# 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

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

## 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

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