Properties

Label 2.20.a
Level 2
Weight 20
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: \( k \) = \( 20 \)
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 \(M_{20}(\Gamma_0(2))\).

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

\(2q \) \(\mathstrut -\mathstrut 66120q^{3} \) \(\mathstrut +\mathstrut 524288q^{4} \) \(\mathstrut +\mathstrut 989820q^{5} \) \(\mathstrut -\mathstrut 20447232q^{6} \) \(\mathstrut +\mathstrut 52177840q^{7} \) \(\mathstrut +\mathstrut 658846314q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 66120q^{3} \) \(\mathstrut +\mathstrut 524288q^{4} \) \(\mathstrut +\mathstrut 989820q^{5} \) \(\mathstrut -\mathstrut 20447232q^{6} \) \(\mathstrut +\mathstrut 52177840q^{7} \) \(\mathstrut +\mathstrut 658846314q^{9} \) \(\mathstrut -\mathstrut 6197084160q^{10} \) \(\mathstrut +\mathstrut 18120369384q^{11} \) \(\mathstrut -\mathstrut 17332961280q^{12} \) \(\mathstrut +\mathstrut 1276754860q^{13} \) \(\mathstrut -\mathstrut 72279392256q^{14} \) \(\mathstrut +\mathstrut 208962833040q^{15} \) \(\mathstrut +\mathstrut 137438953472q^{16} \) \(\mathstrut -\mathstrut 1123052861340q^{17} \) \(\mathstrut +\mathstrut 1351970979840q^{18} \) \(\mathstrut -\mathstrut 497744747240q^{19} \) \(\mathstrut +\mathstrut 259475374080q^{20} \) \(\mathstrut +\mathstrut 1093896907584q^{21} \) \(\mathstrut -\mathstrut 2805257994240q^{22} \) \(\mathstrut -\mathstrut 1558014516720q^{23} \) \(\mathstrut -\mathstrut 5360119185408q^{24} \) \(\mathstrut +\mathstrut 35592433931150q^{25} \) \(\mathstrut -\mathstrut 34573670940672q^{26} \) \(\mathstrut +\mathstrut 2340400843440q^{27} \) \(\mathstrut +\mathstrut 13678107688960q^{28} \) \(\mathstrut -\mathstrut 126085309807860q^{29} \) \(\mathstrut +\mathstrut 194756062740480q^{30} \) \(\mathstrut +\mathstrut 73528521137344q^{31} \) \(\mathstrut -\mathstrut 489654350059680q^{33} \) \(\mathstrut -\mathstrut 164688713220096q^{34} \) \(\mathstrut +\mathstrut 880165751260320q^{35} \) \(\mathstrut +\mathstrut 172712608137216q^{36} \) \(\mathstrut -\mathstrut 220384762054340q^{37} \) \(\mathstrut -\mathstrut 1089278257397760q^{38} \) \(\mathstrut +\mathstrut 1306163651014608q^{39} \) \(\mathstrut -\mathstrut 1624528430039040q^{40} \) \(\mathstrut -\mathstrut 159817982504076q^{41} \) \(\mathstrut +\mathstrut 1856110508113920q^{42} \) \(\mathstrut +\mathstrut 1375186650672040q^{43} \) \(\mathstrut +\mathstrut 4750146111799296q^{44} \) \(\mathstrut -\mathstrut 15654227352447060q^{45} \) \(\mathstrut +\mathstrut 4258570240524288q^{46} \) \(\mathstrut +\mathstrut 11861897154251040q^{47} \) \(\mathstrut -\mathstrut 4543731801784320q^{48} \) \(\mathstrut -\mathstrut 11471945302016814q^{49} \) \(\mathstrut -\mathstrut 6133997843251200q^{50} \) \(\mathstrut +\mathstrut 43550987411484144q^{51} \) \(\mathstrut +\mathstrut 334693626019840q^{52} \) \(\mathstrut -\mathstrut 46640312146730340q^{53} \) \(\mathstrut -\mathstrut 27666922450452480q^{54} \) \(\mathstrut +\mathstrut 42126101503752240q^{55} \) \(\mathstrut -\mathstrut 18947609003556864q^{56} \) \(\mathstrut +\mathstrut 58937293382267040q^{57} \) \(\mathstrut +\mathstrut 34472506561658880q^{58} \) \(\mathstrut -\mathstrut 83330198435011320q^{59} \) \(\mathstrut +\mathstrut 54778352904437760q^{60} \) \(\mathstrut -\mathstrut 132955759859505716q^{61} \) \(\mathstrut +\mathstrut 97497308693790720q^{62} \) \(\mathstrut -\mathstrut 169196834444461200q^{63} \) \(\mathstrut +\mathstrut 36028797018963968q^{64} \) \(\mathstrut +\mathstrut 409292669266505640q^{65} \) \(\mathstrut -\mathstrut 92513869070598144q^{66} \) \(\mathstrut -\mathstrut 5143890515758280q^{67} \) \(\mathstrut -\mathstrut 294401569283112960q^{68} \) \(\mathstrut -\mathstrut 114576279457684032q^{69} \) \(\mathstrut -\mathstrut 197447026904924160q^{70} \) \(\mathstrut +\mathstrut 180975996855926064q^{71} \) \(\mathstrut +\mathstrut 354411080539176960q^{72} \) \(\mathstrut +\mathstrut 703681071936580660q^{73} \) \(\mathstrut -\mathstrut 364822243968024576q^{74} \) \(\mathstrut -\mathstrut 937459949877022200q^{75} \) \(\mathstrut -\mathstrut 130480799020482560q^{76} \) \(\mathstrut +\mathstrut 859479344831540160q^{77} \) \(\mathstrut +\mathstrut 1129952509883842560q^{78} \) \(\mathstrut -\mathstrut 2579721560929264160q^{79} \) \(\mathstrut +\mathstrut 68019912462827520q^{80} \) \(\mathstrut +\mathstrut 235884640246337682q^{81} \) \(\mathstrut +\mathstrut 2016621232631316480q^{82} \) \(\mathstrut -\mathstrut 353130710209828200q^{83} \) \(\mathstrut +\mathstrut 286758510941700096q^{84} \) \(\mathstrut +\mathstrut 1390810498655755320q^{85} \) \(\mathstrut -\mathstrut 5131367195035041792q^{86} \) \(\mathstrut +\mathstrut 2823952586343155280q^{87} \) \(\mathstrut -\mathstrut 735381551642050560q^{88} \) \(\mathstrut -\mathstrut 913988698539478380q^{89} \) \(\mathstrut -\mathstrut 1372359070549278720q^{90} \) \(\mathstrut +\mathstrut 4799704578625074464q^{91} \) \(\mathstrut -\mathstrut 408424157471047680q^{92} \) \(\mathstrut -\mathstrut 6233247947858430720q^{93} \) \(\mathstrut +\mathstrut 6320574270101520384q^{94} \) \(\mathstrut +\mathstrut 12628930149584974800q^{95} \) \(\mathstrut -\mathstrut 1405123083739594752q^{96} \) \(\mathstrut -\mathstrut 24626536111267629500q^{97} \) \(\mathstrut -\mathstrut 3771382564430807040q^{98} \) \(\mathstrut -\mathstrut 1264593396103377912q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.20.a.a \(1\) \(4.576\) \(\Q\) None \(-512\) \(-13092\) \(6546750\) \(96674264\) \(+\) \(q-2^{9}q^{2}-13092q^{3}+2^{18}q^{4}+6546750q^{5}+\cdots\)
2.20.a.b \(1\) \(4.576\) \(\Q\) None \(512\) \(-53028\) \(-5556930\) \(-44496424\) \(-\) \(q+2^{9}q^{2}-53028q^{3}+2^{18}q^{4}-5556930q^{5}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)