Properties

Label 6.12.a
Level 6
Weight 12
Character orbit a
Rep. character \(\chi_{6}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 3
Sturm bound 12
Trace bound 3

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Defining parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 6.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(12\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(6))\).

Total New Old
Modular forms 13 3 10
Cusp forms 9 3 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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(1\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut +\mathstrut 243q^{3} \) \(\mathstrut +\mathstrut 3072q^{4} \) \(\mathstrut -\mathstrut 2334q^{5} \) \(\mathstrut +\mathstrut 7776q^{6} \) \(\mathstrut +\mathstrut 55392q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut +\mathstrut 177147q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut +\mathstrut 243q^{3} \) \(\mathstrut +\mathstrut 3072q^{4} \) \(\mathstrut -\mathstrut 2334q^{5} \) \(\mathstrut +\mathstrut 7776q^{6} \) \(\mathstrut +\mathstrut 55392q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut +\mathstrut 177147q^{9} \) \(\mathstrut +\mathstrut 307008q^{10} \) \(\mathstrut -\mathstrut 1699116q^{11} \) \(\mathstrut +\mathstrut 248832q^{12} \) \(\mathstrut +\mathstrut 217266q^{13} \) \(\mathstrut +\mathstrut 335360q^{14} \) \(\mathstrut -\mathstrut 3369438q^{15} \) \(\mathstrut +\mathstrut 3145728q^{16} \) \(\mathstrut +\mathstrut 8603622q^{17} \) \(\mathstrut -\mathstrut 1889568q^{18} \) \(\mathstrut +\mathstrut 7200204q^{19} \) \(\mathstrut -\mathstrut 2390016q^{20} \) \(\mathstrut -\mathstrut 21757248q^{21} \) \(\mathstrut +\mathstrut 5811840q^{22} \) \(\mathstrut -\mathstrut 1057800q^{23} \) \(\mathstrut +\mathstrut 7962624q^{24} \) \(\mathstrut +\mathstrut 37532181q^{25} \) \(\mathstrut -\mathstrut 165855424q^{26} \) \(\mathstrut +\mathstrut 14348907q^{27} \) \(\mathstrut +\mathstrut 56721408q^{28} \) \(\mathstrut +\mathstrut 70389066q^{29} \) \(\mathstrut +\mathstrut 164275776q^{30} \) \(\mathstrut -\mathstrut 146569368q^{31} \) \(\mathstrut -\mathstrut 33554432q^{32} \) \(\mathstrut -\mathstrut 214136460q^{33} \) \(\mathstrut +\mathstrut 255268800q^{34} \) \(\mathstrut +\mathstrut 1123978944q^{35} \) \(\mathstrut +\mathstrut 181398528q^{36} \) \(\mathstrut -\mathstrut 1480918566q^{37} \) \(\mathstrut -\mathstrut 925913728q^{38} \) \(\mathstrut -\mathstrut 611837550q^{39} \) \(\mathstrut +\mathstrut 314376192q^{40} \) \(\mathstrut +\mathstrut 899560158q^{41} \) \(\mathstrut +\mathstrut 1208452608q^{42} \) \(\mathstrut -\mathstrut 273486252q^{43} \) \(\mathstrut -\mathstrut 1739894784q^{44} \) \(\mathstrut -\mathstrut 137820366q^{45} \) \(\mathstrut +\mathstrut 1348363008q^{46} \) \(\mathstrut +\mathstrut 1552169280q^{47} \) \(\mathstrut +\mathstrut 254803968q^{48} \) \(\mathstrut +\mathstrut 2904631227q^{49} \) \(\mathstrut -\mathstrut 3482708192q^{50} \) \(\mathstrut -\mathstrut 544856058q^{51} \) \(\mathstrut +\mathstrut 222480384q^{52} \) \(\mathstrut -\mathstrut 1324585278q^{53} \) \(\mathstrut +\mathstrut 459165024q^{54} \) \(\mathstrut +\mathstrut 1121307192q^{55} \) \(\mathstrut +\mathstrut 343408640q^{56} \) \(\mathstrut -\mathstrut 5621075028q^{57} \) \(\mathstrut -\mathstrut 408318912q^{58} \) \(\mathstrut +\mathstrut 4411180500q^{59} \) \(\mathstrut -\mathstrut 3450304512q^{60} \) \(\mathstrut -\mathstrut 12529017966q^{61} \) \(\mathstrut +\mathstrut 14322308864q^{62} \) \(\mathstrut +\mathstrut 3270842208q^{63} \) \(\mathstrut +\mathstrut 3221225472q^{64} \) \(\mathstrut -\mathstrut 16758434292q^{65} \) \(\mathstrut -\mathstrut 4947682176q^{66} \) \(\mathstrut +\mathstrut 27115273932q^{67} \) \(\mathstrut +\mathstrut 8810108928q^{68} \) \(\mathstrut +\mathstrut 25109273592q^{69} \) \(\mathstrut -\mathstrut 28315634688q^{70} \) \(\mathstrut -\mathstrut 25932460536q^{71} \) \(\mathstrut -\mathstrut 1934917632q^{72} \) \(\mathstrut +\mathstrut 6082420686q^{73} \) \(\mathstrut +\mathstrut 26916324416q^{74} \) \(\mathstrut +\mathstrut 16692865317q^{75} \) \(\mathstrut +\mathstrut 7373008896q^{76} \) \(\mathstrut -\mathstrut 28048880640q^{77} \) \(\mathstrut -\mathstrut 19034606016q^{78} \) \(\mathstrut -\mathstrut 58308979368q^{79} \) \(\mathstrut -\mathstrut 2447376384q^{80} \) \(\mathstrut +\mathstrut 10460353203q^{81} \) \(\mathstrut -\mathstrut 52338500928q^{82} \) \(\mathstrut +\mathstrut 37956458316q^{83} \) \(\mathstrut -\mathstrut 22279421952q^{84} \) \(\mathstrut +\mathstrut 121299146244q^{85} \) \(\mathstrut +\mathstrut 48461612672q^{86} \) \(\mathstrut -\mathstrut 40525895862q^{87} \) \(\mathstrut +\mathstrut 5951324160q^{88} \) \(\mathstrut -\mathstrut 210662658q^{89} \) \(\mathstrut +\mathstrut 18128515392q^{90} \) \(\mathstrut -\mathstrut 49315648704q^{91} \) \(\mathstrut -\mathstrut 1083187200q^{92} \) \(\mathstrut -\mathstrut 7597286136q^{93} \) \(\mathstrut +\mathstrut 127169639424q^{94} \) \(\mathstrut +\mathstrut 13965964680q^{95} \) \(\mathstrut +\mathstrut 8153726976q^{96} \) \(\mathstrut -\mathstrut 49488374586q^{97} \) \(\mathstrut -\mathstrut 150071184672q^{98} \) \(\mathstrut -\mathstrut 100331100684q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\) 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
6.12.a.a \(1\) \(4.610\) \(\Q\) None \(-32\) \(-243\) \(5766\) \(72464\) \(+\) \(+\) \(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+5766q^{5}+\cdots\)
6.12.a.b \(1\) \(4.610\) \(\Q\) None \(-32\) \(243\) \(-11730\) \(-50008\) \(+\) \(-\) \(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-11730q^{5}+\cdots\)
6.12.a.c \(1\) \(4.610\) \(\Q\) None \(32\) \(243\) \(3630\) \(32936\) \(-\) \(-\) \(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+3630q^{5}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(6)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)