# Properties

 Label 147.2.a Level $147$ Weight $2$ Character orbit 147.a Rep. character $\chi_{147}(1,\cdot)$ Character field $\Q$ Dimension $7$ Newform subspaces $5$ Sturm bound $37$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$37$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(147))$$.

Total New Old
Modular forms 26 7 19
Cusp forms 11 7 4
Eisenstein series 15 0 15

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

$$3$$$$7$$FrickeDim
$$+$$$$+$$$+$$$2$$
$$+$$$$-$$$-$$$2$$
$$-$$$$+$$$-$$$3$$
Plus space$$+$$$$2$$
Minus space$$-$$$$5$$

## Trace form

 $$7 q - q^{2} - q^{3} + 7 q^{4} + 2 q^{5} + q^{6} - 9 q^{8} + 7 q^{9} + O(q^{10})$$ $$7 q - q^{2} - q^{3} + 7 q^{4} + 2 q^{5} + q^{6} - 9 q^{8} + 7 q^{9} - 2 q^{10} - 8 q^{11} + q^{12} + 2 q^{13} + 2 q^{15} + 3 q^{16} + 6 q^{17} - q^{18} - 4 q^{19} - 2 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{24} + q^{25} - 2 q^{26} - q^{27} - 10 q^{29} - 6 q^{30} - 9 q^{32} - 4 q^{33} - 6 q^{34} + 7 q^{36} - 4 q^{37} + 4 q^{38} + 16 q^{39} + 6 q^{40} - 2 q^{41} + 6 q^{43} - 20 q^{44} + 2 q^{45} - 24 q^{46} + q^{48} + 25 q^{50} + 2 q^{51} - 2 q^{52} + 22 q^{53} + q^{54} + 8 q^{55} + 6 q^{57} + 18 q^{58} - 12 q^{59} - 14 q^{60} + 2 q^{61} - 37 q^{64} + 24 q^{65} + 4 q^{66} - 6 q^{67} - 6 q^{68} - 20 q^{71} - 9 q^{72} + 6 q^{73} + 22 q^{74} + q^{75} + 4 q^{76} - 18 q^{78} + 14 q^{79} - 2 q^{80} + 7 q^{81} + 2 q^{82} + 12 q^{83} + 4 q^{85} + 56 q^{86} + 2 q^{87} + 36 q^{88} + 14 q^{89} - 2 q^{90} + 56 q^{92} + 2 q^{93} - 28 q^{95} + 5 q^{96} - 18 q^{97} - 8 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(147))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
147.2.a.a $1$ $1.174$ $$\Q$$ None $$-1$$ $$-1$$ $$2$$ $$0$$ $+$ $-$ $$q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+3q^{8}+\cdots$$
147.2.a.b $1$ $1.174$ $$\Q$$ None $$2$$ $$-1$$ $$2$$ $$0$$ $+$ $-$ $$q+2q^{2}-q^{3}+2q^{4}+2q^{5}-2q^{6}+\cdots$$
147.2.a.c $1$ $1.174$ $$\Q$$ None $$2$$ $$1$$ $$-2$$ $$0$$ $-$ $+$ $$q+2q^{2}+q^{3}+2q^{4}-2q^{5}+2q^{6}+\cdots$$
147.2.a.d $2$ $1.174$ $$\Q(\sqrt{2})$$ None $$-2$$ $$-2$$ $$-4$$ $$0$$ $+$ $+$ $$q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots$$
147.2.a.e $2$ $1.174$ $$\Q(\sqrt{2})$$ None $$-2$$ $$2$$ $$4$$ $$0$$ $-$ $+$ $$q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(2+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(147))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(147)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(21))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(49))$$$$^{\oplus 2}$$