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Properties

Label 2.32.a
Level $2$
Weight $32$
Character orbit 2.a
Rep. character $\chi_{2}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
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: \(\Q\)
Newform subspaces: \( 2 \)
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 and the Fricke involution.

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

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{32}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
2.32.a.a 2.a 1.a $1$ $12.175$ \(\Q\) None \(32768\) \(-19984212\) \(42951708750\) \(-16\!\cdots\!76\) $-$ $\mathrm{SU}(2)$ \(q+2^{15}q^{2}-19984212q^{3}+2^{30}q^{4}+\cdots\)
2.32.a.b 2.a 1.a $2$ $12.175$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-65536\) \(16716504\) \(-26763065700\) \(-23\!\cdots\!08\) $+$ $\mathrm{SU}(2)$ \(q-2^{15}q^{2}+(8358252-\beta )q^{3}+2^{30}q^{4}+\cdots\)

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

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