# Properties

 Label 77.4.a Level $77$ Weight $4$ Character orbit 77.a Rep. character $\chi_{77}(1,\cdot)$ Character field $\Q$ Dimension $16$ Newform subspaces $5$ Sturm bound $32$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$77 = 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 77.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$32$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(77))$$.

Total New Old
Modular forms 26 16 10
Cusp forms 22 16 6
Eisenstein series 4 0 4

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
$$+$$$$+$$$$+$$$$4$$
$$+$$$$-$$$$-$$$$4$$
$$-$$$$+$$$$-$$$$2$$
$$-$$$$-$$$$+$$$$6$$
Plus space$$+$$$$10$$
Minus space$$-$$$$6$$

## Trace form

 $$16 q - 4 q^{2} + 4 q^{3} + 96 q^{4} - 24 q^{5} + 36 q^{6} - 60 q^{8} + 148 q^{9} + O(q^{10})$$ $$16 q - 4 q^{2} + 4 q^{3} + 96 q^{4} - 24 q^{5} + 36 q^{6} - 60 q^{8} + 148 q^{9} - 112 q^{10} + 44 q^{11} - 92 q^{12} - 168 q^{13} + 56 q^{14} + 132 q^{15} + 280 q^{16} + 152 q^{17} - 60 q^{18} + 88 q^{19} - 28 q^{20} + 44 q^{22} + 212 q^{23} - 196 q^{24} + 164 q^{25} - 548 q^{26} + 76 q^{27} - 360 q^{29} - 224 q^{30} - 512 q^{31} - 84 q^{32} - 176 q^{33} - 320 q^{34} - 56 q^{35} + 632 q^{36} + 676 q^{37} - 1036 q^{38} - 480 q^{39} - 384 q^{40} - 184 q^{41} + 28 q^{42} + 296 q^{43} + 440 q^{44} + 836 q^{45} - 760 q^{46} - 436 q^{47} - 1660 q^{48} + 784 q^{49} + 60 q^{50} - 416 q^{51} + 188 q^{52} + 200 q^{53} + 3464 q^{54} - 396 q^{55} + 672 q^{56} + 2672 q^{57} + 376 q^{58} - 2260 q^{59} - 3560 q^{60} + 160 q^{61} + 196 q^{62} - 952 q^{63} + 1848 q^{64} - 104 q^{65} + 436 q^{67} + 1812 q^{68} + 884 q^{69} + 532 q^{70} + 3420 q^{71} - 1412 q^{72} - 360 q^{73} - 200 q^{74} + 2064 q^{75} - 1512 q^{76} + 308 q^{77} - 4584 q^{78} - 784 q^{79} + 2540 q^{80} + 1664 q^{81} + 3304 q^{82} + 1992 q^{83} + 980 q^{84} - 1528 q^{85} + 360 q^{86} + 768 q^{87} + 132 q^{88} - 2508 q^{89} - 11544 q^{90} - 112 q^{91} + 1768 q^{92} + 2940 q^{93} + 1564 q^{94} + 576 q^{95} - 572 q^{96} - 828 q^{97} - 196 q^{98} + 968 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(77))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7 11
77.4.a.a $1$ $4.543$ $$\Q$$ None $$3$$ $$4$$ $$12$$ $$7$$ $-$ $-$ $$q+3q^{2}+4q^{3}+q^{4}+12q^{5}+12q^{6}+\cdots$$
77.4.a.b $2$ $4.543$ $$\Q(\sqrt{2})$$ None $$-2$$ $$-4$$ $$-4$$ $$14$$ $-$ $+$ $$q+(-1+\beta )q^{2}-2q^{3}+(1-2\beta )q^{4}+\cdots$$
77.4.a.c $4$ $4.543$ 4.4.509800.1 None $$-4$$ $$-12$$ $$-18$$ $$-28$$ $+$ $-$ $$q+(-1+\beta _{2})q^{2}+(-3+\beta _{3})q^{3}+(6+\cdots)q^{4}+\cdots$$
77.4.a.d $4$ $4.543$ 4.4.522072.1 None $$-2$$ $$14$$ $$10$$ $$-28$$ $+$ $+$ $$q+\beta _{2}q^{2}+(4+\beta _{1})q^{3}+(6-\beta _{2}-2\beta _{3})q^{4}+\cdots$$
77.4.a.e $5$ $4.543$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$1$$ $$2$$ $$-24$$ $$35$$ $-$ $-$ $$q+\beta _{1}q^{2}-\beta _{3}q^{3}+(9+\beta _{1}+\beta _{2})q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(77))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(77)) \simeq$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 2}$$