# Properties

 Label 147.4.a Level $147$ Weight $4$ Character orbit 147.a Rep. character $\chi_{147}(1,\cdot)$ Character field $\Q$ Dimension $20$ Newform subspaces $13$ Sturm bound $74$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial) Character field: $$\Q$$ Newform subspaces: $$13$$ Sturm bound: $$74$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(147))$$.

Total New Old
Modular forms 64 20 44
Cusp forms 48 20 28
Eisenstein series 16 0 16

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.
$$+$$$$+$$$$+$$$$5$$
$$+$$$$-$$$$-$$$$5$$
$$-$$$$+$$$$-$$$$3$$
$$-$$$$-$$$$+$$$$7$$
Plus space$$+$$$$12$$
Minus space$$-$$$$8$$

## Trace form

 $$20q + 4q^{2} + 60q^{4} + 16q^{5} + 12q^{6} + 48q^{8} + 180q^{9} + O(q^{10})$$ $$20q + 4q^{2} + 60q^{4} + 16q^{5} + 12q^{6} + 48q^{8} + 180q^{9} + 28q^{10} - 40q^{11} - 24q^{12} + 80q^{13} - 84q^{15} + 332q^{16} - 120q^{17} + 36q^{18} - 40q^{19} - 172q^{20} - 132q^{22} - 40q^{23} + 324q^{24} + 728q^{25} - 172q^{26} + 376q^{29} + 288q^{30} + 168q^{31} + 168q^{32} + 96q^{33} - 276q^{34} + 540q^{36} - 508q^{37} - 1360q^{38} - 756q^{39} + 924q^{40} + 1016q^{41} - 828q^{43} - 1072q^{44} + 144q^{45} - 520q^{46} - 552q^{47} - 816q^{48} - 292q^{50} + 456q^{51} + 1420q^{52} - 784q^{53} + 108q^{54} - 952q^{55} - 276q^{57} - 124q^{58} - 792q^{59} - 204q^{60} - 592q^{61} + 1824q^{62} + 156q^{64} - 168q^{65} + 1896q^{66} + 148q^{67} - 492q^{68} - 144q^{69} + 608q^{71} + 432q^{72} + 72q^{73} + 1196q^{74} + 624q^{75} + 832q^{76} - 1692q^{78} + 1936q^{79} - 3484q^{80} + 1620q^{81} - 1492q^{82} - 2208q^{83} + 3904q^{85} - 9836q^{86} + 1032q^{87} - 5484q^{88} - 920q^{89} + 252q^{90} - 2744q^{92} + 1788q^{93} + 1344q^{94} - 416q^{95} + 204q^{96} + 744q^{97} - 360q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(147))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 3 7
147.4.a.a $$1$$ $$8.673$$ $$\Q$$ None $$-3$$ $$-3$$ $$3$$ $$0$$ $$+$$ $$-$$ $$q-3q^{2}-3q^{3}+q^{4}+3q^{5}+9q^{6}+\cdots$$
147.4.a.b $$1$$ $$8.673$$ $$\Q$$ None $$-3$$ $$3$$ $$-3$$ $$0$$ $$-$$ $$+$$ $$q-3q^{2}+3q^{3}+q^{4}-3q^{5}-9q^{6}+\cdots$$
147.4.a.c $$1$$ $$8.673$$ $$\Q$$ None $$-3$$ $$3$$ $$18$$ $$0$$ $$-$$ $$-$$ $$q-3q^{2}+3q^{3}+q^{4}+18q^{5}-9q^{6}+\cdots$$
147.4.a.d $$1$$ $$8.673$$ $$\Q$$ None $$-1$$ $$-3$$ $$12$$ $$0$$ $$+$$ $$-$$ $$q-q^{2}-3q^{3}-7q^{4}+12q^{5}+3q^{6}+\cdots$$
147.4.a.e $$1$$ $$8.673$$ $$\Q$$ None $$-1$$ $$3$$ $$-12$$ $$0$$ $$-$$ $$-$$ $$q-q^{2}+3q^{3}-7q^{4}-12q^{5}-3q^{6}+\cdots$$
147.4.a.f $$1$$ $$8.673$$ $$\Q$$ None $$4$$ $$-3$$ $$-18$$ $$0$$ $$+$$ $$-$$ $$q+4q^{2}-3q^{3}+8q^{4}-18q^{5}-12q^{6}+\cdots$$
147.4.a.g $$1$$ $$8.673$$ $$\Q$$ None $$4$$ $$3$$ $$4$$ $$0$$ $$-$$ $$-$$ $$q+4q^{2}+3q^{3}+8q^{4}+4q^{5}+12q^{6}+\cdots$$
147.4.a.h $$1$$ $$8.673$$ $$\Q$$ None $$4$$ $$3$$ $$18$$ $$0$$ $$-$$ $$-$$ $$q+4q^{2}+3q^{3}+8q^{4}+18q^{5}+12q^{6}+\cdots$$
147.4.a.i $$2$$ $$8.673$$ $$\Q(\sqrt{57})$$ None $$-3$$ $$-6$$ $$-6$$ $$0$$ $$+$$ $$-$$ $$q+(-1-\beta )q^{2}-3q^{3}+(7+3\beta )q^{4}+\cdots$$
147.4.a.j $$2$$ $$8.673$$ $$\Q(\sqrt{2})$$ None $$2$$ $$-6$$ $$20$$ $$0$$ $$+$$ $$+$$ $$q+(1+\beta )q^{2}-3q^{3}+(-5+2\beta )q^{4}+\cdots$$
147.4.a.k $$2$$ $$8.673$$ $$\Q(\sqrt{2})$$ None $$2$$ $$6$$ $$-20$$ $$0$$ $$-$$ $$+$$ $$q+(1+\beta )q^{2}+3q^{3}+(-5+2\beta )q^{4}+\cdots$$
147.4.a.l $$3$$ $$8.673$$ 3.3.57516.1 None $$1$$ $$-9$$ $$11$$ $$0$$ $$+$$ $$+$$ $$q+\beta _{1}q^{2}-3q^{3}+(8+\beta _{1}+\beta _{2})q^{4}+\cdots$$
147.4.a.m $$3$$ $$8.673$$ 3.3.57516.1 None $$1$$ $$9$$ $$-11$$ $$0$$ $$-$$ $$-$$ $$q+\beta _{1}q^{2}+3q^{3}+(8+\beta _{1}+\beta _{2})q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(147))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(147)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(21))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(49))$$$$^{\oplus 2}$$