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Properties

Label 6.8.a
Level $6$
Weight $8$
Character orbit 6.a
Rep. character $\chi_{6}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
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: \(\Q\)
Newform subspaces: \( 1 \)
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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(6))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
6.8.a.a 6.a 1.a $1$ $1.874$ \(\Q\) None \(8\) \(27\) \(-114\) \(-1576\) $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-114q^{5}+\cdots\)

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

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