# Properties

 Label 333.2.a Level $333$ Weight $2$ Character orbit 333.a Rep. character $\chi_{333}(1,\cdot)$ Character field $\Q$ Dimension $15$ Newform subspaces $7$ Sturm bound $76$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$333 = 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 333.a (trivial) Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$76$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(333))$$.

Total New Old
Modular forms 42 15 27
Cusp forms 35 15 20
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.

$$3$$$$37$$FrickeDim
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$4$$
$$-$$$$+$$$$-$$$$5$$
$$-$$$$-$$$$+$$$$4$$
Plus space$$+$$$$6$$
Minus space$$-$$$$9$$

## Trace form

 $$15 q - q^{2} + 11 q^{4} - 2 q^{7} - 3 q^{8} + O(q^{10})$$ $$15 q - q^{2} + 11 q^{4} - 2 q^{7} - 3 q^{8} - 10 q^{10} - 2 q^{11} + 10 q^{14} + 3 q^{16} - 8 q^{17} - 2 q^{19} + 26 q^{20} - 2 q^{22} + 4 q^{23} + 5 q^{25} + 18 q^{26} - 8 q^{28} - 14 q^{29} - 7 q^{32} - 2 q^{34} + 2 q^{35} + q^{37} - 8 q^{38} - 22 q^{40} - 12 q^{41} - 10 q^{43} - 20 q^{44} + 4 q^{46} + 22 q^{47} + 25 q^{49} - 27 q^{50} - 2 q^{52} - 14 q^{55} + 32 q^{56} - 10 q^{58} - 16 q^{59} + 14 q^{61} - 12 q^{62} + 11 q^{64} - 40 q^{65} - 16 q^{67} - 38 q^{68} + 20 q^{70} + 6 q^{71} - 20 q^{73} - 5 q^{74} + 30 q^{77} + 26 q^{79} + 30 q^{80} + 8 q^{82} + 26 q^{83} - 12 q^{85} - 20 q^{86} - 20 q^{88} - 32 q^{89} - 2 q^{91} + 48 q^{92} - 30 q^{94} + 56 q^{95} - 2 q^{97} - 87 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(333))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 37
333.2.a.a $1$ $2.659$ $$\Q$$ None $$-1$$ $$0$$ $$2$$ $$-4$$ $+$ $+$ $$q-q^{2}-q^{4}+2q^{5}-4q^{7}+3q^{8}-2q^{10}+\cdots$$
333.2.a.b $1$ $2.659$ $$\Q$$ None $$0$$ $$0$$ $$0$$ $$-1$$ $-$ $-$ $$q-2q^{4}-q^{7}-3q^{11}-4q^{13}+4q^{16}+\cdots$$
333.2.a.c $1$ $2.659$ $$\Q$$ None $$1$$ $$0$$ $$-2$$ $$-4$$ $+$ $+$ $$q+q^{2}-q^{4}-2q^{5}-4q^{7}-3q^{8}-2q^{10}+\cdots$$
333.2.a.d $1$ $2.659$ $$\Q$$ None $$2$$ $$0$$ $$2$$ $$-1$$ $-$ $+$ $$q+2q^{2}+2q^{4}+2q^{5}-q^{7}+4q^{10}+\cdots$$
333.2.a.e $3$ $2.659$ 3.3.148.1 None $$-3$$ $$0$$ $$-4$$ $$-4$$ $-$ $-$ $$q+(-1-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots$$
333.2.a.f $4$ $2.659$ 4.4.27648.1 None $$0$$ $$0$$ $$0$$ $$8$$ $+$ $-$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+2q^{7}+\cdots$$
333.2.a.g $4$ $2.659$ 4.4.6224.1 None $$0$$ $$0$$ $$2$$ $$4$$ $-$ $+$ $$q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(333))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(333)) \simeq$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(37))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(111))$$$$^{\oplus 2}$$