Properties

Label 4.36.a
Level 4
Weight 36
Character orbit a
Rep. character \(\chi_{4}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 1
Sturm bound 18
Trace bound 0

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Defining parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 36 \)
Character orbit: \([\chi]\) = 4.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{36}(\Gamma_0(4))\).

Total New Old
Modular forms 19 3 16
Cusp forms 16 3 13
Eisenstein series 3 0 3

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

\(2\)Dim.
\(-\)\(3\)

Trace form

\(3q \) \(\mathstrut +\mathstrut 50908884q^{3} \) \(\mathstrut +\mathstrut 280720890q^{5} \) \(\mathstrut -\mathstrut 5549296289016q^{7} \) \(\mathstrut -\mathstrut 4115694118097313q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 50908884q^{3} \) \(\mathstrut +\mathstrut 280720890q^{5} \) \(\mathstrut -\mathstrut 5549296289016q^{7} \) \(\mathstrut -\mathstrut 4115694118097313q^{9} \) \(\mathstrut -\mathstrut 420463958869151460q^{11} \) \(\mathstrut -\mathstrut 7370701315414736574q^{13} \) \(\mathstrut +\mathstrut 35737802623493622360q^{15} \) \(\mathstrut +\mathstrut 616802274448822103958q^{17} \) \(\mathstrut -\mathstrut 3523177807465210008348q^{19} \) \(\mathstrut -\mathstrut 35614415089970552016672q^{21} \) \(\mathstrut -\mathstrut 97200569155131383968872q^{23} \) \(\mathstrut -\mathstrut 1397444430311587979767275q^{25} \) \(\mathstrut -\mathstrut 13813263585892101497770872q^{27} \) \(\mathstrut -\mathstrut 72104888547394509330484398q^{29} \) \(\mathstrut -\mathstrut 305306760334252700869662432q^{31} \) \(\mathstrut -\mathstrut 1206118755141042924734246640q^{33} \) \(\mathstrut -\mathstrut 4139993873456897291989313040q^{35} \) \(\mathstrut -\mathstrut 10467805589252853184477978086q^{37} \) \(\mathstrut -\mathstrut 21306312046147482048067771848q^{39} \) \(\mathstrut -\mathstrut 28614809221544804827249675602q^{41} \) \(\mathstrut -\mathstrut 20661331953141752160177135300q^{43} \) \(\mathstrut +\mathstrut 109445534106618058172074201410q^{45} \) \(\mathstrut +\mathstrut 431671100729347757519313060528q^{47} \) \(\mathstrut +\mathstrut 1202271820749265164710734860219q^{49} \) \(\mathstrut +\mathstrut 1812312130969213433267629787496q^{51} \) \(\mathstrut +\mathstrut 2094590682902326557681205190442q^{53} \) \(\mathstrut -\mathstrut 424018895462656971242660883000q^{55} \) \(\mathstrut -\mathstrut 7056738639317962169249569687824q^{57} \) \(\mathstrut -\mathstrut 21898322952557962385462881418676q^{59} \) \(\mathstrut -\mathstrut 46553596324122391351852871731854q^{61} \) \(\mathstrut -\mathstrut 60534256794589553738854966512984q^{63} \) \(\mathstrut -\mathstrut 47301026213190020793582172528260q^{65} \) \(\mathstrut +\mathstrut 75982094975394264303461043768084q^{67} \) \(\mathstrut +\mathstrut 266168516453886472772566159456416q^{69} \) \(\mathstrut +\mathstrut 591751031973420639599491172643144q^{71} \) \(\mathstrut +\mathstrut 840249720714294658198657086588366q^{73} \) \(\mathstrut +\mathstrut 844328642366224046122007927037900q^{75} \) \(\mathstrut +\mathstrut 366867399909875520359123617491360q^{77} \) \(\mathstrut -\mathstrut 1512021839065233136089826558058544q^{79} \) \(\mathstrut -\mathstrut 4715323858342810015899488694567333q^{81} \) \(\mathstrut -\mathstrut 8944825848970974146478286242709212q^{83} \) \(\mathstrut -\mathstrut 11518175153366806194794571426747180q^{85} \) \(\mathstrut -\mathstrut 9434346897547249340232797055364104q^{87} \) \(\mathstrut +\mathstrut 1625382197094823895014829864559198q^{89} \) \(\mathstrut +\mathstrut 28933989451866564710063788057094832q^{91} \) \(\mathstrut +\mathstrut 43351982414907351295090224432607104q^{93} \) \(\mathstrut +\mathstrut 96155354088435090124008638354927160q^{95} \) \(\mathstrut +\mathstrut 83007021412015685382745763811847014q^{97} \) \(\mathstrut +\mathstrut 91112511762343802272300511308928460q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(4))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
4.36.a.a \(3\) \(31.038\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(50908884\) \(280720890\) \(-5\!\cdots\!16\) \(-\) \(q+(16969628+\beta _{1})q^{3}+(93573630+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_0(4))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_0(4)) \cong \) \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)