Properties

Label 22.12.a
Level 22
Weight 12
Character orbit a
Rep. character \(\chi_{22}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newforms 4
Sturm bound 36
Trace bound 3

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Defining parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 22.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(36\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(22))\).

Total New Old
Modular forms 35 11 24
Cusp forms 31 11 20
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.

\(2\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(5\)

Trace form

\(11q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut -\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 11264q^{4} \) \(\mathstrut +\mathstrut 12082q^{5} \) \(\mathstrut -\mathstrut 2944q^{6} \) \(\mathstrut -\mathstrut 45224q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut +\mathstrut 915375q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut -\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 11264q^{4} \) \(\mathstrut +\mathstrut 12082q^{5} \) \(\mathstrut -\mathstrut 2944q^{6} \) \(\mathstrut -\mathstrut 45224q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut +\mathstrut 915375q^{9} \) \(\mathstrut -\mathstrut 363456q^{10} \) \(\mathstrut +\mathstrut 161051q^{11} \) \(\mathstrut -\mathstrut 20480q^{12} \) \(\mathstrut -\mathstrut 875894q^{13} \) \(\mathstrut -\mathstrut 335360q^{14} \) \(\mathstrut +\mathstrut 472536q^{15} \) \(\mathstrut +\mathstrut 11534336q^{16} \) \(\mathstrut -\mathstrut 13102170q^{17} \) \(\mathstrut +\mathstrut 1427296q^{18} \) \(\mathstrut -\mathstrut 7740548q^{19} \) \(\mathstrut +\mathstrut 12371968q^{20} \) \(\mathstrut +\mathstrut 6889024q^{21} \) \(\mathstrut +\mathstrut 5153632q^{22} \) \(\mathstrut +\mathstrut 93614464q^{23} \) \(\mathstrut -\mathstrut 3014656q^{24} \) \(\mathstrut +\mathstrut 102028181q^{25} \) \(\mathstrut -\mathstrut 20038336q^{26} \) \(\mathstrut -\mathstrut 175002488q^{27} \) \(\mathstrut -\mathstrut 46309376q^{28} \) \(\mathstrut +\mathstrut 286587034q^{29} \) \(\mathstrut -\mathstrut 199855104q^{30} \) \(\mathstrut -\mathstrut 309055976q^{31} \) \(\mathstrut -\mathstrut 33554432q^{32} \) \(\mathstrut +\mathstrut 99851620q^{33} \) \(\mathstrut +\mathstrut 519375552q^{34} \) \(\mathstrut +\mathstrut 16432992q^{35} \) \(\mathstrut +\mathstrut 937344000q^{36} \) \(\mathstrut +\mathstrut 648119698q^{37} \) \(\mathstrut -\mathstrut 167456128q^{38} \) \(\mathstrut -\mathstrut 1959228072q^{39} \) \(\mathstrut -\mathstrut 372178944q^{40} \) \(\mathstrut +\mathstrut 391758926q^{41} \) \(\mathstrut +\mathstrut 295677696q^{42} \) \(\mathstrut -\mathstrut 2183911004q^{43} \) \(\mathstrut +\mathstrut 164916224q^{44} \) \(\mathstrut +\mathstrut 749597026q^{45} \) \(\mathstrut -\mathstrut 1551425280q^{46} \) \(\mathstrut -\mathstrut 3684676472q^{47} \) \(\mathstrut -\mathstrut 20971520q^{48} \) \(\mathstrut +\mathstrut 3700166595q^{49} \) \(\mathstrut -\mathstrut 172337376q^{50} \) \(\mathstrut +\mathstrut 380497640q^{51} \) \(\mathstrut -\mathstrut 896915456q^{52} \) \(\mathstrut -\mathstrut 9108206598q^{53} \) \(\mathstrut +\mathstrut 7972846592q^{54} \) \(\mathstrut -\mathstrut 4378332486q^{55} \) \(\mathstrut -\mathstrut 343408640q^{56} \) \(\mathstrut -\mathstrut 3849141920q^{57} \) \(\mathstrut +\mathstrut 11874670656q^{58} \) \(\mathstrut +\mathstrut 12074756484q^{59} \) \(\mathstrut +\mathstrut 483876864q^{60} \) \(\mathstrut +\mathstrut 16775683690q^{61} \) \(\mathstrut -\mathstrut 23914092544q^{62} \) \(\mathstrut +\mathstrut 24940101224q^{63} \) \(\mathstrut +\mathstrut 11811160064q^{64} \) \(\mathstrut -\mathstrut 9892919524q^{65} \) \(\mathstrut +\mathstrut 5009330304q^{66} \) \(\mathstrut -\mathstrut 19655325860q^{67} \) \(\mathstrut -\mathstrut 13416622080q^{68} \) \(\mathstrut -\mathstrut 21210197368q^{69} \) \(\mathstrut +\mathstrut 3837970176q^{70} \) \(\mathstrut +\mathstrut 43068243344q^{71} \) \(\mathstrut +\mathstrut 1461551104q^{72} \) \(\mathstrut -\mathstrut 19855221026q^{73} \) \(\mathstrut -\mathstrut 20581162432q^{74} \) \(\mathstrut -\mathstrut 33234655484q^{75} \) \(\mathstrut -\mathstrut 7926321152q^{76} \) \(\mathstrut +\mathstrut 16268727816q^{77} \) \(\mathstrut +\mathstrut 63217593344q^{78} \) \(\mathstrut -\mathstrut 98359292336q^{79} \) \(\mathstrut +\mathstrut 12668895232q^{80} \) \(\mathstrut +\mathstrut 127815100707q^{81} \) \(\mathstrut -\mathstrut 30084194880q^{82} \) \(\mathstrut +\mathstrut 97303283084q^{83} \) \(\mathstrut +\mathstrut 7054360576q^{84} \) \(\mathstrut -\mathstrut 71855645292q^{85} \) \(\mathstrut -\mathstrut 63811647360q^{86} \) \(\mathstrut -\mathstrut 99384630104q^{87} \) \(\mathstrut +\mathstrut 5277319168q^{88} \) \(\mathstrut +\mathstrut 218796781046q^{89} \) \(\mathstrut -\mathstrut 83004021440q^{90} \) \(\mathstrut -\mathstrut 41718652624q^{91} \) \(\mathstrut +\mathstrut 95861211136q^{92} \) \(\mathstrut -\mathstrut 76691269368q^{93} \) \(\mathstrut +\mathstrut 62528030976q^{94} \) \(\mathstrut -\mathstrut 133726589800q^{95} \) \(\mathstrut -\mathstrut 3087007744q^{96} \) \(\mathstrut -\mathstrut 197647267154q^{97} \) \(\mathstrut -\mathstrut 354675106080q^{98} \) \(\mathstrut +\mathstrut 141202591607q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(22))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11
22.12.a.a \(2\) \(16.904\) \(\Q(\sqrt{331}) \) None \(64\) \(-426\) \(2290\) \(-86324\) \(-\) \(+\) \(q+2^{5}q^{2}+(-213+3\beta )q^{3}+2^{10}q^{4}+\cdots\)
22.12.a.b \(3\) \(16.904\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-96\) \(-70\) \(-5624\) \(-30576\) \(+\) \(-\) \(q-2^{5}q^{2}+(-23+\beta _{2})q^{3}+2^{10}q^{4}+\cdots\)
22.12.a.c \(3\) \(16.904\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-96\) \(106\) \(17344\) \(13204\) \(+\) \(+\) \(q-2^{5}q^{2}+(35+\beta _{1})q^{3}+2^{10}q^{4}+(5771+\cdots)q^{5}+\cdots\)
22.12.a.d \(3\) \(16.904\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(96\) \(370\) \(-1928\) \(58472\) \(-\) \(-\) \(q+2^{5}q^{2}+(123-\beta _{1})q^{3}+2^{10}q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(22))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(22)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)