# Properties

 Label 37.2.a Level $37$ Weight $2$ Character orbit 37.a Rep. character $\chi_{37}(1,\cdot)$ Character field $\Q$ Dimension $2$ Newform subspaces $2$ Sturm bound $6$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$6$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(37))$$.

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.

$$37$$Dim
$$+$$$$1$$
$$-$$$$1$$

## Trace form

 $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{5} + 6 q^{6} - 2 q^{7} + 4 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{5} + 6 q^{6} - 2 q^{7} + 4 q^{9} + 4 q^{10} - 2 q^{11} - 8 q^{12} - 6 q^{13} + 2 q^{14} + 6 q^{15} + 6 q^{17} - 12 q^{18} + 2 q^{19} - 4 q^{20} + 2 q^{21} + 10 q^{22} + 8 q^{23} - 6 q^{25} + 4 q^{26} - 14 q^{27} - 12 q^{30} - 8 q^{31} + 8 q^{32} + 18 q^{33} + 2 q^{35} + 16 q^{36} + 2 q^{39} - 18 q^{41} - 6 q^{42} + 10 q^{43} - 16 q^{44} - 12 q^{45} - 4 q^{46} - 6 q^{47} + 16 q^{48} - 12 q^{49} + 2 q^{50} + 6 q^{51} + 4 q^{52} - 2 q^{53} + 18 q^{54} + 10 q^{55} + 2 q^{57} - 12 q^{58} + 20 q^{59} + 12 q^{60} + 8 q^{62} - 4 q^{63} - 16 q^{64} + 4 q^{65} - 30 q^{66} + 4 q^{67} - 12 q^{68} - 4 q^{70} - 6 q^{71} + 10 q^{73} + 2 q^{74} - 2 q^{75} - 4 q^{76} + 2 q^{77} - 12 q^{78} - 6 q^{79} + 8 q^{80} + 10 q^{81} + 18 q^{82} - 6 q^{83} + 8 q^{84} - 4 q^{86} - 24 q^{87} + 10 q^{89} + 24 q^{90} + 6 q^{91} - 8 q^{92} + 8 q^{93} + 18 q^{94} - 24 q^{96} + 12 q^{97} + 12 q^{98} - 36 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(37))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 37
37.2.a.a $1$ $0.295$ $$\Q$$ None $$-2$$ $$-3$$ $$-2$$ $$-1$$ $+$ $$q-2q^{2}-3q^{3}+2q^{4}-2q^{5}+6q^{6}+\cdots$$
37.2.a.b $1$ $0.295$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$-1$$ $-$ $$q+q^{3}-2q^{4}-q^{7}-2q^{9}+3q^{11}+\cdots$$