DEGREE 3 WEIGHT 18

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eigenvalues | standard Euler factors | spinor Euler factors | identified eigenforms | special values


DIMENSIONS
     dim M183)= 8
     dim S183) = 4


DETERMINING INDICES
Introduce the notation
    [a b c d e f]=
 
a d e
d b f
e f c
 
We use the following set of determining Fourier coefficient indices.
t1½ [0 0 0 0 0 0]
t2½ [2 0 0 0 0 0]
t3½ [2 2 0 1 0 0]
t4½ [2 2 0 0 0 0]
t5½ [2 2 2 1 1 1]
t6½ [2 2 2 1 0 0]
t7½ [2 2 2 0 0 0]
t8½ [4 4 4 2 2 2]


PULLBACK OF EISENSTEIN SERIES
Let E(6)18 denote the monic Eisenstein series of degree 6 and weight 18.
The rescaled Eisenstein series
     E(6)18 = -88361515511741542241620438735422234638389/1933845185986560 E(6)18
has rational Fourier coefficients that are near-integers in that their denominator prime factors are at most 35.
Consider the Witt map W3,3 and the pullback W*3,3E(6)18 ∈ M183) ⊗ M183).
The columns of the 8 x 8 matrix [a(ti x tj; W*3,3E(6)18)] are determining truncations of an M183)-basis gt1,...,gt8. Each gt is the cofficient function of e(tr(t,z)) in the Fourier expansion of W*3,3E(6)18(w,z). For convenience let gj = gtj for j=1,...,8. The basis elements g1,...,g8 are not Hecke eigenforms. The matrix [a(ti x tj; W*3,3E(6)18)] is as follows (switch to factored form).
t1t2t3t4t5t6t7t8
t1 -88361515511741542241620438735422234638389/19338451859865602014304956157055240650612960435457967/67315691520-1676966135848108719283695971538017/8414461440-3030655468563872673655395910067629/1319915523414105522856211405374241569/10999296105119910515024127823838029321/515592223750233651427526874011669707553/10999296961029263619615285198271191047877597281057/10999296
t2 2014304956157055240650612960435457967/673156915201624433971675241850680181793116281/44520960-5509635668739119404276766813431/5565120-6170510849871612808924851519947/8729671581889083002258925480101/21824472528535444824419918533903/3412185228922458697085918771459237/21824-1335024185923399801529550660511398515547/21824
t3 -1676966135848108719283695971538017/8414461440-5509635668739119404276766813431/5565120-1657797180436310610022852866919/69564024132153334864745357813796997/1091240608413339658966590197749/2728824659028463571537981469176/341-10856459316476224278615179787/27281289552554921151219361278218930692597/2728
t4 -3030655468563872673655395910067629/131991552-6170510849871612808924851519947/8729624132153334864745357813796997/10912-1750602966372722478880125778305/43648-27052707731868889355227185/10912458695705235266432284820890/3412813162253375958988076561224655/10912545646506567655785425943088972619807695/10912
t5 3414105522856211405374241569/1099929671581889083002258925480101/2182440608413339658966590197749/2728-27052707731868889355227185/1091240478365588210294777147935/2728-7216336328070128002646040/341-20570036933559517999799265/2728-2522480163293255673384621072265185/2728
t6 105119910515024127823838029321/515592472528535444824419918533903/341824659028463571537981469176/341458695705235266432284820890/341-7216336328070128002646040/341657320517748565276612851200/341-3353430442644761020685031960/341-174448011704278127472905882404348440/341
t7 223750233651427526874011669707553/109992962185228922458697085918771459237/21824-10856459316476224278615179787/27282813162253375958988076561224655/10912-20570036933559517999799265/2728-3353430442644761020685031960/341-298097086009144631006254913505/2728-128321399478199872671629989808743141345/2728
t8 961029263619615285198271191047877597281057/10999296-1335024185923399801529550660511398515547/218241289552554921151219361278218930692597/2728545646506567655785425943088972619807695/10912-2522480163293255673384621072265185/2728-174448011704278127472905882404348440/341-128321399478199872671629989808743141345/2728-455010527897205370786515640340014845315544846305/2728


EIGENFORM NUMBER FIELD GENERATORS
  • a = Sqrt[2356201]
  • b = Sqrt[18295489]

  • INDICES FOR COMPUTING HECKE ACTION
    Fourier coefficients for the following indices are needed to determine the action of the Hecke operator T(2) at the determining indices t1,...,t8 and also the action of Ti(4) for i=0,1,2,3 at t5.
    u1½ [0 0 0 0 0 0]
    u2½ [2 0 0 0 0 0]
    u3½ [4 0 0 0 0 0]
    u4½ [2 2 0 0 0 0]
    u5½ [2 2 0 1 0 0]
    u6½ [4 4 0 0 0 0]
    u7½ [4 4 0 2 0 0]
    u8½ [2 2 2 1 1 1]
    u9½ [2 2 2 1 0 0]
    u10½ [2 2 2 0 0 0]
    u11½ [2 2 4 0 1 1]
    u12½ [2 2 4 0 0 0]
    u13½ [2 2 6 1 1 1]
    u14½ [4 4 4 2 2 2]
    u15½ [4 4 4 2 0 0]
    u16½ [2 6 6 0 0 2]
    u17½ [4 4 4 0 0 0]
    u18½ [8 8 8 4 4 4]


    EIGENFORM BASIS FROM PULLBACK-GENUS BASIS
    The M183) eigenform basis f1,...,f8 comprises algebraic integer linear combinations of the rational near-integer non-eigenform basis g1,...,g8 from above. Specifically fi = Σj=18 bi,jgj for each i, where the matrix (bi,j) is as follows.
    g1g2g3g4g5g6g7g8
    f1 -10000000
    f2 2872843867000000
    f3 -67202900494684938176544288+37955905036352555428512*a-201526200865237315820133985+118857413553114631525765*a-671176985626006311572317321-365075163419234868705971*a-246149264514486832554912121/2+(165068985928173128799229*a)/20000
    f4 -67202900494684938176544288-37955905036352555428512*a-201526200865237315820133985-118857413553114631525765*a-671176985626006311572317321+365075163419234868705971*a-246149264514486832554912121/2-(165068985928173128799229*a)/20000
    f5 -18148524847419393399055188077280-65113969784846999719016160*b-660111424834562398043235398520-948603906535624873336440*b-51585879553605230902116523200+2668594030421019484449600*b-4312489122944918274618005148+1814735135417189226268644*b463403724294314947933910150496+31018590135968683126262112*b-4859687800234813657721720544+2650475706009755747411232*b-293651514672119719899580808+665940241930039604789624*b-18532153914277914679169/2-(66811140722575393*b)/2
    f6 -18148524847419393399055188077280+65113969784846999719016160*b-660111424834562398043235398520+948603906535624873336440*b-51585879553605230902116523200-2668594030421019484449600*b-4312489122944918274618005148-1814735135417189226268644*b463403724294314947933910150496-31018590135968683126262112*b-4859687800234813657721720544-2650475706009755747411232*b-293651514672119719899580808-665940241930039604789624*b-18532153914277914679169/2+(66811140722575393*b)/2
    f7 -20837366772937707580789429277280-1851326226135283704281909280*a-754915161542151534384468242520-66294205253652082337480520*a-255941903099833651040759179200-138562108814665629761659200*a2421786374117932735401055092+3240113827377212055224892*a-999158411773547942293881041184-350386966180544450004530784*a-187865501178198915244904637024-127952855362729897952622624*a4291147970447657373130841432+2636557439741971618332232*a-21278201754931208656549/2-(1890492852727583399*a)/2
    f8 -20837366772937707580789429277280+1851326226135283704281909280*a-754915161542151534384468242520+66294205253652082337480520*a-255941903099833651040759179200+138562108814665629761659200*a2421786374117932735401055092-3240113827377212055224892*a-999158411773547942293881041184+350386966180544450004530784*a-187865501178198915244904637024+127952855362729897952622624*a4291147970447657373130841432-2636557439741971618332232*a-21278201754931208656549/2+(1890492852727583399*a)/2


    EIGENVALUES
    The T(2) and Ti(4) (i=0,1,2,3) eigenvalues of the eigenforms are shown in the following table.
    f λ2(f) λ0,4(f) λ1,4(f) λ2,4(f) λ3,4(f)
    f1 28149000932556979230278295859274766069383169211564439727809236682342480704813204525547524398046511104
    f2 -113392327118437899933754624830139622430224358826537026119270434587997685227192324398046511104
    f3 4448981592-2359368*a24503174988911586432+1806864483987072*a-2962603862614867968-7599854784872448*a1152903297166737408-57982058496*a4398046511104
    f4 4448981592+2359368*a24503174988911586432-1806864483987072*a-2962603862614867968+7599854784872448*a1152903297166737408+57982058496*a4398046511104
    f5 -62453952+6336*b946120373379072-203838529536*b694537653583872-99222552576*b67433134030848-19327352832*b4398046511104
    f6 -62453952-6336*b946120373379072+203838529536*b694537653583872+99222552576*b67433134030848+19327352832*b4398046511104
    f7 29325888-15552*a-192598426902528-352093298688*a535321034883072+103223918592*a-38480759488512-57982058496*a4398046511104
    f8 29325888+15552*a-192598426902528+352093298688*a535321034883072-103223918592*a-38480759488512+57982058496*a4398046511104


    EIGENFORM FOURIER COEFFICIENTS
    The eigenform Fourier coefficients at the indices ui are as follows (switch to factored denominator form).
    ua(u; f1)a(u; f2)a(u; f3)a(u; f4)a(u; f5)a(u; f6)a(u; f7)a(u; f8)
    u1 88361515511741542241620438735422234638389/19338451859865600000000
    u2 -2014304956157055240650612960435457967/673156915201296529820104899193136235570157500/527000000
    u3 -88006997839457900519265930854385594036197/22438563840-684567745015386773975932381043160000/527000000
    u4 3030655468563872673655395910067629/131991552-116571112696744183199062139175000/3119239773718219724700714365411918600771317504000000000-13390539062068831545090199005730214204160000000000*a19239773718219724700714365411918600771317504000000000+13390539062068831545090199005730214204160000000000*a0000
    u5 1676966135848108719283695971538017/8414461440-25904691710387596266458253150000/52726183849574890555482242231773164483214611219200000000/17+(15764269822595704935234115823179464690643200000000*a)/1726183849574890555482242231773164483214611219200000000/17-(15764269822595704935234115823179464690643200000000*a)/170000
    u6 26033529482955009513398966449053367700858285/1319915524041412299203538252909475635532095600000/313622905497836173997967432494709905792911202664448000000000-2325708047119038355526209166228583659521425408000000000*a3622905497836173997967432494709905792911202664448000000000+2325708047119038355526209166228583659521425408000000000*a0000
    u7 4801786375523081102119675711902242596456139/2804820480896393932169298761359853755373138400000/527880576130024313712097607079937223549465145717555200000000/17+(255046215882053672925424749413586770667238195200000000*a)/17880576130024313712097607079937223549465145717555200000000/17-(255046215882053672925424749413586770667238195200000000*a)/170000
    u8 -3414105522856211405374241569/109992964736770942055204747713590000/31-10367728895481441866995206852553536717315840000000-5237417334574675405008218617941452924160000000*a-10367728895481441866995206852553536717315840000000+5237417334574675405008218617941452924160000000*a6993566905565904068126239947857906803880755200000000+441937405143396528262704150406732814745600000000*b6993566905565904068126239947857906803880755200000000-441937405143396528262704150406732814745600000000*b-13804504127284323538703609751622976431718400000000000-4499872614502271909480909851173951897600000000000*a-13804504127284323538703609751622976431718400000000000+4499872614502271909480909851173951897600000000000*a
    u9 -105119910515024127823838029321/5155922065971596740324659747204480000/31-2081657021577790679137863827252467081406760960000000-599419224945384488224625329363703984855040000000*a-2081657021577790679137863827252467081406760960000000+599419224945384488224625329363703984855040000000*a-21404355235619786050719538075403551318946611200000000+9297959276185087539925196250459989763686400000000*b-21404355235619786050719538075403551318946611200000000-9297959276185087539925196250459989763686400000000*b-850622980325563673860309394489694304704921600000000000-581642208433201128311316974239141488230400000000000*a-850622980325563673860309394489694304704921600000000000+581642208433201128311316974239141488230400000000000*a
    u10 -223750233651427526874011669707553/10999296154280114289242946164513370420000/31-50603945630577723561028289375986729474132684800000000+35403884201959915423341002238840570811084800000000*a-50603945630577723561028289375986729474132684800000000-35403884201959915423341002238840570811084800000000*a-244694430578264573266186711101724675817865216000000000+509964063288431649248116696469873834852352000000000*b-244694430578264573266186711101724675817865216000000000-509964063288431649248116696469873834852352000000000*b4009890102902764931144196407030188181815296000000000000+2524563228770904477262427930408219836416000000000000*a4009890102902764931144196407030188181815296000000000000-2524563228770904477262427930408219836416000000000000*a
    u11 -6889243575423136265190872927610377/51559267290504797182584578438666528640000/31-10914929441026156725136156596430283656615540531200000000+7464492475988843851205810552434245283579699200000000*a-10914929441026156725136156596430283656615540531200000000-7464492475988843851205810552434245283579699200000000*a895072008925181781408071061400509772229246976000000000-5566925356679100255135407396048093023567872000000000*b895072008925181781408071061400509772229246976000000000+5566925356679100255135407396048093023567872000000000*b-101233319579449959241074611595546887945453568000000000000-67087872313983507835763254170490993901568000000000000*a-101233319579449959241074611595546887945453568000000000000+67087872313983507835763254170490993901568000000000000*a
    u12 -698271205523526663051020961491830237/5237766754272131299679008849782086866020000/31-1104987368912533172838826408439656811903562857717760000000+773788294593053041360328986547744247485476085760000000*a-1104987368912533172838826408439656811903562857717760000000-773788294593053041360328986547744247485476085760000000*a71536843487995970238542434107205055666698636492800000000-139815604146781161056219238878143368272648601600000000*b71536843487995970238542434107205055666698636492800000000+139815604146781161056219238878143368272648601600000000*b1111842023019233053259053390909163059458500198400000000000+786102620619940083978949107799542835288473600000000000*a1111842023019233053259053390909163059458500198400000000000-786102620619940083978949107799542835288473600000000000*a
    u13 -14663471569174514019400777585032968993/1099929610192755648940521567031330531895430000/31-10302893751766345971930722811337478647140512015435520000000-6203433856881133723322344236514176090367715892480000000*a-10302893751766345971930722811337478647140512015435520000000+6203433856881133723322344236514176090367715892480000000*a383247309592530662024349817664725339404664150425600000000+15782929785051947232666637151338726012275916800000000*b383247309592530662024349817664725339404664150425600000000-15782929785051947232666637151338726012275916800000000*b-49331307752797299965775223915167124261385011200000000000+224975301021924300091590909232313154089779200000000000*a-49331307752797299965775223915167124261385011200000000000-224975301021924300091590909232313154089779200000000000*a
    u14 -961029263619615285198271191047877597281057/10999296-5376190252249584996300200191654113120000/31-15352081265956541816067216170114047654230428565504000000000-15352081265956541816067216170114047654230428565504000000000-377528264246887113990326142518103581896131949363200000000000-377528264246887113990326142518103581896131949363200000000000-371333717908254700983040575795399935896219287552000000000000-371333717908254700983040575795399935896219287552000000000000
    u15 -1409025111951270577467990201715114354595621/24552-2349268196932602886377130709135371130880000/31-4856014014386640969374855724476306822184717715655884800000000+1510800397653418739633508650394446603010368294092800000000*a-4856014014386640969374855724476306822184717715655884800000000-1510800397653418739633508650394446603010368294092800000000*a2326618884788042914359847643986874335877937311514624000000000-169061266842722893270814194801453343189791408128000000000*b2326618884788042914359847643986874335877937311514624000000000+169061266842722893270814194801453343189791408128000000000*b6319852500112508278976434018467260300592710418432000000000000+2766720529392289239831091922043723991272652800000000000000*a6319852500112508278976434018467260300592710418432000000000000-2766720529392289239831091922043723991272652800000000000000*a
    u16 -62980091850030039985905547209991617392064674593/1099929614593241820300705970247869090731758778526980000/31972051282886358186814942285720400311045662264317036462080000000+728958876074487988954751515719507335303657182296145920000000*a972051282886358186814942285720400311045662264317036462080000000-728958876074487988954751515719507335303657182296145920000000*a43166744659199577029498774831135805462536514036144537600000000-1288861663971896832952253017343032038219534512947200000000*b43166744659199577029498774831135805462536514036144537600000000+1288861663971896832952253017343032038219534512947200000000*b-64779191195666637384422335529154309085108901196595200000000000-14254043120572165990535648695968466813045230796800000000000*a-64779191195666637384422335529154309085108901196595200000000000+14254043120572165990535648695968466813045230796800000000000*a
    u17 -62982013849902497959327915101800375192693979937/10999296-175605794013233168688156285189318693583680000/31-313304414124269142332000637594825859272709986012282224640000000+200849321850752441205638581381073732831978023271792640000000*a-313304414124269142332000637594825859272709986012282224640000000-200849321850752441205638581381073732831978023271792640000000*a52346351332451441769723899263616147835065794894705459200000000-20604275243424376596790826205851783068003454051942400000000*b52346351332451441769723899263616147835065794894705459200000000+20604275243424376596790826205851783068003454051942400000000*b-54569164240769304976788296971552430060417313039974400000000000-55940381544185831281264657446478320978041084313600000000000*a-54569164240769304976788296971552430060417313039974400000000000+55940381544185831281264657446478320978041084313600000000000*a
    u18 -270505689973472570836031250773878273643077550221421656865/109992963232662329686440506182212534186193826012466624000000/31-153524897924619349956420176203563486051674348358705570565324800000000-65375895863988699841577683977056500057966378802877117235200000000*a-153524897924619349956420176203563486051674348358705570565324800000000+65375895863988699841577683977056500057966378802877117235200000000*a7654463510812314835533253785503673257501237486147447291904000000000-822671161484229612769587806556610632118260410181746688000000000*b7654463510812314835533253785503673257501237486147447291904000000000+822671161484229612769587806556610632118260410181746688000000000*b780926068322816387781391705517087920836709953312915456000000000000+4680146161678809195562534492516145472978555884273664000000000000*a780926068322816387781391705517087920836709953312915456000000000000-4680146161678809195562534492516145472978555884273664000000000000*a


    STANDARD EULER FACTORS
    The standard 2-Euler factors Q2(f,  x,   st) are given by the following table (switch to expanded form).
    fQ2(f,  x,   st)/(1-x)
    f1(1-131072*x)*(1-65536*x)*(1-32768*x)*(1-x/32768)*(1-x/65536)*(1-x/131072)
    f2(1-65536*x)*(1-32768*x)*(1-x/32768)*(1-x/65536)*(1-(65*x)/512+x^2)
    f3(1-32768*x)*(1-x/32768)*(1+(7605/16384+(9*a)/16384)*x+x^2/2)*(1+(7605/8192+(9*a)/8192)*x+2*x^2)
    f4(1-32768*x)*(1-x/32768)*(1+(7605/16384-(9*a)/16384)*x+x^2/2)*(1+(7605/8192-(9*a)/8192)*x+2*x^2)
    f5(1+(-4995/16384+(3*b)/16384)*x+x^2/2)*(1-(65*x)/512+x^2)*(1+(-4995/8192+(3*b)/8192)*x+2*x^2)
    f6(1+(-4995/16384-(3*b)/16384)*x+x^2/2)*(1-(65*x)/512+x^2)*(1+(-4995/8192-(3*b)/8192)*x+2*x^2)
    f7(1+(7605/16384+(9*a)/16384)*x+x^2/2)*(1+(295*x)/512+x^2)*(1+(7605/8192+(9*a)/8192)*x+2*x^2)
    f8(1+(7605/16384-(9*a)/16384)*x+x^2/2)*(1+(295*x)/512+x^2)*(1+(7605/8192-(9*a)/8192)*x+2*x^2)


    SPINOR EULER FACTORS
    The spinor 2-Euler factors Q2(f,  x,   spin) are given by the following table (switch to expanded form).
    fQ2(f,  x,  spin)
    f1(1-281474976710656*x)*(1-8589934592*x)*(1-4294967296*x)*(1-2147483648*x)*(1-131072*x)*(1-65536*x)*(1-32768*x)*(1-x)
    f2(1+528*x+131072*x^2)*(1+17301504*x+140737488355328*x^2)*(1+34603008*x+562949953421312*x^2)*(1+1133871366144*x+604462909807314587353088*x^2)
    f3(1-4294967296*x)*(1-2147483648*x)*(1-131072*x)*(1-65536*x)*(1+(60840+72*a)*x+8589934592*x^2)*(1+(1993605120+2359296*a)*x+9223372036854775808*x^2)
    f4(1-4294967296*x)*(1-2147483648*x)*(1-131072*x)*(1-65536*x)*(1+(60840-72*a)*x+8589934592*x^2)*(1+(1993605120-2359296*a)*x+9223372036854775808*x^2)
    f5(1+17301504*x+140737488355328*x^2)*(1+34603008*x+562949953421312*x^2)*(1+(10549440-6336*b)*x+(433373914857472-62851645440*b)*x^2+(2969403378310462832640-1783425452438716416*b)*x^3+79228162514264337593543950336*x^4)
    f6(1+17301504*x+140737488355328*x^2)*(1+34603008*x+562949953421312*x^2)*(1+(10549440+6336*b)*x+(433373914857472+62851645440*b)*x^2+(2969403378310462832640+1783425452438716416*b)*x^3+79228162514264337593543950336*x^4)
    f7(1-14155776*x+140737488355328*x^2)*(1-28311552*x+562949953421312*x^2)*(1+(13141440+15552*a)*x+(359359213207552+287079137280*a)*x^2+(3698986517944483184640+4377498837804122112*a)*x^3+79228162514264337593543950336*x^4)
    f8(1-14155776*x+140737488355328*x^2)*(1-28311552*x+562949953421312*x^2)*(1+(13141440-15552*a)*x+(359359213207552-287079137280*a)*x^2+(3698986517944483184640-4377498837804122112*a)*x^3+79228162514264337593543950336*x^4)


    IDENTIFIED EIGENFORMS
    Based on matching standard and spinor 2-Euler factors we identify the following eigenforms.
  • f1 = E 18 : Basic Siegel Eisenstein series
  • f2 = K1 18 1 : Klingen lift from degree 1
  • f3 = K2 18 1 li : Klingen lift of degree 2 lift
  • f4 = K2 18 2 li : Klingen lift of degree 2 lift
  • f5 = M1 18 1 1 : Miyawaki lift of type 1
  • f6 = M1 18 2 1 : Miyawaki lift of type 1
  • f7 = M2 18 1 1 : Conjectural Miyawaki lift of type 2
  • f8 = M2 18 2 1 : Conjectural Miyawaki lift of type 2

  • SPECIAL VALUES (return to top)

    The special values ci such that W*3,3E(6)18i=18 ci fi⊗fi are given in the following table (switch to factored denominator form).
    c1-1933845185986560/88361515511741542241620438735422234638389
    c2527/56874873618541612905307245756099052500
    c3-260516305465134039321405658307269/1392747843306888023205107560193427337095935802143179698609472356021139028298349775015788670767595520000000000000-(185075913360897737242322869336210081*a)/3281593861147532867163897638435313685092781032945562149043337374729363799615600038241976281851279331819520000000000000
    c4-260516305465134039321405658307269/1392747843306888023205107560193427337095935802143179698609472356021139028298349775015788670767595520000000000000+(185075913360897737242322869336210081*a)/3281593861147532867163897638435313685092781032945562149043337374729363799615600038241976281851279331819520000000000000
    c547241888338514346677413/653323864632863892193806880848740218573616084637454626656661394595253675687526953488744448000000000000000-(1232243313434233045361029*b)/11952879578828050378128979656692437332771188766707620109996055321542123075750716814756835672195072000000000000000
    c647241888338514346677413/653323864632863892193806880848740218573616084637454626656661394595253675687526953488744448000000000000000+(1232243313434233045361029*b)/11952879578828050378128979656692437332771188766707620109996055321542123075750716814756835672195072000000000000000
    c74565530545101801323753/63225954154553014157699575735615462447886234204780836214024426744406878577583907143680000000000000000000-(6882343785622625101761803*a)/148973056404911966511385898047832888235171992919538811068320568319598231711381779595845959680000000000000000000
    c84565530545101801323753/63225954154553014157699575735615462447886234204780836214024426744406878577583907143680000000000000000000+(6882343785622625101761803*a)/148973056404911966511385898047832888235171992919538811068320568319598231711381779595845959680000000000000000000


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