# Properties

 Label 39.2.a Level $39$ Weight $2$ Character orbit 39.a Rep. character $\chi_{39}(1,\cdot)$ Character field $\Q$ Dimension $3$ Newform subspaces $2$ Sturm bound $9$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 39.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$9$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(39))$$.

Total New Old
Modular forms 6 3 3
Cusp forms 3 3 0
Eisenstein series 3 0 3

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

$$3$$$$13$$FrickeDim
$$+$$$$-$$$-$$$1$$
$$-$$$$+$$$-$$$2$$
Plus space$$+$$$$0$$
Minus space$$-$$$$3$$

## Trace form

 $$3 q - q^{2} + q^{3} + q^{4} + 2 q^{5} - 3 q^{6} - 4 q^{7} - 9 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q - q^{2} + q^{3} + q^{4} + 2 q^{5} - 3 q^{6} - 4 q^{7} - 9 q^{8} + 3 q^{9} - 6 q^{10} + 3 q^{12} - q^{13} + 4 q^{14} - 2 q^{15} + 5 q^{16} + 6 q^{17} - q^{18} + 14 q^{20} + 4 q^{21} + 8 q^{22} - 8 q^{23} - 3 q^{24} + 5 q^{25} + 3 q^{26} + q^{27} - 12 q^{28} - 6 q^{29} - 10 q^{30} - 4 q^{31} + 11 q^{32} - 8 q^{33} + 14 q^{34} - 24 q^{35} + q^{36} - 6 q^{37} - 8 q^{38} - 3 q^{39} - 14 q^{40} + 22 q^{41} + 12 q^{42} - 4 q^{43} - 8 q^{44} + 2 q^{45} + 8 q^{46} - 12 q^{47} + 7 q^{48} + 11 q^{49} - 7 q^{50} + 2 q^{51} - 3 q^{52} + 2 q^{53} - 3 q^{54} + 8 q^{55} + 20 q^{56} - 14 q^{58} + 16 q^{59} + 18 q^{60} + 2 q^{61} + 20 q^{62} - 4 q^{63} - 7 q^{64} + 2 q^{65} - 30 q^{68} - 8 q^{69} + 8 q^{70} + 4 q^{71} - 9 q^{72} + 14 q^{73} - 14 q^{74} + 7 q^{75} + 16 q^{76} - 16 q^{77} + q^{78} + 8 q^{79} - 2 q^{80} + 3 q^{81} - 18 q^{82} - 20 q^{84} - 28 q^{85} - 36 q^{86} + 14 q^{87} + 22 q^{89} - 6 q^{90} - 4 q^{91} - 8 q^{92} - 12 q^{93} - 4 q^{94} + 16 q^{95} + q^{96} + 6 q^{97} + 7 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(39))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 13
39.2.a.a $1$ $0.311$ $$\Q$$ None $$1$$ $$-1$$ $$2$$ $$-4$$ $+$ $-$ $$q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-4q^{7}+\cdots$$
39.2.a.b $2$ $0.311$ $$\Q(\sqrt{2})$$ None $$-2$$ $$2$$ $$0$$ $$0$$ $-$ $+$ $$q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-2\beta q^{5}+\cdots$$