# Properties

 Label 33.6.a Level $33$ Weight $6$ Character orbit 33.a Rep. character $\chi_{33}(1,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $5$ Sturm bound $24$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 33.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$24$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(33))$$.

Total New Old
Modular forms 22 8 14
Cusp forms 18 8 10
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.

$$3$$$$11$$FrickeDim
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$3$$
$$-$$$$+$$$$-$$$$2$$
$$-$$$$-$$$$+$$$$1$$
Plus space$$+$$$$3$$
Minus space$$-$$$$5$$

## Trace form

 $$8 q + 8 q^{2} - 18 q^{3} + 108 q^{4} + 32 q^{5} + 180 q^{6} - 36 q^{7} - 108 q^{8} + 648 q^{9} + O(q^{10})$$ $$8 q + 8 q^{2} - 18 q^{3} + 108 q^{4} + 32 q^{5} + 180 q^{6} - 36 q^{7} - 108 q^{8} + 648 q^{9} + 544 q^{10} - 864 q^{12} - 480 q^{13} + 1508 q^{14} - 900 q^{15} + 948 q^{16} - 4612 q^{17} + 648 q^{18} - 916 q^{19} + 5788 q^{20} + 2484 q^{21} - 968 q^{22} - 2480 q^{23} + 540 q^{24} + 15816 q^{25} - 14308 q^{26} - 1458 q^{27} - 24176 q^{28} - 6180 q^{29} - 2736 q^{30} + 1752 q^{31} + 13612 q^{32} - 2178 q^{33} - 3392 q^{34} + 9992 q^{35} + 8748 q^{36} - 27904 q^{37} + 8304 q^{38} - 10476 q^{39} + 10968 q^{40} + 39188 q^{41} - 15372 q^{42} + 30356 q^{43} - 4840 q^{44} + 2592 q^{45} + 54896 q^{46} + 68152 q^{47} + 12528 q^{48} + 12744 q^{49} - 137624 q^{50} + 2448 q^{51} - 20696 q^{52} - 16776 q^{53} + 14580 q^{54} - 24200 q^{55} + 5508 q^{56} + 32220 q^{57} + 7320 q^{58} + 33280 q^{59} - 20052 q^{60} - 51208 q^{61} + 49592 q^{62} - 2916 q^{63} - 180532 q^{64} - 128992 q^{65} - 17424 q^{66} + 58648 q^{67} - 111848 q^{68} - 10440 q^{69} + 217864 q^{70} + 35064 q^{71} - 8748 q^{72} + 30456 q^{73} + 23304 q^{74} - 98766 q^{75} - 117712 q^{76} + 29524 q^{77} + 123084 q^{78} - 128372 q^{79} + 59164 q^{80} + 52488 q^{81} + 86560 q^{82} + 51576 q^{83} + 41328 q^{84} + 87160 q^{85} + 220296 q^{86} - 31320 q^{87} + 68244 q^{88} + 40368 q^{89} + 44064 q^{90} + 42536 q^{91} + 210140 q^{92} - 118152 q^{93} - 128896 q^{94} - 234168 q^{95} - 44460 q^{96} - 267584 q^{97} - 228120 q^{98} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(33))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
33.6.a.a $1$ $5.293$ $$\Q$$ None $$-2$$ $$-9$$ $$46$$ $$148$$ $+$ $-$ $$q-2q^{2}-9q^{3}-28q^{4}+46q^{5}+18q^{6}+\cdots$$
33.6.a.b $1$ $5.293$ $$\Q$$ None $$1$$ $$9$$ $$-92$$ $$-26$$ $-$ $-$ $$q+q^{2}+9q^{3}-31q^{4}-92q^{5}+9q^{6}+\cdots$$
33.6.a.c $2$ $5.293$ $$\Q(\sqrt{177})$$ None $$-5$$ $$-18$$ $$58$$ $$-286$$ $+$ $+$ $$q+(-2-\beta )q^{2}-9q^{3}+(2^{4}+5\beta )q^{4}+\cdots$$
33.6.a.d $2$ $5.293$ $$\Q(\sqrt{313})$$ None $$1$$ $$-18$$ $$-38$$ $$-18$$ $+$ $-$ $$q+\beta q^{2}-9q^{3}+(46+\beta )q^{4}+(-24+\cdots)q^{5}+\cdots$$
33.6.a.e $2$ $5.293$ $$\Q(\sqrt{33})$$ None $$13$$ $$18$$ $$58$$ $$146$$ $-$ $+$ $$q+(7-\beta )q^{2}+9q^{3}+(5^{2}-13\beta )q^{4}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(33))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(33)) \simeq$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(3))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(11))$$$$^{\oplus 2}$$