# Properties

 Label 55.2.a Level $55$ Weight $2$ Character orbit 55.a Rep. character $\chi_{55}(1,\cdot)$ Character field $\Q$ Dimension $3$ Newform subspaces $2$ Sturm bound $12$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 55.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$12$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(55))$$.

Total New Old
Modular forms 8 3 5
Cusp forms 5 3 2
Eisenstein series 3 0 3

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

$$5$$$$11$$FrickeDim
$$+$$$$-$$$-$$$2$$
$$-$$$$+$$$-$$$1$$
Plus space$$+$$$$0$$
Minus space$$-$$$$3$$

## Trace form

 $$3 q + 3 q^{2} + q^{4} - q^{5} - 8 q^{6} - 4 q^{7} + 3 q^{8} + 7 q^{9} + O(q^{10})$$ $$3 q + 3 q^{2} + q^{4} - q^{5} - 8 q^{6} - 4 q^{7} + 3 q^{8} + 7 q^{9} - q^{10} + q^{11} - 16 q^{12} - 6 q^{13} - 4 q^{14} + 5 q^{16} + 14 q^{17} + 7 q^{18} - 4 q^{19} - 3 q^{20} + q^{22} + 4 q^{23} - 8 q^{24} + 3 q^{25} + 2 q^{26} - 4 q^{28} + 10 q^{29} + 8 q^{30} - 8 q^{31} - q^{32} + 22 q^{34} + 4 q^{35} + 13 q^{36} - 6 q^{37} - 4 q^{38} - 16 q^{39} - 9 q^{40} + 14 q^{41} + 16 q^{42} - 8 q^{43} + 3 q^{44} - 13 q^{45} - 4 q^{46} - 12 q^{47} - 13 q^{49} + 3 q^{50} - 16 q^{51} + 6 q^{52} + 10 q^{53} - 16 q^{54} - 3 q^{55} - 12 q^{56} - 6 q^{58} - 4 q^{59} + 16 q^{60} - 6 q^{61} - 8 q^{62} - 20 q^{63} - 7 q^{64} + 10 q^{65} - 8 q^{66} - 8 q^{67} + 18 q^{68} + 16 q^{69} + 4 q^{70} + 8 q^{71} + 39 q^{72} + 6 q^{73} - 22 q^{74} + 4 q^{76} - 4 q^{77} + 16 q^{78} + 16 q^{79} - 7 q^{80} + 11 q^{81} + 14 q^{82} - 16 q^{83} + 32 q^{84} - 2 q^{85} - 8 q^{86} + 32 q^{87} + 9 q^{88} + 6 q^{89} - 13 q^{90} + 16 q^{91} - 20 q^{92} - 4 q^{94} - 4 q^{95} - 8 q^{96} + 6 q^{97} - 13 q^{98} + 13 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(55))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 11
55.2.a.a $1$ $0.439$ $$\Q$$ None $$1$$ $$0$$ $$1$$ $$0$$ $-$ $+$ $$q+q^{2}-q^{4}+q^{5}-3q^{8}-3q^{9}+q^{10}+\cdots$$
55.2.a.b $2$ $0.439$ $$\Q(\sqrt{2})$$ None $$2$$ $$0$$ $$-2$$ $$-4$$ $+$ $-$ $$q+(1+\beta )q^{2}-2\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(55))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(55)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 2}$$