# Properties

 Label 19.6.a Level $19$ Weight $6$ Character orbit 19.a Rep. character $\chi_{19}(1,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $4$ Sturm bound $10$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 19.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$10$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(19))$$.

Total New Old
Modular forms 10 8 2
Cusp forms 8 8 0
Eisenstein series 2 0 2

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

$$19$$Dim.
$$+$$$$3$$
$$-$$$$5$$

## Trace form

 $$8q - 6q^{2} + 2q^{3} + 122q^{4} - 13q^{5} + 106q^{6} - 37q^{7} - 504q^{8} + 712q^{9} + O(q^{10})$$ $$8q - 6q^{2} + 2q^{3} + 122q^{4} - 13q^{5} + 106q^{6} - 37q^{7} - 504q^{8} + 712q^{9} + 368q^{10} - 401q^{11} - 360q^{12} - 1014q^{13} + 1228q^{14} - 2566q^{15} + 890q^{16} + 2453q^{17} - 4974q^{18} + 722q^{19} - 4568q^{20} + 3286q^{21} + 1648q^{22} + 4768q^{23} + 1902q^{24} + 8911q^{25} + 2138q^{26} - 5092q^{27} - 10674q^{28} + 1520q^{29} + 22316q^{30} + 11324q^{31} - 6144q^{32} - 12994q^{33} - 37036q^{34} - 507q^{35} - 788q^{36} - 844q^{37} + 4332q^{38} - 16320q^{39} + 21900q^{40} - 12712q^{41} - 45574q^{42} + 36739q^{43} + 10976q^{44} + 2003q^{45} - 908q^{46} + 35505q^{47} + 40080q^{48} + 12535q^{49} + 20054q^{50} - 75506q^{51} - 13060q^{52} - 50462q^{53} + 138310q^{54} - 37683q^{55} + 66564q^{56} + 6498q^{57} + 24962q^{58} - 2186q^{59} - 249524q^{60} - 123553q^{61} + 159748q^{62} - 53493q^{63} - 103918q^{64} + 132744q^{65} + 162092q^{66} + 49600q^{67} + 226978q^{68} + 93336q^{69} - 180756q^{70} - 80058q^{71} - 335160q^{72} + 86233q^{73} - 237200q^{74} + 90676q^{75} + 28880q^{76} + 24835q^{77} + 130900q^{78} - 307768q^{79} - 366860q^{80} + 439324q^{81} + 31628q^{82} - 116560q^{83} + 268452q^{84} + 68709q^{85} + 40956q^{86} + 311484q^{87} + 438504q^{88} - 230842q^{89} - 238876q^{90} - 405932q^{91} + 463610q^{92} - 331008q^{93} + 14328q^{94} + 108661q^{95} - 241378q^{96} + 243218q^{97} - 508862q^{98} - 432977q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(19))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 19
19.6.a.a $$1$$ $$3.047$$ $$\Q$$ None $$-6$$ $$4$$ $$54$$ $$248$$ $$-$$ $$q-6q^{2}+4q^{3}+4q^{4}+54q^{5}-24q^{6}+\cdots$$
19.6.a.b $$1$$ $$3.047$$ $$\Q$$ None $$-2$$ $$-1$$ $$-24$$ $$-167$$ $$+$$ $$q-2q^{2}-q^{3}-28q^{4}-24q^{5}+2q^{6}+\cdots$$
19.6.a.c $$2$$ $$3.047$$ $$\Q(\sqrt{177})$$ None $$-7$$ $$-7$$ $$-133$$ $$72$$ $$+$$ $$q+(-3-\beta )q^{2}+(-5+3\beta )q^{3}+(21+\cdots)q^{4}+\cdots$$
19.6.a.d $$4$$ $$3.047$$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$9$$ $$6$$ $$90$$ $$-190$$ $$-$$ $$q+(3-\beta _{1}+\beta _{2})q^{2}+(3+3\beta _{2})q^{3}+(23+\cdots)q^{4}+\cdots$$