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Properties

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

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Defining parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 19.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
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.

\(19\)Dim.
\(+\)\(3\)
\(-\)\(5\)

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(19))\) into irreducible Hecke orbits

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 \(9\) \(6\) \(90\) \(-190\) \(-\) \(q+(3-\beta _{1}+\beta _{2})q^{2}+(3+3\beta _{2})q^{3}+(23+\cdots)q^{4}+\cdots\)