Properties

Label 2.32.a
Level 2
Weight 32
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 2
Sturm bound 8
Trace bound 1

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 32 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{32}(\Gamma_0(2))\).

Total New Old
Modular forms 9 3 6
Cusp forms 7 3 4
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.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 32768q^{2} \) \(\mathstrut -\mathstrut 3267708q^{3} \) \(\mathstrut +\mathstrut 3221225472q^{4} \) \(\mathstrut +\mathstrut 16188643050q^{5} \) \(\mathstrut -\mathstrut 1202609061888q^{6} \) \(\mathstrut -\mathstrut 40339061108184q^{7} \) \(\mathstrut -\mathstrut 35184372088832q^{8} \) \(\mathstrut +\mathstrut 506242286116911q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 32768q^{2} \) \(\mathstrut -\mathstrut 3267708q^{3} \) \(\mathstrut +\mathstrut 3221225472q^{4} \) \(\mathstrut +\mathstrut 16188643050q^{5} \) \(\mathstrut -\mathstrut 1202609061888q^{6} \) \(\mathstrut -\mathstrut 40339061108184q^{7} \) \(\mathstrut -\mathstrut 35184372088832q^{8} \) \(\mathstrut +\mathstrut 506242286116911q^{9} \) \(\mathstrut +\mathstrut 2284413729177600q^{10} \) \(\mathstrut +\mathstrut 17943304773520236q^{11} \) \(\mathstrut -\mathstrut 3508674748219392q^{12} \) \(\mathstrut -\mathstrut 80729290322112558q^{13} \) \(\mathstrut +\mathstrut 218508267127832576q^{14} \) \(\mathstrut +\mathstrut 5798205411634092600q^{15} \) \(\mathstrut +\mathstrut 3458764513820540928q^{16} \) \(\mathstrut -\mathstrut 15354666131035291434q^{17} \) \(\mathstrut -\mathstrut 30895361889498464256q^{18} \) \(\mathstrut +\mathstrut 9000412427840451540q^{19} \) \(\mathstrut +\mathstrut 17382423116591923200q^{20} \) \(\mathstrut +\mathstrut 69911217564690499296q^{21} \) \(\mathstrut -\mathstrut 1060338734973089611776q^{22} \) \(\mathstrut +\mathstrut 204552987528325429752q^{23} \) \(\mathstrut -\mathstrut 1291291647670550003712q^{24} \) \(\mathstrut +\mathstrut 14240501297278632193125q^{25} \) \(\mathstrut -\mathstrut 15052897630340094558208q^{26} \) \(\mathstrut +\mathstrut 42863898178419275249640q^{27} \) \(\mathstrut -\mathstrut 43313737052748949487616q^{28} \) \(\mathstrut -\mathstrut 5152219418241122842590q^{29} \) \(\mathstrut -\mathstrut 246248817245506849996800q^{30} \) \(\mathstrut +\mathstrut 212506076123285467809696q^{31} \) \(\mathstrut -\mathstrut 37778931862957161709568q^{32} \) \(\mathstrut +\mathstrut 734753859823846189785744q^{33} \) \(\mathstrut -\mathstrut 563488351545479717781504q^{34} \) \(\mathstrut -\mathstrut 673495410613761865525200q^{35} \) \(\mathstrut +\mathstrut 543573515681101894385664q^{36} \) \(\mathstrut -\mathstrut 627427520423108034277494q^{37} \) \(\mathstrut +\mathstrut 6854220706376434238750720q^{38} \) \(\mathstrut -\mathstrut 1657568986782505644897768q^{39} \) \(\mathstrut +\mathstrut 2452870564337798243942400q^{40} \) \(\mathstrut -\mathstrut 38580414046444764597457074q^{41} \) \(\mathstrut +\mathstrut 19758171719029294397128704q^{42} \) \(\mathstrut -\mathstrut 23235521310401328869775348q^{43} \) \(\mathstrut +\mathstrut 19266476796107525103550464q^{44} \) \(\mathstrut +\mathstrut 95941684717265189250464850q^{45} \) \(\mathstrut -\mathstrut 26752574090090454617161728q^{46} \) \(\mathstrut +\mathstrut 41615413662513099737014896q^{47} \) \(\mathstrut -\mathstrut 3767410823975830718251008q^{48} \) \(\mathstrut +\mathstrut 89013404657201283819879579q^{49} \) \(\mathstrut -\mathstrut 650904485047296466104320000q^{50} \) \(\mathstrut +\mathstrut 322159788711896680360393416q^{51} \) \(\mathstrut -\mathstrut 86682415440690685560225792q^{52} \) \(\mathstrut +\mathstrut 132201013676122738960543962q^{53} \) \(\mathstrut -\mathstrut 309696020134895796317061120q^{54} \) \(\mathstrut +\mathstrut 792108207243600797640954600q^{55} \) \(\mathstrut +\mathstrut 234621465304918191320858624q^{56} \) \(\mathstrut -\mathstrut 1346223147874794463777211280q^{57} \) \(\mathstrut +\mathstrut 2411246737580100730728284160q^{58} \) \(\mathstrut -\mathstrut 5753814512127517114030614180q^{59} \) \(\mathstrut +\mathstrut 6225775654614661408908902400q^{60} \) \(\mathstrut +\mathstrut 4774788086222347937804104386q^{61} \) \(\mathstrut +\mathstrut 3704488326858799189200994304q^{62} \) \(\mathstrut -\mathstrut 6011028060421807597353897528q^{63} \) \(\mathstrut +\mathstrut 3713820117856140824697372672q^{64} \) \(\mathstrut -\mathstrut 46779813940115369976955152900q^{65} \) \(\mathstrut -\mathstrut 14636421813031570904389779456q^{66} \) \(\mathstrut +\mathstrut 37537572598145362063271260356q^{67} \) \(\mathstrut -\mathstrut 16486947218448856832714735616q^{68} \) \(\mathstrut +\mathstrut 114807594119435471176160702112q^{69} \) \(\mathstrut -\mathstrut 25320471334662660110902886400q^{70} \) \(\mathstrut +\mathstrut 29743241884330124318650592616q^{71} \) \(\mathstrut -\mathstrut 33173642228370167455436242944q^{72} \) \(\mathstrut -\mathstrut 6993517670034468868152687378q^{73} \) \(\mathstrut -\mathstrut 143866840612066043738862321664q^{74} \) \(\mathstrut +\mathstrut 14581319241373402412513617500q^{75} \) \(\mathstrut +\mathstrut 9664119257021674817543208960q^{76} \) \(\mathstrut -\mathstrut 188875652446259997073599662688q^{77} \) \(\mathstrut +\mathstrut 408000501136764191282028085248q^{78} \) \(\mathstrut -\mathstrut 107419046085410287397483731440q^{79} \) \(\mathstrut +\mathstrut 18664234702749176280435916800q^{80} \) \(\mathstrut -\mathstrut 638480712110453330899602053637q^{81} \) \(\mathstrut +\mathstrut 655047217844750861554293080064q^{82} \) \(\mathstrut +\mathstrut 129892408077704251994568001812q^{83} \) \(\mathstrut +\mathstrut 75066598265971614709557755904q^{84} \) \(\mathstrut -\mathstrut 752165420623931823652095914700q^{85} \) \(\mathstrut +\mathstrut 1145428249152260544065651474432q^{86} \) \(\mathstrut +\mathstrut 647298811205047465573329356760q^{87} \) \(\mathstrut -\mathstrut 1138530047347857830663814119424q^{88} \) \(\mathstrut +\mathstrut 2498232351187298658428880727230q^{89} \) \(\mathstrut -\mathstrut 3758319261146831194135506124800q^{90} \) \(\mathstrut +\mathstrut 2654072415403402092024535236336q^{91} \) \(\mathstrut +\mathstrut 219637097933313398607496347648q^{92} \) \(\mathstrut -\mathstrut 8761476664309588878199887106176q^{93} \) \(\mathstrut -\mathstrut 7369141721635260156957065478144q^{94} \) \(\mathstrut +\mathstrut 12338734824597537076522949391000q^{95} \) \(\mathstrut -\mathstrut 1386513849085741712068929650688q^{96} \) \(\mathstrut +\mathstrut 1522573849695598754720081798886q^{97} \) \(\mathstrut +\mathstrut 5318064748220693286020267606016q^{98} \) \(\mathstrut +\mathstrut 17045580509844366582504033951132q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{32}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.32.a.a \(1\) \(12.175\) \(\Q\) None \(32768\) \(-19984212\) \(42951708750\) \(-1\!\cdots\!76\) \(-\) \(q+2^{15}q^{2}-19984212q^{3}+2^{30}q^{4}+\cdots\)
2.32.a.b \(2\) \(12.175\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-65536\) \(16716504\) \(-26763065700\) \(-2\!\cdots\!08\) \(+\) \(q-2^{15}q^{2}+(8358252-\beta )q^{3}+2^{30}q^{4}+\cdots\)

Decomposition of \(S_{32}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{32}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{32}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)