Properties

Label 6.8.a
Level 6
Weight 8
Character orbit a
Rep. character \(\chi_{6}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newforms 1
Sturm bound 8
Trace bound 0

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Defining parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 6.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(6))\).

Total New Old
Modular forms 9 1 8
Cusp forms 5 1 4
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.

\(2\)\(3\)FrickeDim.
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(0\)

Trace form

\(q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 114q^{5} \) \(\mathstrut +\mathstrut 216q^{6} \) \(\mathstrut -\mathstrut 1576q^{7} \) \(\mathstrut +\mathstrut 512q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 27q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 114q^{5} \) \(\mathstrut +\mathstrut 216q^{6} \) \(\mathstrut -\mathstrut 1576q^{7} \) \(\mathstrut +\mathstrut 512q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut -\mathstrut 912q^{10} \) \(\mathstrut +\mathstrut 7332q^{11} \) \(\mathstrut +\mathstrut 1728q^{12} \) \(\mathstrut -\mathstrut 3802q^{13} \) \(\mathstrut -\mathstrut 12608q^{14} \) \(\mathstrut -\mathstrut 3078q^{15} \) \(\mathstrut +\mathstrut 4096q^{16} \) \(\mathstrut -\mathstrut 6606q^{17} \) \(\mathstrut +\mathstrut 5832q^{18} \) \(\mathstrut +\mathstrut 24860q^{19} \) \(\mathstrut -\mathstrut 7296q^{20} \) \(\mathstrut -\mathstrut 42552q^{21} \) \(\mathstrut +\mathstrut 58656q^{22} \) \(\mathstrut +\mathstrut 41448q^{23} \) \(\mathstrut +\mathstrut 13824q^{24} \) \(\mathstrut -\mathstrut 65129q^{25} \) \(\mathstrut -\mathstrut 30416q^{26} \) \(\mathstrut +\mathstrut 19683q^{27} \) \(\mathstrut -\mathstrut 100864q^{28} \) \(\mathstrut -\mathstrut 41610q^{29} \) \(\mathstrut -\mathstrut 24624q^{30} \) \(\mathstrut +\mathstrut 33152q^{31} \) \(\mathstrut +\mathstrut 32768q^{32} \) \(\mathstrut +\mathstrut 197964q^{33} \) \(\mathstrut -\mathstrut 52848q^{34} \) \(\mathstrut +\mathstrut 179664q^{35} \) \(\mathstrut +\mathstrut 46656q^{36} \) \(\mathstrut -\mathstrut 36466q^{37} \) \(\mathstrut +\mathstrut 198880q^{38} \) \(\mathstrut -\mathstrut 102654q^{39} \) \(\mathstrut -\mathstrut 58368q^{40} \) \(\mathstrut -\mathstrut 639078q^{41} \) \(\mathstrut -\mathstrut 340416q^{42} \) \(\mathstrut -\mathstrut 156412q^{43} \) \(\mathstrut +\mathstrut 469248q^{44} \) \(\mathstrut -\mathstrut 83106q^{45} \) \(\mathstrut +\mathstrut 331584q^{46} \) \(\mathstrut -\mathstrut 433776q^{47} \) \(\mathstrut +\mathstrut 110592q^{48} \) \(\mathstrut +\mathstrut 1660233q^{49} \) \(\mathstrut -\mathstrut 521032q^{50} \) \(\mathstrut -\mathstrut 178362q^{51} \) \(\mathstrut -\mathstrut 243328q^{52} \) \(\mathstrut +\mathstrut 786078q^{53} \) \(\mathstrut +\mathstrut 157464q^{54} \) \(\mathstrut -\mathstrut 835848q^{55} \) \(\mathstrut -\mathstrut 806912q^{56} \) \(\mathstrut +\mathstrut 671220q^{57} \) \(\mathstrut -\mathstrut 332880q^{58} \) \(\mathstrut +\mathstrut 745140q^{59} \) \(\mathstrut -\mathstrut 196992q^{60} \) \(\mathstrut -\mathstrut 1660618q^{61} \) \(\mathstrut +\mathstrut 265216q^{62} \) \(\mathstrut -\mathstrut 1148904q^{63} \) \(\mathstrut +\mathstrut 262144q^{64} \) \(\mathstrut +\mathstrut 433428q^{65} \) \(\mathstrut +\mathstrut 1583712q^{66} \) \(\mathstrut -\mathstrut 3290836q^{67} \) \(\mathstrut -\mathstrut 422784q^{68} \) \(\mathstrut +\mathstrut 1119096q^{69} \) \(\mathstrut +\mathstrut 1437312q^{70} \) \(\mathstrut +\mathstrut 5716152q^{71} \) \(\mathstrut +\mathstrut 373248q^{72} \) \(\mathstrut +\mathstrut 2659898q^{73} \) \(\mathstrut -\mathstrut 291728q^{74} \) \(\mathstrut -\mathstrut 1758483q^{75} \) \(\mathstrut +\mathstrut 1591040q^{76} \) \(\mathstrut -\mathstrut 11555232q^{77} \) \(\mathstrut -\mathstrut 821232q^{78} \) \(\mathstrut +\mathstrut 3807440q^{79} \) \(\mathstrut -\mathstrut 466944q^{80} \) \(\mathstrut +\mathstrut 531441q^{81} \) \(\mathstrut -\mathstrut 5112624q^{82} \) \(\mathstrut +\mathstrut 2229468q^{83} \) \(\mathstrut -\mathstrut 2723328q^{84} \) \(\mathstrut +\mathstrut 753084q^{85} \) \(\mathstrut -\mathstrut 1251296q^{86} \) \(\mathstrut -\mathstrut 1123470q^{87} \) \(\mathstrut +\mathstrut 3753984q^{88} \) \(\mathstrut +\mathstrut 5991210q^{89} \) \(\mathstrut -\mathstrut 664848q^{90} \) \(\mathstrut +\mathstrut 5991952q^{91} \) \(\mathstrut +\mathstrut 2652672q^{92} \) \(\mathstrut +\mathstrut 895104q^{93} \) \(\mathstrut -\mathstrut 3470208q^{94} \) \(\mathstrut -\mathstrut 2834040q^{95} \) \(\mathstrut +\mathstrut 884736q^{96} \) \(\mathstrut -\mathstrut 4060126q^{97} \) \(\mathstrut +\mathstrut 13281864q^{98} \) \(\mathstrut +\mathstrut 5345028q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(6))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
6.8.a.a \(1\) \(1.874\) \(\Q\) None \(8\) \(27\) \(-114\) \(-1576\) \(-\) \(-\) \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-114q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(6)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)