Properties

Label 2.22.a
Level 2
Weight 22
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 2
Sturm bound 5
Trace bound 2

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 22 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(5\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(2))\).

Total New Old
Modular forms 6 2 4
Cusp forms 4 2 2
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators.

\(2\)Dim.
\(+\)\(1\)
\(-\)\(1\)

Trace form

\(2q \) \(\mathstrut +\mathstrut 130920q^{3} \) \(\mathstrut +\mathstrut 2097152q^{4} \) \(\mathstrut -\mathstrut 23718420q^{5} \) \(\mathstrut -\mathstrut 12582912q^{6} \) \(\mathstrut +\mathstrut 574223440q^{7} \) \(\mathstrut -\mathstrut 12275185734q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 130920q^{3} \) \(\mathstrut +\mathstrut 2097152q^{4} \) \(\mathstrut -\mathstrut 23718420q^{5} \) \(\mathstrut -\mathstrut 12582912q^{6} \) \(\mathstrut +\mathstrut 574223440q^{7} \) \(\mathstrut -\mathstrut 12275185734q^{9} \) \(\mathstrut +\mathstrut 34477178880q^{10} \) \(\mathstrut -\mathstrut 20036715336q^{11} \) \(\mathstrut +\mathstrut 137279569920q^{12} \) \(\mathstrut -\mathstrut 1045474008260q^{13} \) \(\mathstrut +\mathstrut 2335363301376q^{14} \) \(\mathstrut -\mathstrut 1759470846480q^{15} \) \(\mathstrut +\mathstrut 2199023255552q^{16} \) \(\mathstrut -\mathstrut 7946413934940q^{17} \) \(\mathstrut -\mathstrut 1647354839040q^{18} \) \(\mathstrut +\mathstrut 34056523599880q^{19} \) \(\mathstrut -\mathstrut 24870565969920q^{20} \) \(\mathstrut +\mathstrut 23576486574144q^{21} \) \(\mathstrut -\mathstrut 198143052349440q^{22} \) \(\mathstrut +\mathstrut 275998560315120q^{23} \) \(\mathstrut -\mathstrut 13194139533312q^{24} \) \(\mathstrut -\mathstrut 105587771970850q^{25} \) \(\mathstrut +\mathstrut 763057026367488q^{26} \) \(\mathstrut -\mathstrut 2163118970450160q^{27} \) \(\mathstrut +\mathstrut 602116917821440q^{28} \) \(\mathstrut +\mathstrut 1648319193086940q^{29} \) \(\mathstrut +\mathstrut 2406099525304320q^{30} \) \(\mathstrut -\mathstrut 4025217119435456q^{31} \) \(\mathstrut -\mathstrut 122745071797920q^{33} \) \(\mathstrut -\mathstrut 14808624685645824q^{34} \) \(\mathstrut +\mathstrut 31583536312739040q^{35} \) \(\mathstrut -\mathstrut 12871465156214784q^{36} \) \(\mathstrut +\mathstrut 18166192255567660q^{37} \) \(\mathstrut -\mathstrut 12296613569495040q^{38} \) \(\mathstrut -\mathstrut 73015070738904528q^{39} \) \(\mathstrut +\mathstrut 36151942321274880q^{40} \) \(\mathstrut +\mathstrut 21101722895757204q^{41} \) \(\mathstrut +\mathstrut 149260180201144320q^{42} \) \(\mathstrut -\mathstrut 157459727760364040q^{43} \) \(\mathstrut -\mathstrut 21010018820161536q^{44} \) \(\mathstrut +\mathstrut 118491491854692540q^{45} \) \(\mathstrut -\mathstrut 17400697687375872q^{46} \) \(\mathstrut -\mathstrut 642515922620105760q^{47} \) \(\mathstrut +\mathstrut 143948062308433920q^{48} \) \(\mathstrut +\mathstrut 1648407099408845874q^{49} \) \(\mathstrut -\mathstrut 817744209090969600q^{50} \) \(\mathstrut -\mathstrut 431320508067297456q^{51} \) \(\mathstrut -\mathstrut 1096258953685237760q^{52} \) \(\mathstrut +\mathstrut 1470670856960699340q^{53} \) \(\mathstrut +\mathstrut 101014647015997440q^{54} \) \(\mathstrut -\mathstrut 3019852165744949040q^{55} \) \(\mathstrut +\mathstrut 2448805909103640576q^{56} \) \(\mathstrut +\mathstrut 2303119716265115040q^{57} \) \(\mathstrut +\mathstrut 3191216599172382720q^{58} \) \(\mathstrut -\mathstrut 6736451234462070120q^{59} \) \(\mathstrut -\mathstrut 1844938902318612480q^{60} \) \(\mathstrut +\mathstrut 364336743223117084q^{61} \) \(\mathstrut +\mathstrut 2322545873591992320q^{62} \) \(\mathstrut -\mathstrut 5358824269905078000q^{63} \) \(\mathstrut +\mathstrut 2305843009213693952q^{64} \) \(\mathstrut +\mathstrut 24943153326978577320q^{65} \) \(\mathstrut -\mathstrut 12844384093873373184q^{66} \) \(\mathstrut -\mathstrut 13954116530216070680q^{67} \) \(\mathstrut -\mathstrut 8332418938243645440q^{68} \) \(\mathstrut +\mathstrut 18171269944352010432q^{69} \) \(\mathstrut -\mathstrut 17796761688326799360q^{70} \) \(\mathstrut +\mathstrut 4322530771479236304q^{71} \) \(\mathstrut -\mathstrut 1727376747701207040q^{72} \) \(\mathstrut +\mathstrut 11112960893524425940q^{73} \) \(\mathstrut +\mathstrut 45152071124533641216q^{74} \) \(\mathstrut -\mathstrut 2005310298666023400q^{75} \) \(\mathstrut +\mathstrut 35710853290267770880q^{76} \) \(\mathstrut -\mathstrut 226402501182977528640q^{77} \) \(\mathstrut +\mathstrut 56527266668127191040q^{78} \) \(\mathstrut +\mathstrut 133177052109561844000q^{79} \) \(\mathstrut -\mathstrut 26078678582474833920q^{80} \) \(\mathstrut -\mathstrut 13801077277696652238q^{81} \) \(\mathstrut -\mathstrut 72015436582087557120q^{82} \) \(\mathstrut +\mathstrut 207172716519439614600q^{83} \) \(\mathstrut +\mathstrut 24721737985969618944q^{84} \) \(\mathstrut -\mathstrut 149215598230637549160q^{85} \) \(\mathstrut -\mathstrut 111936012458259382272q^{86} \) \(\mathstrut +\mathstrut 88751674784436796080q^{87} \) \(\mathstrut -\mathstrut 207768049260366397440q^{88} \) \(\mathstrut +\mathstrut 77778919424044267380q^{89} \) \(\mathstrut -\mathstrut 192070560187479490560q^{90} \) \(\mathstrut +\mathstrut 549563307151695419744q^{91} \) \(\mathstrut +\mathstrut 289405466380987269120q^{92} \) \(\mathstrut -\mathstrut 277425987879796903680q^{93} \) \(\mathstrut +\mathstrut 263647117032718073856q^{94} \) \(\mathstrut -\mathstrut 606039792323431352400q^{95} \) \(\mathstrut -\mathstrut 13835058055282163712q^{96} \) \(\mathstrut -\mathstrut 1296645454449453731900q^{97} \) \(\mathstrut +\mathstrut 1341020348565883453440q^{98} \) \(\mathstrut +\mathstrut 278622531605875217112q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.22.a.a \(1\) \(5.590\) \(\Q\) None \(-1024\) \(71604\) \(-28693770\) \(-853202392\) \(+\) \(q-2^{10}q^{2}+71604q^{3}+2^{20}q^{4}-28693770q^{5}+\cdots\)
2.22.a.b \(1\) \(5.590\) \(\Q\) None \(1024\) \(59316\) \(4975350\) \(1427425832\) \(-\) \(q+2^{10}q^{2}+59316q^{3}+2^{20}q^{4}+4975350q^{5}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

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