# Properties

 Label 165.2.a Level $165$ Weight $2$ Character orbit 165.a Rep. character $\chi_{165}(1,\cdot)$ Character field $\Q$ Dimension $7$ Newform subspaces $3$ Sturm bound $48$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$48$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(165))$$.

Total New Old
Modular forms 28 7 21
Cusp forms 21 7 14
Eisenstein series 7 0 7

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

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

## Trace form

 $$7 q - 3 q^{2} + 3 q^{3} + 9 q^{4} - q^{5} + q^{6} - 15 q^{8} + 7 q^{9} + O(q^{10})$$ $$7 q - 3 q^{2} + 3 q^{3} + 9 q^{4} - q^{5} + q^{6} - 15 q^{8} + 7 q^{9} + q^{10} - q^{11} + 5 q^{12} + 2 q^{13} - 16 q^{14} + 3 q^{15} + 9 q^{16} - 10 q^{17} - 3 q^{18} + 4 q^{19} + q^{20} + 8 q^{21} + q^{22} - 8 q^{23} - 3 q^{24} + 7 q^{25} + 14 q^{26} + 3 q^{27} + 8 q^{28} - 14 q^{29} - 3 q^{30} - 23 q^{32} + 3 q^{33} - 30 q^{34} + 9 q^{36} + 10 q^{37} + 4 q^{38} + 2 q^{39} - 3 q^{40} - 10 q^{41} - 8 q^{42} - 4 q^{43} + q^{44} - q^{45} + 8 q^{46} - 16 q^{47} - 3 q^{48} + 15 q^{49} - 3 q^{50} + 6 q^{51} - 58 q^{52} - 22 q^{53} + q^{54} + 7 q^{55} + 20 q^{57} + 6 q^{58} + 4 q^{59} + 5 q^{60} - 14 q^{61} + 40 q^{62} + q^{64} - 6 q^{65} - 3 q^{66} + 12 q^{67} + 50 q^{68} + 8 q^{69} - 8 q^{70} + 24 q^{71} - 15 q^{72} - 10 q^{73} - 2 q^{74} + 3 q^{75} + 28 q^{76} - 18 q^{78} - 8 q^{79} + 17 q^{80} + 7 q^{81} + 2 q^{82} + 4 q^{83} - 16 q^{84} + 6 q^{85} + 36 q^{86} - 6 q^{87} - 3 q^{88} - 26 q^{89} + q^{90} - 16 q^{91} + 8 q^{92} + 16 q^{94} + 12 q^{95} - 35 q^{96} + 14 q^{97} + 61 q^{98} - q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(165))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 11
165.2.a.a $2$ $1.318$ $$\Q(\sqrt{2})$$ None $$-2$$ $$-2$$ $$-2$$ $$-4$$ $+$ $+$ $+$ $$q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}-q^{5}+\cdots$$
165.2.a.b $2$ $1.318$ $$\Q(\sqrt{3})$$ None $$0$$ $$2$$ $$-2$$ $$4$$ $-$ $+$ $+$ $$q+\beta q^{2}+q^{3}+q^{4}-q^{5}+\beta q^{6}+2q^{7}+\cdots$$
165.2.a.c $3$ $1.318$ 3.3.148.1 None $$-1$$ $$3$$ $$3$$ $$0$$ $-$ $-$ $-$ $$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(165))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(165)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(33))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(55))$$$$^{\oplus 2}$$