13
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
$$+$$$$+$$$$+$$$$2$$
$$-$$$$+$$$$-$$$$2$$
$$-$$$$-$$$$+$$$$3$$
Plus space$$+$$$$5$$
Minus space$$-$$$$2$$

## Trace form

 $$7 q + 3 q^{2} + 41 q^{4} - 7 q^{7} + 51 q^{8} + O(q^{10})$$ $$7 q + 3 q^{2} + 41 q^{4} - 7 q^{7} + 51 q^{8} + 32 q^{10} - 12 q^{11} + 112 q^{13} + 63 q^{14} - 19 q^{16} - 174 q^{17} - 110 q^{19} - 60 q^{20} - 292 q^{22} - 12 q^{23} + 5 q^{25} - 144 q^{26} - 35 q^{28} + 270 q^{29} + 156 q^{31} + 651 q^{32} - 462 q^{34} + 168 q^{35} + 30 q^{37} - 1470 q^{38} - 456 q^{40} + 834 q^{41} + 632 q^{43} - 1320 q^{44} - 396 q^{46} - 876 q^{47} + 343 q^{49} + 1329 q^{50} + 188 q^{52} + 1506 q^{53} + 64 q^{55} + 567 q^{56} + 1586 q^{58} - 1602 q^{59} + 1684 q^{61} + 1836 q^{62} - 551 q^{64} - 672 q^{65} + 1300 q^{67} - 114 q^{68} - 392 q^{70} + 1212 q^{71} - 522 q^{73} + 534 q^{74} - 502 q^{76} + 672 q^{77} - 1840 q^{79} - 4140 q^{80} - 134 q^{82} - 906 q^{83} - 1728 q^{85} - 1752 q^{86} - 3180 q^{88} - 1650 q^{89} - 1036 q^{91} + 3528 q^{92} + 2436 q^{94} + 1104 q^{95} + 1146 q^{97} + 147 q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(63))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
63.4.a.a $1$ $3.717$ $$\Q$$ None $$-4$$ $$0$$ $$4$$ $$-7$$ $-$ $+$ $$q-4q^{2}+8q^{4}+4q^{5}-7q^{7}-2^{4}q^{10}+\cdots$$
63.4.a.b $1$ $3.717$ $$\Q$$ None $$1$$ $$0$$ $$-16$$ $$-7$$ $-$ $+$ $$q+q^{2}-7q^{4}-2^{4}q^{5}-7q^{7}-15q^{8}+\cdots$$
63.4.a.c $1$ $3.717$ $$\Q$$ None $$3$$ $$0$$ $$18$$ $$7$$ $-$ $-$ $$q+3q^{2}+q^{4}+18q^{5}+7q^{7}-21q^{8}+\cdots$$
63.4.a.d $2$ $3.717$ $$\Q(\sqrt{19})$$ None $$0$$ $$0$$ $$0$$ $$-14$$ $+$ $+$ $$q+\beta q^{2}+11q^{4}+2\beta q^{5}-7q^{7}+3\beta q^{8}+\cdots$$
63.4.a.e $2$ $3.717$ $$\Q(\sqrt{57})$$ None $$3$$ $$0$$ $$-6$$ $$14$$ $-$ $-$ $$q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(-2-2\beta )q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(63))$$ into lower level spaces

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