# Properties

 Label 177.2.a Level $177$ Weight $2$ Character orbit 177.a Rep. character $\chi_{177}(1,\cdot)$ Character field $\Q$ Dimension $9$ Newform subspaces $4$ Sturm bound $40$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 177.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$40$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(177))$$.

Total New Old
Modular forms 22 9 13
Cusp forms 19 9 10
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.

$$3$$$$59$$FrickeDim.
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$3$$
$$-$$$$+$$$$-$$$$2$$
$$-$$$$-$$$$+$$$$2$$
Plus space$$+$$$$4$$
Minus space$$-$$$$5$$

## Trace form

 $$9q - 3q^{2} - q^{3} + 3q^{4} - 6q^{5} - q^{6} - 4q^{7} - 3q^{8} + 9q^{9} + O(q^{10})$$ $$9q - 3q^{2} - q^{3} + 3q^{4} - 6q^{5} - q^{6} - 4q^{7} - 3q^{8} + 9q^{9} + 10q^{10} + q^{12} - 6q^{13} + 20q^{14} - 2q^{15} + 3q^{16} - 2q^{17} - 3q^{18} - 8q^{19} - 14q^{20} - 8q^{21} - 16q^{22} - 8q^{23} - 9q^{24} - q^{25} + 2q^{26} - q^{27} + 4q^{28} - 2q^{29} + 10q^{30} + 4q^{31} - 19q^{32} + 4q^{33} + 2q^{34} + 16q^{35} + 3q^{36} - 22q^{37} + 32q^{38} + 2q^{39} + 26q^{40} + 2q^{41} - 8q^{42} + 12q^{43} - 6q^{45} - 4q^{46} + 4q^{47} + 17q^{48} + 29q^{49} - 37q^{50} - 2q^{51} - 46q^{52} + 10q^{53} - q^{54} + 4q^{55} + 20q^{56} - 12q^{57} + 18q^{58} + q^{59} - 6q^{60} - 2q^{61} - 24q^{62} - 4q^{63} - 9q^{64} + 8q^{65} - 4q^{66} + 20q^{67} + 2q^{68} + 4q^{69} - 4q^{70} + 36q^{71} - 3q^{72} + 10q^{73} - 26q^{74} + q^{75} - 36q^{76} + 12q^{77} + 26q^{78} - 34q^{80} + 9q^{81} + 54q^{82} - 16q^{84} - 40q^{85} + 4q^{86} - 10q^{87} - 36q^{88} - 30q^{89} + 10q^{90} + 8q^{91} - 4q^{92} - 20q^{93} + 24q^{94} - 12q^{95} - 29q^{96} + 6q^{97} + 17q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(177))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 3 59
177.2.a.a $$2$$ $$1.413$$ $$\Q(\sqrt{5})$$ None $$-3$$ $$2$$ $$-6$$ $$-7$$ $$-$$ $$-$$ $$q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}-3q^{5}+\cdots$$
177.2.a.b $$2$$ $$1.413$$ $$\Q(\sqrt{5})$$ None $$-1$$ $$-2$$ $$0$$ $$-7$$ $$+$$ $$+$$ $$q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots$$
177.2.a.c $$2$$ $$1.413$$ $$\Q(\sqrt{5})$$ None $$1$$ $$2$$ $$2$$ $$1$$ $$-$$ $$+$$ $$q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots$$
177.2.a.d $$3$$ $$1.413$$ 3.3.229.1 None $$0$$ $$-3$$ $$-2$$ $$9$$ $$+$$ $$-$$ $$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(177))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(177)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(59))$$$$^{\oplus 2}$$