# Properties

 Label 48.4.a Level $48$ Weight $4$ Character orbit 48.a Rep. character $\chi_{48}(1,\cdot)$ Character field $\Q$ Dimension $3$ Newform subspaces $3$ Sturm bound $32$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 48.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$32$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(48))$$.

Total New Old
Modular forms 30 3 27
Cusp forms 18 3 15
Eisenstein series 12 0 12

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

$$2$$$$3$$FrickeDim
$$+$$$$+$$$+$$$1$$
$$-$$$$+$$$-$$$1$$
$$-$$$$-$$$+$$$1$$
Plus space$$+$$$$2$$
Minus space$$-$$$$1$$

## Trace form

 $$3 q - 3 q^{3} + 2 q^{5} + 32 q^{7} + 27 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 2 q^{5} + 32 q^{7} + 27 q^{9} - 20 q^{11} - 46 q^{13} + 30 q^{15} - 26 q^{17} - 12 q^{19} - 248 q^{23} + 181 q^{25} - 27 q^{27} - 342 q^{29} + 24 q^{31} - 12 q^{33} + 576 q^{35} + 58 q^{37} + 366 q^{39} + 414 q^{41} - 404 q^{43} + 18 q^{45} - 576 q^{47} - 133 q^{49} - 678 q^{51} + 482 q^{53} + 968 q^{55} - 84 q^{57} + 748 q^{59} - 334 q^{61} + 288 q^{63} - 628 q^{65} - 780 q^{67} - 264 q^{69} - 1288 q^{71} - 754 q^{73} - 1077 q^{75} + 768 q^{77} + 40 q^{79} + 243 q^{81} + 3188 q^{83} + 68 q^{85} + 1206 q^{87} - 66 q^{89} - 1088 q^{91} + 456 q^{93} - 3208 q^{95} + 966 q^{97} - 180 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(48))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
48.4.a.a $1$ $2.832$ $$\Q$$ None $$0$$ $$-3$$ $$-18$$ $$-8$$ $-$ $+$ $$q-3q^{3}-18q^{5}-8q^{7}+9q^{9}-6^{2}q^{11}+\cdots$$
48.4.a.b $1$ $2.832$ $$\Q$$ None $$0$$ $$-3$$ $$14$$ $$24$$ $+$ $+$ $$q-3q^{3}+14q^{5}+24q^{7}+9q^{9}+28q^{11}+\cdots$$
48.4.a.c $1$ $2.832$ $$\Q$$ None $$0$$ $$3$$ $$6$$ $$16$$ $-$ $-$ $$q+3q^{3}+6q^{5}+2^{4}q^{7}+9q^{9}-12q^{11}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(48))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(48)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(12))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(24))$$$$^{\oplus 2}$$