# Properties

 Label 31.2.a Level $31$ Weight $2$ Character orbit 31.a Rep. character $\chi_{31}(1,\cdot)$ Character field $\Q$ Dimension $2$ Newform subspaces $1$ Sturm bound $5$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 31.a (trivial) Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$5$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(31))$$.

Total New Old
Modular forms 3 3 0
Cusp forms 2 2 0
Eisenstein series 1 1 0

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

$$31$$Dim.
$$-$$$$2$$

## Trace form

 $$2 q + q^{2} - 2 q^{3} - q^{4} + 2 q^{5} - 6 q^{6} - 4 q^{7} + 6 q^{9} + O(q^{10})$$ $$2 q + q^{2} - 2 q^{3} - q^{4} + 2 q^{5} - 6 q^{6} - 4 q^{7} + 6 q^{9} + q^{10} + 4 q^{11} - 4 q^{12} - 2 q^{13} + 3 q^{14} - 2 q^{15} - 3 q^{16} + 6 q^{17} + 13 q^{18} - q^{20} - 6 q^{21} + 2 q^{22} - 2 q^{23} + 10 q^{24} - 8 q^{25} - 6 q^{26} - 20 q^{27} + 7 q^{28} + 10 q^{29} - 6 q^{30} + 2 q^{31} - 9 q^{32} - 4 q^{33} - 2 q^{34} - 4 q^{35} + 7 q^{36} - 4 q^{37} - 5 q^{38} + 12 q^{39} + 14 q^{41} + 2 q^{42} - 2 q^{43} - 2 q^{44} + 6 q^{45} + 14 q^{46} - 4 q^{47} + 18 q^{48} + 4 q^{49} - 4 q^{50} + 4 q^{51} - 4 q^{52} - 12 q^{53} - 20 q^{54} + 4 q^{55} - 10 q^{56} + 10 q^{57} - 4 q^{60} - 6 q^{61} + q^{62} + 8 q^{63} + 4 q^{64} - 2 q^{65} - 12 q^{66} + 16 q^{67} - 8 q^{68} - 28 q^{69} + 3 q^{70} + 4 q^{71} - 20 q^{72} + 8 q^{73} - 2 q^{74} + 8 q^{75} - 5 q^{76} - 8 q^{77} + 16 q^{78} - 10 q^{79} - 3 q^{80} + 22 q^{81} + 7 q^{82} - 12 q^{83} + 8 q^{84} + 6 q^{85} + 4 q^{86} + 10 q^{89} + 13 q^{90} - 6 q^{91} + 16 q^{92} - 2 q^{93} + 8 q^{94} + 4 q^{96} - 14 q^{97} - 18 q^{98} + 12 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(31))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 31
31.2.a.a $2$ $0.248$ $$\Q(\sqrt{5})$$ None $$1$$ $$-2$$ $$2$$ $$-4$$ $-$ $$q+\beta q^{2}-2\beta q^{3}+(-1+\beta )q^{4}+q^{5}+\cdots$$