# Properties

 Label 361.2.a Level $361$ Weight $2$ Character orbit 361.a Rep. character $\chi_{361}(1,\cdot)$ Character field $\Q$ Dimension $20$ Newform subspaces $9$ Sturm bound $63$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 361.a (trivial) Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$63$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(361))$$.

Total New Old
Modular forms 41 37 4
Cusp forms 22 20 2
Eisenstein series 19 17 2

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

$$19$$Dim
$$+$$$$8$$
$$-$$$$12$$

## Trace form

 $$20 q + 2 q^{3} + 14 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$20 q + 2 q^{3} + 14 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 4 q^{12} + 4 q^{13} + 6 q^{15} - 10 q^{16} + 4 q^{17} - 8 q^{20} - 2 q^{21} + 4 q^{23} + 8 q^{24} - 18 q^{25} + 8 q^{26} - 4 q^{27} + 2 q^{28} - 6 q^{29} + 8 q^{30} + 4 q^{31} + 6 q^{33} + 8 q^{35} - 14 q^{36} - 2 q^{37} - 20 q^{39} + 6 q^{41} - 30 q^{42} + 6 q^{43} + 18 q^{44} + 2 q^{45} + 10 q^{47} + 8 q^{48} - 22 q^{49} - 6 q^{51} - 8 q^{52} - 12 q^{53} + 14 q^{54} - 14 q^{58} + 6 q^{59} - 12 q^{60} + 6 q^{61} + 6 q^{62} + 12 q^{63} - 28 q^{64} + 12 q^{65} - 42 q^{66} + 4 q^{67} + 4 q^{68} - 6 q^{71} + 16 q^{73} + 12 q^{74} + 8 q^{75} - 4 q^{77} - 8 q^{79} - 30 q^{80} - 36 q^{81} + 10 q^{82} - 4 q^{83} + 4 q^{84} + 24 q^{85} + 22 q^{87} - 12 q^{89} - 4 q^{91} - 56 q^{92} + 6 q^{96} - 8 q^{97} + 18 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(361))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 19
361.2.a.a $1$ $2.883$ $$\Q$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$-1$$ $$3$$ $+$ $$q-2q^{4}-q^{5}+3q^{7}-3q^{9}-5q^{11}+\cdots$$
361.2.a.b $1$ $2.883$ $$\Q$$ None $$0$$ $$2$$ $$3$$ $$-1$$ $-$ $$q+2q^{3}-2q^{4}+3q^{5}-q^{7}+q^{9}+3q^{11}+\cdots$$
361.2.a.c $2$ $2.883$ $$\Q(\sqrt{5})$$ None $$-1$$ $$-3$$ $$2$$ $$6$$ $-$ $$q-\beta q^{2}+(-2+\beta )q^{3}+(-1+\beta )q^{4}+\cdots$$
361.2.a.d $2$ $2.883$ $$\Q(\sqrt{5})$$ None $$0$$ $$-4$$ $$1$$ $$-2$$ $-$ $$q+(1-2\beta )q^{2}-2q^{3}+3q^{4}+(1-\beta )q^{5}+\cdots$$
361.2.a.e $2$ $2.883$ $$\Q(\sqrt{5})$$ None $$0$$ $$4$$ $$1$$ $$-2$$ $-$ $$q+(1-2\beta )q^{2}+2q^{3}+3q^{4}+\beta q^{5}+\cdots$$
361.2.a.f $2$ $2.883$ $$\Q(\sqrt{5})$$ None $$1$$ $$3$$ $$2$$ $$6$$ $-$ $$q+\beta q^{2}+(2-\beta )q^{3}+(-1+\beta )q^{4}+2\beta q^{5}+\cdots$$
361.2.a.g $3$ $2.883$ $$\Q(\zeta_{18})^+$$ None $$-3$$ $$-3$$ $$-3$$ $$0$$ $+$ $$q+(-1+\beta _{1})q^{2}+(-1+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots$$
361.2.a.h $3$ $2.883$ $$\Q(\zeta_{18})^+$$ None $$3$$ $$3$$ $$-3$$ $$0$$ $-$ $$q+(1-\beta _{1})q^{2}+(1-\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots$$
361.2.a.i $4$ $2.883$ $$\Q(\zeta_{20})^+$$ None $$0$$ $$0$$ $$-4$$ $$-8$$ $+$ $$q+\beta _{1}q^{2}-\beta _{1}q^{3}+(1+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(361))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(361)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(19))$$$$^{\oplus 2}$$