Properties

Label 6.10.a
Level 6
Weight 10
Character orbit a
Rep. character \(\chi_{6}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newforms 1
Sturm bound 10
Trace bound 0

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Defining parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 6.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(10\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(6))\).

Total New Old
Modular forms 11 1 10
Cusp forms 7 1 6
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\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\(q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 81q^{3} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut +\mathstrut 2694q^{5} \) \(\mathstrut -\mathstrut 1296q^{6} \) \(\mathstrut -\mathstrut 3544q^{7} \) \(\mathstrut -\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 81q^{3} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut +\mathstrut 2694q^{5} \) \(\mathstrut -\mathstrut 1296q^{6} \) \(\mathstrut -\mathstrut 3544q^{7} \) \(\mathstrut -\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut -\mathstrut 43104q^{10} \) \(\mathstrut +\mathstrut 29580q^{11} \) \(\mathstrut +\mathstrut 20736q^{12} \) \(\mathstrut -\mathstrut 44818q^{13} \) \(\mathstrut +\mathstrut 56704q^{14} \) \(\mathstrut +\mathstrut 218214q^{15} \) \(\mathstrut +\mathstrut 65536q^{16} \) \(\mathstrut -\mathstrut 101934q^{17} \) \(\mathstrut -\mathstrut 104976q^{18} \) \(\mathstrut -\mathstrut 895084q^{19} \) \(\mathstrut +\mathstrut 689664q^{20} \) \(\mathstrut -\mathstrut 287064q^{21} \) \(\mathstrut -\mathstrut 473280q^{22} \) \(\mathstrut -\mathstrut 1113000q^{23} \) \(\mathstrut -\mathstrut 331776q^{24} \) \(\mathstrut +\mathstrut 5304511q^{25} \) \(\mathstrut +\mathstrut 717088q^{26} \) \(\mathstrut +\mathstrut 531441q^{27} \) \(\mathstrut -\mathstrut 907264q^{28} \) \(\mathstrut -\mathstrut 2357346q^{29} \) \(\mathstrut -\mathstrut 3491424q^{30} \) \(\mathstrut +\mathstrut 175808q^{31} \) \(\mathstrut -\mathstrut 1048576q^{32} \) \(\mathstrut +\mathstrut 2395980q^{33} \) \(\mathstrut +\mathstrut 1630944q^{34} \) \(\mathstrut -\mathstrut 9547536q^{35} \) \(\mathstrut +\mathstrut 1679616q^{36} \) \(\mathstrut -\mathstrut 2919418q^{37} \) \(\mathstrut +\mathstrut 14321344q^{38} \) \(\mathstrut -\mathstrut 3630258q^{39} \) \(\mathstrut -\mathstrut 11034624q^{40} \) \(\mathstrut +\mathstrut 26218794q^{41} \) \(\mathstrut +\mathstrut 4593024q^{42} \) \(\mathstrut -\mathstrut 18762964q^{43} \) \(\mathstrut +\mathstrut 7572480q^{44} \) \(\mathstrut +\mathstrut 17675334q^{45} \) \(\mathstrut +\mathstrut 17808000q^{46} \) \(\mathstrut -\mathstrut 20966160q^{47} \) \(\mathstrut +\mathstrut 5308416q^{48} \) \(\mathstrut -\mathstrut 27793671q^{49} \) \(\mathstrut -\mathstrut 84872176q^{50} \) \(\mathstrut -\mathstrut 8256654q^{51} \) \(\mathstrut -\mathstrut 11473408q^{52} \) \(\mathstrut +\mathstrut 57251574q^{53} \) \(\mathstrut -\mathstrut 8503056q^{54} \) \(\mathstrut +\mathstrut 79688520q^{55} \) \(\mathstrut +\mathstrut 14516224q^{56} \) \(\mathstrut -\mathstrut 72501804q^{57} \) \(\mathstrut +\mathstrut 37717536q^{58} \) \(\mathstrut +\mathstrut 33587580q^{59} \) \(\mathstrut +\mathstrut 55862784q^{60} \) \(\mathstrut +\mathstrut 82260830q^{61} \) \(\mathstrut -\mathstrut 2812928q^{62} \) \(\mathstrut -\mathstrut 23252184q^{63} \) \(\mathstrut +\mathstrut 16777216q^{64} \) \(\mathstrut -\mathstrut 120739692q^{65} \) \(\mathstrut -\mathstrut 38335680q^{66} \) \(\mathstrut -\mathstrut 188455804q^{67} \) \(\mathstrut -\mathstrut 26095104q^{68} \) \(\mathstrut -\mathstrut 90153000q^{69} \) \(\mathstrut +\mathstrut 152760576q^{70} \) \(\mathstrut +\mathstrut 80924040q^{71} \) \(\mathstrut -\mathstrut 26873856q^{72} \) \(\mathstrut -\mathstrut 236140918q^{73} \) \(\mathstrut +\mathstrut 46710688q^{74} \) \(\mathstrut +\mathstrut 429665391q^{75} \) \(\mathstrut -\mathstrut 229141504q^{76} \) \(\mathstrut -\mathstrut 104831520q^{77} \) \(\mathstrut +\mathstrut 58084128q^{78} \) \(\mathstrut +\mathstrut 526909808q^{79} \) \(\mathstrut +\mathstrut 176553984q^{80} \) \(\mathstrut +\mathstrut 43046721q^{81} \) \(\mathstrut -\mathstrut 419500704q^{82} \) \(\mathstrut +\mathstrut 18346452q^{83} \) \(\mathstrut -\mathstrut 73488384q^{84} \) \(\mathstrut -\mathstrut 274610196q^{85} \) \(\mathstrut +\mathstrut 300207424q^{86} \) \(\mathstrut -\mathstrut 190945026q^{87} \) \(\mathstrut -\mathstrut 121159680q^{88} \) \(\mathstrut +\mathstrut 690643098q^{89} \) \(\mathstrut -\mathstrut 282805344q^{90} \) \(\mathstrut +\mathstrut 158834992q^{91} \) \(\mathstrut -\mathstrut 284928000q^{92} \) \(\mathstrut +\mathstrut 14240448q^{93} \) \(\mathstrut +\mathstrut 335458560q^{94} \) \(\mathstrut -\mathstrut 2411356296q^{95} \) \(\mathstrut -\mathstrut 84934656q^{96} \) \(\mathstrut -\mathstrut 438251038q^{97} \) \(\mathstrut +\mathstrut 444698736q^{98} \) \(\mathstrut +\mathstrut 194074380q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(6))\) 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
6.10.a.a \(1\) \(3.090\) \(\Q\) None \(-16\) \(81\) \(2694\) \(-3544\) \(+\) \(-\) \(q-2^{4}q^{2}+3^{4}q^{3}+2^{8}q^{4}+2694q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(6)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)