# Properties

 Label 169.2.b Level $169$ Weight $2$ Character orbit 169.b Rep. character $\chi_{169}(168,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $2$ Sturm bound $30$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$30$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(169, [\chi])$$.

Total New Old
Modular forms 22 18 4
Cusp forms 8 8 0
Eisenstein series 14 10 4

## Trace form

 $$8 q - 2 q^{4} - 4 q^{9} + O(q^{10})$$ $$8 q - 2 q^{4} - 4 q^{9} - 4 q^{10} - 4 q^{12} - 10 q^{14} - 6 q^{16} - 2 q^{17} + 6 q^{22} - 2 q^{23} + 14 q^{25} - 6 q^{27} + 4 q^{29} + 14 q^{30} + 8 q^{35} + 12 q^{36} - 12 q^{38} + 16 q^{42} - 10 q^{43} - 18 q^{48} + 22 q^{49} - 10 q^{51} - 4 q^{53} + 12 q^{55} + 8 q^{56} + 10 q^{61} - 14 q^{62} + 20 q^{64} + 10 q^{66} - 36 q^{68} - 12 q^{69} + 14 q^{74} - 22 q^{75} - 16 q^{77} - 2 q^{79} - 24 q^{81} + 10 q^{82} - 24 q^{87} - 30 q^{88} + 30 q^{90} + 12 q^{92} + 22 q^{94} - 18 q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(169, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
169.2.b.a $2$ $1.349$ $$\Q(\sqrt{-3})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q-\zeta_{6}q^{2}+2q^{3}-q^{4}+\zeta_{6}q^{5}-2\zeta_{6}q^{6}+\cdots$$
169.2.b.b $6$ $1.349$ 6.0.153664.1 None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(\beta _{3}+\beta _{5})q^{2}+(-1+\beta _{4})q^{3}+(1-2\beta _{2}+\cdots)q^{4}+\cdots$$