# Properties

 Label 539.2.e Level $539$ Weight $2$ Character orbit 539.e Rep. character $\chi_{539}(67,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $68$ Newform subspaces $15$ Sturm bound $112$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 539.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$15$$ Sturm bound: $$112$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(539, [\chi])$$.

Total New Old
Modular forms 128 68 60
Cusp forms 96 68 28
Eisenstein series 32 0 32

## Trace form

 $$68 q + 2 q^{3} - 36 q^{4} + 4 q^{5} + 8 q^{6} + 12 q^{8} - 40 q^{9} + O(q^{10})$$ $$68 q + 2 q^{3} - 36 q^{4} + 4 q^{5} + 8 q^{6} + 12 q^{8} - 40 q^{9} - 6 q^{10} - 6 q^{12} + 16 q^{13} - 20 q^{15} - 44 q^{16} - 6 q^{17} + 6 q^{18} - 2 q^{19} - 40 q^{20} + 24 q^{23} + 8 q^{24} - 30 q^{25} + 10 q^{26} - 16 q^{27} + 16 q^{29} + 24 q^{30} + 6 q^{31} + 8 q^{32} + 4 q^{33} + 32 q^{34} + 52 q^{36} - 32 q^{37} + 8 q^{38} - 22 q^{39} - 8 q^{41} + 28 q^{43} - 4 q^{44} - 6 q^{45} - 26 q^{46} - 6 q^{47} + 16 q^{48} + 2 q^{52} - 2 q^{53} - 26 q^{54} - 16 q^{55} - 48 q^{57} - 18 q^{58} + 8 q^{59} + 68 q^{60} - 12 q^{61} - 20 q^{62} + 76 q^{64} + 18 q^{65} - 2 q^{66} - 36 q^{67} - 16 q^{68} - 36 q^{69} - 28 q^{71} + 94 q^{72} - 14 q^{73} + 36 q^{74} + 34 q^{75} + 96 q^{76} - 128 q^{78} - 16 q^{79} - 18 q^{80} - 2 q^{81} + 38 q^{82} + 52 q^{83} + 112 q^{85} + 6 q^{86} + 6 q^{87} + 24 q^{88} - 14 q^{89} + 52 q^{90} - 180 q^{92} - 32 q^{93} - 10 q^{94} + 28 q^{95} + 30 q^{96} - 108 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(539, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
539.2.e.a $2$ $4.304$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-2$$ $$2$$ $$0$$ $$q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots$$
539.2.e.b $2$ $4.304$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$2$$ $$-2$$ $$0$$ $$q-\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots$$
539.2.e.c $2$ $4.304$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-1$$ $$0$$ $$q+(-3+3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+\cdots$$
539.2.e.d $2$ $4.304$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-3$$ $$0$$ $$q+(-1+\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots$$
539.2.e.e $2$ $4.304$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$3$$ $$0$$ $$q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots$$
539.2.e.f $2$ $4.304$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$1$$ $$0$$ $$q+(3-3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots$$
539.2.e.g $2$ $4.304$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-1$$ $$1$$ $$0$$ $$q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots$$
539.2.e.h $2$ $4.304$ $$\Q(\sqrt{-3})$$ None $$2$$ $$1$$ $$-1$$ $$0$$ $$q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots$$
539.2.e.i $4$ $4.304$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$-2$$ $$4$$ $$0$$ $$q-\beta _{2}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+3\beta _{1}q^{4}+\cdots$$
539.2.e.j $4$ $4.304$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$2$$ $$-4$$ $$0$$ $$q-\beta _{2}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+3\beta _{1}q^{4}+\cdots$$
539.2.e.k $4$ $4.304$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
539.2.e.l $6$ $4.304$ 6.0.1783323.2 None $$0$$ $$-1$$ $$-2$$ $$0$$ $$q+(-\beta _{3}+\beta _{5})q^{2}-\beta _{1}q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots$$
539.2.e.m $6$ $4.304$ $$\Q(\zeta_{18})$$ None $$0$$ $$3$$ $$6$$ $$0$$ $$q+(-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+\cdots$$
539.2.e.n $8$ $4.304$ 8.0.6927565824.3 None $$-2$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\beta _{3}-\beta _{4}-\beta _{5})q^{2}+\beta _{2}q^{3}+\cdots$$
539.2.e.o $20$ $4.304$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$-2$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{8}q^{2}+(-\beta _{1}+\beta _{10})q^{3}+(-2\beta _{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(539, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(539, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(77, [\chi])$$$$^{\oplus 2}$$