# Properties

 Label 775.2.a Level $775$ Weight $2$ Character orbit 775.a Rep. character $\chi_{775}(1,\cdot)$ Character field $\Q$ Dimension $47$ Newform subspaces $12$ Sturm bound $160$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 775.a (trivial) Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$160$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(775))$$.

Total New Old
Modular forms 86 47 39
Cusp forms 75 47 28
Eisenstein series 11 0 11

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

$$5$$$$31$$FrickeDim.
$$+$$$$+$$$$+$$$$10$$
$$+$$$$-$$$$-$$$$13$$
$$-$$$$+$$$$-$$$$15$$
$$-$$$$-$$$$+$$$$9$$
Plus space$$+$$$$19$$
Minus space$$-$$$$28$$

## Trace form

 $$47 q + 2 q^{2} + 2 q^{3} + 48 q^{4} - 2 q^{6} + 3 q^{8} + 43 q^{9} + O(q^{10})$$ $$47 q + 2 q^{2} + 2 q^{3} + 48 q^{4} - 2 q^{6} + 3 q^{8} + 43 q^{9} - 4 q^{11} + 8 q^{12} - 12 q^{13} - 3 q^{14} + 50 q^{16} - 8 q^{17} - 2 q^{18} - 4 q^{19} - 26 q^{21} + 22 q^{22} - 10 q^{23} - 14 q^{24} + 4 q^{26} + 8 q^{27} + 9 q^{28} + 12 q^{29} - 3 q^{31} + 28 q^{32} + 16 q^{33} + 20 q^{34} + 40 q^{36} + 2 q^{37} + 21 q^{38} - 32 q^{39} - 4 q^{41} + 26 q^{42} - 18 q^{43} - 10 q^{44} + 26 q^{46} + 28 q^{47} - 2 q^{48} + 13 q^{49} + 8 q^{51} - 38 q^{52} - 14 q^{53} - 4 q^{54} + 38 q^{56} - 18 q^{57} - 18 q^{58} + 12 q^{59} - 32 q^{61} + 4 q^{62} + 8 q^{63} + 39 q^{64} - 8 q^{67} - 34 q^{68} - 12 q^{69} - 20 q^{71} + 11 q^{72} - 22 q^{73} + 32 q^{74} - 51 q^{76} - 20 q^{77} - 40 q^{78} + 6 q^{79} - 29 q^{81} - 61 q^{82} + 44 q^{83} - 156 q^{84} - 8 q^{86} - 12 q^{87} + 36 q^{88} - 44 q^{89} - 30 q^{91} - 20 q^{92} + 2 q^{93} - 104 q^{94} - 140 q^{96} + 8 q^{97} - 35 q^{98} - 8 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(775))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 31
775.2.a.a $1$ $6.188$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$0$$ $+$ $+$ $$q+q^{3}-2q^{4}-2q^{9}-4q^{11}-2q^{12}+\cdots$$
775.2.a.b $1$ $6.188$ $$\Q$$ None $$1$$ $$-2$$ $$0$$ $$-4$$ $+$ $-$ $$q+q^{2}-2q^{3}-q^{4}-2q^{6}-4q^{7}-3q^{8}+\cdots$$
775.2.a.c $1$ $6.188$ $$\Q$$ None $$2$$ $$1$$ $$0$$ $$2$$ $+$ $-$ $$q+2q^{2}+q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots$$
775.2.a.d $2$ $6.188$ $$\Q(\sqrt{5})$$ None $$-1$$ $$2$$ $$0$$ $$4$$ $+$ $-$ $$q-\beta q^{2}+2\beta q^{3}+(-1+\beta )q^{4}+(-2+\cdots)q^{6}+\cdots$$
775.2.a.e $4$ $6.188$ 4.4.8468.1 None $$-1$$ $$-1$$ $$0$$ $$-2$$ $+$ $+$ $$q+\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots$$
775.2.a.f $4$ $6.188$ $$\Q(\zeta_{24})^+$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $-$ $$q+(\beta _{1}+\beta _{3})q^{2}-\beta _{1}q^{3}+(-1-\beta _{2}+\cdots)q^{6}+\cdots$$
775.2.a.g $4$ $6.188$ 4.4.20308.1 None $$1$$ $$1$$ $$0$$ $$0$$ $+$ $-$ $$q+\beta _{1}q^{2}-\beta _{3}q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots$$
775.2.a.h $5$ $6.188$ 5.5.144209.1 None $$-4$$ $$-3$$ $$0$$ $$-2$$ $+$ $+$ $$q+(-1+\beta _{3})q^{2}+(-1+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots$$
775.2.a.i $5$ $6.188$ 5.5.205225.1 None $$-4$$ $$-1$$ $$0$$ $$-6$$ $-$ $-$ $$q+(-1+\beta _{1})q^{2}+\beta _{3}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
775.2.a.j $5$ $6.188$ 5.5.205225.1 None $$4$$ $$1$$ $$0$$ $$6$$ $+$ $-$ $$q+(1-\beta _{1})q^{2}-\beta _{3}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
775.2.a.k $5$ $6.188$ 5.5.144209.1 None $$4$$ $$3$$ $$0$$ $$2$$ $-$ $+$ $$q+(1-\beta _{3})q^{2}+(1-\beta _{2})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots$$
775.2.a.l $10$ $6.188$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $+$ $$q+\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{4}+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(775))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(775)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(31))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(155))$$$$^{\oplus 2}$$