# Properties

 Label 169.2.a Level $169$ Weight $2$ Character orbit 169.a Rep. character $\chi_{169}(1,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $3$ Sturm bound $30$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$30$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(169))$$.

Total New Old
Modular forms 21 19 2
Cusp forms 8 8 0
Eisenstein series 13 11 2

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

$$13$$Dim
$$+$$$$3$$
$$-$$$$5$$

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(169))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 13
169.2.a.a $2$ $1.349$ $$\Q(\sqrt{3})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $-$ $$q+\beta q^{2}+2q^{3}+q^{4}-\beta q^{5}+2\beta q^{6}+\cdots$$
169.2.a.b $3$ $1.349$ $$\Q(\zeta_{14})^+$$ None $$-2$$ $$-2$$ $$-4$$ $$-3$$ $+$ $$q+(-1-\beta _{2})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots$$
169.2.a.c $3$ $1.349$ $$\Q(\zeta_{14})^+$$ None $$2$$ $$-2$$ $$4$$ $$3$$ $-$ $$q+(1-\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots$$