# Properties

 Label 66.4.a Level $66$ Weight $4$ Character orbit 66.a Rep. character $\chi_{66}(1,\cdot)$ Character field $\Q$ Dimension $4$ Newform subspaces $3$ Sturm bound $48$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$66 = 2 \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 66.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$48$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(66))$$.

Total New Old
Modular forms 40 4 36
Cusp forms 32 4 28
Eisenstein series 8 0 8

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

$$2$$$$3$$$$11$$FrickeDim
$$+$$$$-$$$$-$$$+$$$1$$
$$-$$$$+$$$$-$$$+$$$1$$
$$-$$$$-$$$$+$$$+$$$2$$
Plus space$$+$$$$4$$
Minus space$$-$$$$0$$

## Trace form

 $$4 q + 4 q^{2} + 6 q^{3} + 16 q^{4} + 20 q^{5} + 28 q^{7} + 16 q^{8} + 36 q^{9} + O(q^{10})$$ $$4 q + 4 q^{2} + 6 q^{3} + 16 q^{4} + 20 q^{5} + 28 q^{7} + 16 q^{8} + 36 q^{9} + 40 q^{10} + 24 q^{12} + 104 q^{13} + 64 q^{16} - 60 q^{17} + 36 q^{18} - 148 q^{19} + 80 q^{20} - 12 q^{21} - 44 q^{22} - 412 q^{23} - 156 q^{25} - 112 q^{26} + 54 q^{27} + 112 q^{28} - 340 q^{29} - 64 q^{31} + 64 q^{32} - 66 q^{33} - 240 q^{34} - 432 q^{35} + 144 q^{36} + 128 q^{37} - 520 q^{38} + 252 q^{39} + 160 q^{40} - 44 q^{41} - 192 q^{42} - 60 q^{43} + 180 q^{45} - 320 q^{46} + 260 q^{47} + 96 q^{48} + 828 q^{49} + 188 q^{50} - 120 q^{51} + 416 q^{52} + 540 q^{53} + 420 q^{57} + 208 q^{58} + 1080 q^{59} + 1344 q^{61} - 64 q^{62} + 252 q^{63} + 256 q^{64} + 1528 q^{65} - 264 q^{66} - 472 q^{67} - 240 q^{68} - 732 q^{69} - 864 q^{70} + 604 q^{71} + 144 q^{72} - 80 q^{73} + 680 q^{74} - 318 q^{75} - 592 q^{76} + 352 q^{77} - 456 q^{78} + 508 q^{79} + 320 q^{80} + 324 q^{81} - 544 q^{82} - 752 q^{83} - 48 q^{84} - 888 q^{85} + 88 q^{86} - 2328 q^{87} - 176 q^{88} - 248 q^{89} + 360 q^{90} - 2808 q^{91} - 1648 q^{92} + 864 q^{93} - 464 q^{94} - 3680 q^{95} - 2304 q^{97} + 2244 q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(66))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 11
66.4.a.a $1$ $3.894$ $$\Q$$ None $$-2$$ $$3$$ $$0$$ $$14$$ $+$ $-$ $-$ $$q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+14q^{7}+\cdots$$
66.4.a.b $1$ $3.894$ $$\Q$$ None $$2$$ $$-3$$ $$10$$ $$16$$ $-$ $+$ $-$ $$q+2q^{2}-3q^{3}+4q^{4}+10q^{5}-6q^{6}+\cdots$$
66.4.a.c $2$ $3.894$ $$\Q(\sqrt{97})$$ None $$4$$ $$6$$ $$10$$ $$-2$$ $-$ $-$ $+$ $$q+2q^{2}+3q^{3}+4q^{4}+(5-\beta )q^{5}+6q^{6}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(66))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(66)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(6))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(22))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(33))$$$$^{\oplus 2}$$