# Properties

 Label 51.2.a Level $51$ Weight $2$ Character orbit 51.a Rep. character $\chi_{51}(1,\cdot)$ Character field $\Q$ Dimension $3$ Newform subspaces $2$ Sturm bound $12$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$51 = 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 51.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$12$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(51))$$.

Total New Old
Modular forms 8 3 5
Cusp forms 5 3 2
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.

$$3$$$$17$$FrickeDim
$$+$$$$-$$$$-$$$$2$$
$$-$$$$+$$$$-$$$$1$$
Plus space$$+$$$$0$$
Minus space$$-$$$$3$$

## Trace form

 $$3 q - q^{2} - q^{3} + 3 q^{4} + 6 q^{5} + q^{6} - 4 q^{7} - 9 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q - q^{2} - q^{3} + 3 q^{4} + 6 q^{5} + q^{6} - 4 q^{7} - 9 q^{8} + 3 q^{9} - 10 q^{10} - 4 q^{11} - 7 q^{12} + 4 q^{13} + 7 q^{16} + q^{17} - q^{18} + 2 q^{19} + 10 q^{20} - 4 q^{21} - 8 q^{22} + 9 q^{24} + 7 q^{25} + 6 q^{26} - q^{27} + 8 q^{28} + 6 q^{29} + 10 q^{30} - 9 q^{32} - 2 q^{33} - q^{34} - 12 q^{35} + 3 q^{36} - 6 q^{37} + 24 q^{38} - 6 q^{39} - 22 q^{40} - 6 q^{41} - 10 q^{43} + 12 q^{44} + 6 q^{45} - 4 q^{46} - 20 q^{47} + q^{48} - 5 q^{49} - 27 q^{50} - 3 q^{51} + 6 q^{52} + 2 q^{53} + q^{54} - 2 q^{55} - 4 q^{57} + 34 q^{58} + 12 q^{59} - 22 q^{60} + 18 q^{61} - 16 q^{62} - 4 q^{63} - q^{64} - 4 q^{65} + 8 q^{66} + 4 q^{67} + 7 q^{68} + 18 q^{69} + 16 q^{71} - 9 q^{72} - 6 q^{73} + 18 q^{74} + q^{75} - 16 q^{76} + 12 q^{77} - 6 q^{78} - 4 q^{79} + 42 q^{80} + 3 q^{81} + 10 q^{82} - 16 q^{83} + 8 q^{84} - 24 q^{86} + 6 q^{87} - 4 q^{88} + 6 q^{89} - 10 q^{90} + 4 q^{91} - 32 q^{92} + 4 q^{93} + 24 q^{94} - 24 q^{95} + 9 q^{96} - 30 q^{97} + 7 q^{98} - 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(51))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 17
51.2.a.a $1$ $0.407$ $$\Q$$ None $$0$$ $$1$$ $$3$$ $$-4$$ $-$ $+$ $$q+q^{3}-2q^{4}+3q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots$$
51.2.a.b $2$ $0.407$ $$\Q(\sqrt{17})$$ None $$-1$$ $$-2$$ $$3$$ $$0$$ $+$ $-$ $$q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+(1+\beta )q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(51))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(51)) \simeq$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(17))$$$$^{\oplus 2}$$