# Properties

 Label 77.2.a Level $77$ Weight $2$ Character orbit 77.a Rep. character $\chi_{77}(1,\cdot)$ Character field $\Q$ Dimension $5$ Newform subspaces $4$ Sturm bound $16$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$77 = 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 77.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$16$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(77))$$.

Total New Old
Modular forms 10 5 5
Cusp forms 7 5 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.

$$7$$$$11$$FrickeDim
$$+$$$$+$$$+$$$1$$
$$+$$$$-$$$-$$$1$$
$$-$$$$+$$$-$$$3$$
Plus space$$+$$$$1$$
Minus space$$-$$$$4$$

## Trace form

 $$5 q + q^{2} + 2 q^{3} + q^{4} - 4 q^{5} - 8 q^{6} + q^{7} - 3 q^{8} + 11 q^{9} + O(q^{10})$$ $$5 q + q^{2} + 2 q^{3} + q^{4} - 4 q^{5} - 8 q^{6} + q^{7} - 3 q^{8} + 11 q^{9} - 2 q^{10} - 3 q^{11} + 8 q^{12} - 2 q^{13} - q^{14} - 2 q^{15} + 5 q^{16} - 2 q^{17} - 19 q^{18} - 14 q^{20} + 4 q^{21} + q^{22} - 10 q^{23} - 16 q^{24} - 3 q^{25} + 14 q^{26} + 2 q^{27} + 7 q^{28} + 6 q^{29} + 16 q^{30} + 6 q^{31} + 5 q^{32} + 2 q^{33} - 6 q^{34} + 2 q^{35} + 9 q^{36} - 8 q^{37} + 20 q^{38} + 8 q^{39} + 6 q^{40} - 10 q^{41} - 12 q^{42} + 28 q^{43} - 3 q^{44} - 26 q^{45} + 16 q^{46} + 8 q^{47} - 12 q^{48} + 5 q^{49} - q^{50} + 4 q^{51} + 18 q^{52} - 10 q^{53} - 24 q^{54} + 3 q^{56} + 4 q^{57} + 14 q^{58} - 2 q^{59} - 20 q^{60} - 22 q^{61} + 20 q^{62} - 3 q^{63} - 35 q^{64} - 20 q^{65} + 12 q^{66} + 30 q^{67} - 2 q^{68} - 14 q^{69} + 2 q^{70} - 14 q^{71} - 23 q^{72} - 2 q^{73} - 26 q^{74} + 12 q^{75} + 20 q^{76} - 3 q^{77} + 8 q^{78} + 4 q^{79} + 14 q^{80} + 21 q^{81} - 6 q^{82} + 28 q^{83} - 24 q^{85} + 12 q^{86} - 60 q^{87} - 3 q^{88} - 20 q^{89} + 38 q^{90} - 2 q^{91} - 4 q^{92} + 2 q^{93} - 20 q^{94} + 4 q^{95} + 40 q^{96} - 8 q^{97} + q^{98} - 9 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(77))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7 11
77.2.a.a $1$ $0.615$ $$\Q$$ None $$0$$ $$-3$$ $$-1$$ $$-1$$ $+$ $+$ $$q-3q^{3}-2q^{4}-q^{5}-q^{7}+6q^{9}-q^{11}+\cdots$$
77.2.a.b $1$ $0.615$ $$\Q$$ None $$0$$ $$1$$ $$3$$ $$1$$ $-$ $+$ $$q+q^{3}-2q^{4}+3q^{5}+q^{7}-2q^{9}-q^{11}+\cdots$$
77.2.a.c $1$ $0.615$ $$\Q$$ None $$1$$ $$2$$ $$-2$$ $$-1$$ $+$ $-$ $$q+q^{2}+2q^{3}-q^{4}-2q^{5}+2q^{6}-q^{7}+\cdots$$
77.2.a.d $2$ $0.615$ $$\Q(\sqrt{5})$$ None $$0$$ $$2$$ $$-4$$ $$2$$ $-$ $+$ $$q-\beta q^{2}+(1+\beta )q^{3}+3q^{4}-2q^{5}+(-5+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(77))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(77)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 2}$$