# Properties

 Label 115.2.a Level $115$ Weight $2$ Character orbit 115.a Rep. character $\chi_{115}(1,\cdot)$ Character field $\Q$ Dimension $7$ Newform subspaces $3$ Sturm bound $24$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 115.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$24$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(115))$$.

Total New Old
Modular forms 14 7 7
Cusp forms 11 7 4
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.

$$5$$$$23$$FrickeDim.
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$1$$
$$-$$$$+$$$$-$$$$4$$
Plus space$$+$$$$2$$
Minus space$$-$$$$5$$

## Trace form

 $$7 q + q^{2} - 4 q^{3} + 9 q^{4} + q^{5} + 2 q^{6} - 4 q^{7} + 3 q^{8} - q^{9} + O(q^{10})$$ $$7 q + q^{2} - 4 q^{3} + 9 q^{4} + q^{5} + 2 q^{6} - 4 q^{7} + 3 q^{8} - q^{9} + 3 q^{10} + 4 q^{11} - 22 q^{12} - 10 q^{13} - 12 q^{14} + 17 q^{16} - 2 q^{17} + 3 q^{18} - 4 q^{19} - q^{20} + 12 q^{21} - 8 q^{22} - 5 q^{23} - 24 q^{24} + 7 q^{25} + 12 q^{26} - 4 q^{27} - 8 q^{28} + 16 q^{29} - 4 q^{30} - 2 q^{31} - 3 q^{32} - 10 q^{34} - 2 q^{35} + 11 q^{36} + 2 q^{37} + 28 q^{38} + 8 q^{39} + 15 q^{40} + 8 q^{41} + 8 q^{42} - 12 q^{43} - 32 q^{44} + 13 q^{45} + 3 q^{46} + 16 q^{47} - 34 q^{48} + q^{49} + q^{50} - 4 q^{51} - 32 q^{52} + 18 q^{53} - 22 q^{54} + 4 q^{55} - 24 q^{56} + 40 q^{58} + 10 q^{59} - 16 q^{60} - 6 q^{61} - 34 q^{62} - 12 q^{63} + 23 q^{64} + 10 q^{65} + 36 q^{66} + 8 q^{67} + 26 q^{68} + 4 q^{69} - 12 q^{70} - 6 q^{71} + 51 q^{72} - 26 q^{73} + 34 q^{74} - 4 q^{75} - 28 q^{76} + 12 q^{77} + 26 q^{78} + 12 q^{79} - q^{80} - 17 q^{81} + 16 q^{82} - 16 q^{83} + 16 q^{84} - 8 q^{86} - 8 q^{87} + 12 q^{88} + 14 q^{89} + 3 q^{90} - 44 q^{91} - 5 q^{92} - 12 q^{93} + 22 q^{94} - 4 q^{95} - 46 q^{96} - 22 q^{97} + 17 q^{98} + 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(115))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 23
115.2.a.a $1$ $0.918$ $$\Q$$ None $$2$$ $$0$$ $$-1$$ $$1$$ $+$ $-$ $$q+2q^{2}+2q^{4}-q^{5}+q^{7}-3q^{9}-2q^{10}+\cdots$$
115.2.a.b $2$ $0.918$ $$\Q(\sqrt{5})$$ None $$-3$$ $$-2$$ $$-2$$ $$-2$$ $+$ $+$ $$q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}-q^{5}+\cdots$$
115.2.a.c $4$ $0.918$ 4.4.15317.1 None $$2$$ $$-2$$ $$4$$ $$-3$$ $-$ $+$ $$q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(115))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(115)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(23))$$$$^{\oplus 2}$$