Properties

Label 21.12.a
Level 21
Weight 12
Character orbit a
Rep. character \(\chi_{21}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newforms 5
Sturm bound 32
Trace bound 2

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 21.a (trivial)
Character field: \(\Q\)
Newforms: \( 5 \)
Sturm bound: \(32\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(21))\).

Total New Old
Modular forms 32 12 20
Cusp forms 28 12 16
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.

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(7\)
Minus space\(-\)\(5\)

Trace form

\(12q \) \(\mathstrut -\mathstrut 110q^{2} \) \(\mathstrut +\mathstrut 7298q^{4} \) \(\mathstrut +\mathstrut 20864q^{5} \) \(\mathstrut +\mathstrut 22356q^{6} \) \(\mathstrut +\mathstrut 33614q^{7} \) \(\mathstrut -\mathstrut 178530q^{8} \) \(\mathstrut +\mathstrut 708588q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 110q^{2} \) \(\mathstrut +\mathstrut 7298q^{4} \) \(\mathstrut +\mathstrut 20864q^{5} \) \(\mathstrut +\mathstrut 22356q^{6} \) \(\mathstrut +\mathstrut 33614q^{7} \) \(\mathstrut -\mathstrut 178530q^{8} \) \(\mathstrut +\mathstrut 708588q^{9} \) \(\mathstrut -\mathstrut 628628q^{10} \) \(\mathstrut +\mathstrut 479768q^{11} \) \(\mathstrut +\mathstrut 468504q^{12} \) \(\mathstrut +\mathstrut 211816q^{13} \) \(\mathstrut +\mathstrut 2252138q^{14} \) \(\mathstrut +\mathstrut 1945944q^{15} \) \(\mathstrut +\mathstrut 15060050q^{16} \) \(\mathstrut -\mathstrut 17677776q^{17} \) \(\mathstrut -\mathstrut 6495390q^{18} \) \(\mathstrut +\mathstrut 14340400q^{19} \) \(\mathstrut +\mathstrut 80062972q^{20} \) \(\mathstrut +\mathstrut 8168202q^{21} \) \(\mathstrut -\mathstrut 141776528q^{22} \) \(\mathstrut -\mathstrut 36696792q^{23} \) \(\mathstrut +\mathstrut 124008732q^{24} \) \(\mathstrut +\mathstrut 170526692q^{25} \) \(\mathstrut +\mathstrut 20750260q^{26} \) \(\mathstrut +\mathstrut 64438038q^{28} \) \(\mathstrut +\mathstrut 157567448q^{29} \) \(\mathstrut +\mathstrut 83372328q^{30} \) \(\mathstrut -\mathstrut 145925920q^{31} \) \(\mathstrut +\mathstrut 218643110q^{32} \) \(\mathstrut -\mathstrut 381578040q^{33} \) \(\mathstrut -\mathstrut 231045636q^{34} \) \(\mathstrut +\mathstrut 202557964q^{35} \) \(\mathstrut +\mathstrut 430939602q^{36} \) \(\mathstrut -\mathstrut 549593704q^{37} \) \(\mathstrut +\mathstrut 1120674064q^{38} \) \(\mathstrut +\mathstrut 1311947280q^{39} \) \(\mathstrut -\mathstrut 5178070044q^{40} \) \(\mathstrut -\mathstrut 2022176960q^{41} \) \(\mathstrut +\mathstrut 261382464q^{42} \) \(\mathstrut -\mathstrut 53478240q^{43} \) \(\mathstrut +\mathstrut 2698519120q^{44} \) \(\mathstrut +\mathstrut 1231998336q^{45} \) \(\mathstrut +\mathstrut 2478782472q^{46} \) \(\mathstrut -\mathstrut 8003791248q^{47} \) \(\mathstrut -\mathstrut 2121428880q^{48} \) \(\mathstrut +\mathstrut 3389702988q^{49} \) \(\mathstrut -\mathstrut 16753717858q^{50} \) \(\mathstrut +\mathstrut 3351877848q^{51} \) \(\mathstrut +\mathstrut 7515182916q^{52} \) \(\mathstrut +\mathstrut 6139416360q^{53} \) \(\mathstrut +\mathstrut 1320099444q^{54} \) \(\mathstrut +\mathstrut 613335632q^{55} \) \(\mathstrut +\mathstrut 9830582370q^{56} \) \(\mathstrut -\mathstrut 3569250096q^{57} \) \(\mathstrut +\mathstrut 25290484780q^{58} \) \(\mathstrut +\mathstrut 10657359408q^{59} \) \(\mathstrut -\mathstrut 7813297584q^{60} \) \(\mathstrut -\mathstrut 25651586024q^{61} \) \(\mathstrut +\mathstrut 2309408448q^{62} \) \(\mathstrut +\mathstrut 1984873086q^{63} \) \(\mathstrut +\mathstrut 9391808682q^{64} \) \(\mathstrut +\mathstrut 5245084544q^{65} \) \(\mathstrut +\mathstrut 28063197144q^{66} \) \(\mathstrut +\mathstrut 37188880240q^{67} \) \(\mathstrut -\mathstrut 77247208644q^{68} \) \(\mathstrut -\mathstrut 4494378312q^{69} \) \(\mathstrut +\mathstrut 6367903388q^{70} \) \(\mathstrut -\mathstrut 6587245800q^{71} \) \(\mathstrut -\mathstrut 10542017970q^{72} \) \(\mathstrut -\mathstrut 8394109912q^{73} \) \(\mathstrut -\mathstrut 55306181652q^{74} \) \(\mathstrut -\mathstrut 13311415584q^{75} \) \(\mathstrut +\mathstrut 62495317632q^{76} \) \(\mathstrut -\mathstrut 31343038160q^{77} \) \(\mathstrut +\mathstrut 25344923328q^{78} \) \(\mathstrut +\mathstrut 53406163008q^{79} \) \(\mathstrut +\mathstrut 59610226684q^{80} \) \(\mathstrut +\mathstrut 41841412812q^{81} \) \(\mathstrut -\mathstrut 29032629796q^{82} \) \(\mathstrut +\mathstrut 29954364864q^{83} \) \(\mathstrut +\mathstrut 25092716544q^{84} \) \(\mathstrut +\mathstrut 108643471296q^{85} \) \(\mathstrut -\mathstrut 304681471672q^{86} \) \(\mathstrut -\mathstrut 25734127680q^{87} \) \(\mathstrut -\mathstrut 407148400704q^{88} \) \(\mathstrut +\mathstrut 227251464224q^{89} \) \(\mathstrut -\mathstrut 37119854772q^{90} \) \(\mathstrut +\mathstrut 15560525652q^{91} \) \(\mathstrut +\mathstrut 42455083272q^{92} \) \(\mathstrut +\mathstrut 13948460496q^{93} \) \(\mathstrut -\mathstrut 278600413056q^{94} \) \(\mathstrut -\mathstrut 75625756384q^{95} \) \(\mathstrut +\mathstrut 324143639892q^{96} \) \(\mathstrut +\mathstrut 37409914376q^{97} \) \(\mathstrut -\mathstrut 31072277390q^{98} \) \(\mathstrut +\mathstrut 28329820632q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(21))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
21.12.a.a \(1\) \(16.135\) \(\Q\) None \(-62\) \(243\) \(-3310\) \(-16807\) \(-\) \(+\) \(q-62q^{2}+3^{5}q^{3}+1796q^{4}-3310q^{5}+\cdots\)
21.12.a.b \(1\) \(16.135\) \(\Q\) None \(8\) \(243\) \(4390\) \(-16807\) \(-\) \(+\) \(q+8q^{2}+3^{5}q^{3}-1984q^{4}+4390q^{5}+\cdots\)
21.12.a.c \(3\) \(16.135\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-68\) \(-729\) \(3326\) \(-50421\) \(+\) \(+\) \(q+(-23-\beta _{1})q^{2}-3^{5}q^{3}+(664+72\beta _{1}+\cdots)q^{4}+\cdots\)
21.12.a.d \(3\) \(16.135\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-33\) \(-729\) \(3102\) \(50421\) \(+\) \(-\) \(q+(-11-\beta _{2})q^{2}-3^{5}q^{3}+(255+13\beta _{1}+\cdots)q^{4}+\cdots\)
21.12.a.e \(4\) \(16.135\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(45\) \(972\) \(13356\) \(67228\) \(-\) \(-\) \(q+(11+\beta _{1})q^{2}+3^{5}q^{3}+(1196+18\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)