# Properties

 Label 731.2.f Level 731 Weight 2 Character orbit f Rep. character $$\chi_{731}(259,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 128 Newforms 4 Sturm bound 132 Trace bound 3

# Related objects

## Defining parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$17$$ Character field: $$\Q(i)$$ Newforms: $$4$$ Sturm bound: $$132$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(731, [\chi])$$.

Total New Old
Modular forms 136 128 8
Cusp forms 128 128 0
Eisenstein series 8 0 8

## Trace form

 $$128q - 132q^{4} - 8q^{5} + 8q^{6} + O(q^{10})$$ $$128q - 132q^{4} - 8q^{5} + 8q^{6} + 4q^{10} - 10q^{11} - 20q^{12} - 4q^{13} - 12q^{14} + 124q^{16} - 4q^{17} - 4q^{18} + 20q^{20} - 8q^{21} + 16q^{22} + 10q^{23} - 12q^{24} - 24q^{27} - 28q^{28} + 4q^{29} + 64q^{30} - 18q^{31} - 8q^{33} + 32q^{34} + 32q^{35} - 12q^{37} + 16q^{38} - 48q^{39} - 20q^{40} + 34q^{41} + 20q^{44} + 52q^{45} + 4q^{46} - 48q^{47} - 24q^{48} - 20q^{50} + 32q^{51} - 40q^{52} + 52q^{54} + 32q^{55} + 36q^{56} - 52q^{57} - 28q^{58} - 16q^{61} - 24q^{62} + 32q^{63} - 172q^{64} - 4q^{65} - 8q^{67} + 40q^{68} - 16q^{69} - 28q^{71} + 12q^{72} + 36q^{73} - 60q^{74} + 60q^{75} - 28q^{79} - 60q^{80} - 144q^{81} + 24q^{82} + 88q^{84} + 16q^{85} + 20q^{86} + 32q^{88} + 32q^{89} + 128q^{90} - 16q^{91} - 116q^{92} - 28q^{95} + 52q^{96} - 30q^{97} - 76q^{98} - 42q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(731, [\chi])$$ into irreducible Hecke orbits

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
731.2.f.a $$2$$ $$5.837$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$-4$$ $$q-iq^{2}+q^{4}+(-1-i)q^{5}+(-2+2i)q^{7}+\cdots$$
731.2.f.b $$2$$ $$5.837$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+iq^{2}+(2+2i)q^{3}+q^{4}+(-2+2i)q^{6}+\cdots$$
731.2.f.c $$56$$ $$5.837$$ None $$0$$ $$-4$$ $$-2$$ $$4$$
731.2.f.d $$68$$ $$5.837$$ None $$0$$ $$0$$ $$-4$$ $$0$$