# Properties

 Label 88.4.a Level $88$ Weight $4$ Character orbit 88.a Rep. character $\chi_{88}(1,\cdot)$ Character field $\Q$ Dimension $7$ Newform subspaces $4$ Sturm bound $48$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$88 = 2^{3} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 88.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$48$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(88))$$.

Total New Old
Modular forms 40 7 33
Cusp forms 32 7 25
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.

$$2$$$$11$$FrickeDim
$$+$$$$+$$$+$$$1$$
$$+$$$$-$$$-$$$2$$
$$-$$$$+$$$-$$$1$$
$$-$$$$-$$$+$$$3$$
Plus space$$+$$$$4$$
Minus space$$-$$$$3$$

## Trace form

 $$7 q + 6 q^{3} + 4 q^{5} - 36 q^{7} + 73 q^{9} + O(q^{10})$$ $$7 q + 6 q^{3} + 4 q^{5} - 36 q^{7} + 73 q^{9} + 33 q^{11} - 18 q^{13} - 94 q^{15} + 86 q^{17} - 180 q^{19} + 316 q^{21} + 66 q^{23} + 287 q^{25} - 282 q^{27} - 362 q^{29} + 250 q^{31} - 66 q^{33} - 236 q^{35} + 192 q^{37} - 352 q^{39} + 250 q^{41} - 704 q^{43} - 54 q^{45} - 608 q^{47} - 9 q^{49} + 468 q^{51} - 934 q^{53} + 864 q^{57} + 1170 q^{59} - 674 q^{61} + 1904 q^{63} + 668 q^{65} - 718 q^{67} - 1274 q^{69} + 374 q^{71} + 290 q^{73} + 1500 q^{75} - 308 q^{77} + 820 q^{79} + 623 q^{81} + 568 q^{83} - 3360 q^{85} - 760 q^{87} + 1540 q^{89} - 320 q^{91} - 3994 q^{93} + 824 q^{95} - 1672 q^{97} + 891 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(88))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 11
88.4.a.a $1$ $5.192$ $$\Q$$ None $$0$$ $$-1$$ $$-7$$ $$-6$$ $-$ $+$ $$q-q^{3}-7q^{5}-6q^{7}-26q^{9}-11q^{11}+\cdots$$
88.4.a.b $1$ $5.192$ $$\Q$$ None $$0$$ $$7$$ $$9$$ $$2$$ $+$ $+$ $$q+7q^{3}+9q^{5}+2q^{7}+22q^{9}-11q^{11}+\cdots$$
88.4.a.c $2$ $5.192$ $$\Q(\sqrt{5})$$ None $$0$$ $$-2$$ $$-6$$ $$-56$$ $+$ $-$ $$q+(-1-\beta )q^{3}+(-3+4\beta )q^{5}+(-28+\cdots)q^{7}+\cdots$$
88.4.a.d $3$ $5.192$ 3.3.11109.1 None $$0$$ $$2$$ $$8$$ $$24$$ $-$ $-$ $$q+(1-\beta _{1})q^{3}+(3+\beta _{2})q^{5}+(8-\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(88))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(88)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(22))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(44))$$$$^{\oplus 2}$$