DEGREE 3 WEIGHT 14

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


DIMENSIONS
     dim M143)= 3
     dim S143) = 1


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 2 0 1 0 0]
t3½ [2 2 2 1 1 1]


PULLBACK OF EISENSTEIN SERIES
Let E(6)14 denote the monic Eisenstein series of degree 6 and weight 14.
The rescaled Eisenstein series
     E(6)14 = -12080581424652674443/20832215040 E(6)14
has rational Fourier coefficients that are near-integers in that their denominator prime factors are at most 27.
Consider the Witt map W3,3 and the pullback W*3,3E(6)14 ∈ M143) ⊗ M143).
The columns of the 3 x 3 matrix [a(ti x tj; W*3,3E(6)14)] are determining truncations of an M143)-basis gt1,...,gt3. Each gt is the cofficient function of e(tr(t,z)) in the Fourier expansion of W*3,3E(6)14(w,z). For convenience let gj = gtj for j=1,...,3. The basis elements g1,...,g3 are not Hecke eigenforms. The matrix [a(ti x tj; W*3,3E(6)14)] is as follows (switch to factored form).
t1t2t3
t1 -12080581424652674443/20832215040-11218859197184483/119232018361471681153/8832
t2 -11218859197184483/1192320-2009862675060793/62108191964515763/46
t3 18361471681153/88328191964515763/46-671924037015/46


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,...,t3 and also the action of Ti(4) for i=0,1,2,3 at t3.
u1½ [0 0 0 0 0 0]
u2½ [2 2 0 1 0 0]
u3½ [4 4 0 2 0 0]
u4½ [2 2 2 1 1 1]
u5½ [2 2 2 0 0 0]
u6½ [2 2 4 0 0 0]
u7½ [2 2 6 1 1 1]
u8½ [4 4 4 2 2 2]
u9½ [2 6 6 0 0 2]
u10½ [4 4 4 0 0 0]
u11½ [8 8 8 4 4 4]


EIGENFORM BASIS FROM PULLBACK-GENUS BASIS
The M143) eigenform basis f1,...,f3 comprises integer linear combinations of the rational near-integer non-eigenform basis g1,...,g3 from above. Specifically fi = Σj=13 bi,jgj for each i, where the matrix (bi,j) is as follows.
g1g2g3
f1 -100
f2 10675392-6579310
f3 34944-2917-3455


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 68778211329472438655879974224588920179824598329661441231528258437121073741824
f2 2507976020996798438016051629363527680175995554365441073741824
f3 -293760-8714870784046489927680-73987522561073741824


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)
u1 12080581424652674443/2083221504000
u2 11218859197184483/11923201124913316332960000
u3 125526772203391842130081/39744013768938991915430400000
u4 -18361471681153/8832-94974591714710400-396362502144000
u5 -75226949477683841/883217898982533165888000-7927250042880000
u6 -102709867807354897963/294424451529733772038374400133177800720384000
u7 -308054394834058691201/8832-41697939506274329382720091956100497408000
u8 -1262715114805259643875969/8832-2418597082530849040128000132670456717639680000
u9 -5169557528790298430001336961/8832-1689325359518984403257395200-2994395038597251072000
u10 -5172081726501531983973675649/88322994686921115518608336896004835394221355565056000
u11 -86773152975921590746522484699787905/8832-271300257627801326343899504640008719725433918649794560000


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(1-8192*x)*(1-4096*x)*(1-2048*x)*(1-x/2048)*(1-x/4096)*(1-x/8192)
f2(1-2048*x)*(1-x/2048)*(1+(3*x)/512+x^2/2)*(1+(3*x)/256+2*x^2)
f3(1+(3*x)/512+x^2/2)*(1+(55*x)/32+x^2)*(1+(3*x)/256+2*x^2)


SPINOR EULER FACTORS (return to top)

The spinor 2-Euler factors Q2(f,  x,   spin) are given by the following table (switch to expanded form).
fQ2(f,  x,  spin)
f1(1-68719476736*x)*(1-33554432*x)*(1-16777216*x)*(1-8388608*x)*(1-8192*x)*(1-4096*x)*(1-2048*x)*(1-x)
f2(1-16777216*x)*(1-8388608*x)*(1-8192*x)*(1-4096*x)*(1+48*x+33554432*x^2)*(1+98304*x+140737488355328*x^2)
f3(1+98304*x+34359738368*x^2)*(1+196608*x+137438953472*x^2)*(1-1152*x-118106882048*x^2-79164837199872*x^3+4722366482869645213696*x^4)


IDENTIFIED EIGENFORMS
Based on matching standard and spinor 2-Euler factors we identify the following eigenforms.
  • f1 = E 14 : Basic Siegel Eisenstein series
  • f2 = K2 14 1 li : Klingen lift of degree 2 lift
  • f3 = M2 14 1 1 : Conjectural Miyawaki lift of type 2

  • SPECIAL VALUES
    The special values ci such that W*3,3E(6)14i=13 ci fi⊗fi are given in the following table (switch to factored denominator form).
    c1-20832215040/12080581424652674443
    c2-1/74011534312826070576000
    c31/1369432444907520000


    MORE COEFFICIENTS
    Showing more coefficients for all forms | cusp forms | hide more coefficients

    We give the positive definite coefficients of the eigenforms up to determinant 70 (103 coefficients).
    2udet(2u)a(u; f1)a(u; f2)a(u; f3)
    [2 2 2 1 1 1]4 -18361471681153/8832-94974591714710400-396362502144000
    [2 2 2 1 0 0]6 -1340387432724169/4968-28048923987106176002378175012864000
    [2 2 2 0 0 0]8 -75226949477683841/883217898982533165888000-7927250042880000
    [2 2 4 1 1 1]10 -342092578891561543/2760-1497955409786493504000-91163375493120000
    [2 2 4 0 1 1]12 -5491567311870920393/4968418750365762279936000380508002058240000
    [2 2 4 1 0 0]12 -4879028616587656313/4416-13114263861056746080000535089377894400000
    [2 2 4 0 1 0]14 -13576196738969310905/19324516000746581361945600-410631552221184000
    [2 2 4 0 0 0]16 -102709867807354897963/294424451529733772038374400133177800720384000
    [2 2 6 1 1 1]16 -308054394834058691201/8832-41697939506274329382720091956100497408000
    [2 2 6 1 0 0]18 -356169089010898910849/2484-1714792964809522308480000-3210536267366400000
    [2 2 6 0 1 1]20 -2241390595258982910889/44163361299670281611741760003975515896504320000
    [2 4 4 1 1 2]20 -1401553295718727641671/2760-598230868189234452480006500345035161600000
    [2 2 6 0 1 0]22 -879520203944921538583/5521043707338671383641024000-9635572427120640000
    [2 2 8 1 1 1]22 -879520203944921538583/552-190480143003114899088000005013985652121600000
    [2 2 6 0 0 0]24 -22487970822886284457673/4968295534326684658215813120025417934537490432000
    [2 4 4 0 0 2]24 -19989380242159627914361/4416323273171243170963584000-41237554723061760000
    [2 2 8 1 0 0]24 -22482480595961846261449/4968-54105379233281564064307200-10292741455675392000
    [2 4 4 1 1 1]24 -7497820349936907551299/1656-613840396926251090534400-26768737944797184000
    [2 4 4 1 0 1]26 -84878398849505663395361/7176-47660545362362148437760030125928337956864000
    [2 2 8 0 1 1]28 -55621678039557266777785/193218895012391720534952960000-8878520048025600000
    [2 4 4 1 0 0]28 -55621678039557266777785/1932-11967092472623385968640005771038031216640000
    [2 2 10 1 1 1]28 -127073201495114221751953/4416-344026311512799699796320000-15953590711296000000
    [2 4 4 0 0 1]30 -90901182155644624767503/13803229300706630316506496000-76696144164864000000
    [2 2 8 0 1 0]30 -163621513453905752494897/24844319157894697688490201600029418024909127680000
    [2 2 10 1 0 0]30 -90901182155644624767503/1380-787483797572685433739904000123070556915712000000
    [2 2 8 0 0 0]32 -1262098855635138457850497/883293767587710091128763776000-108698452587970560000
    [2 4 4 0 0 0]32 -1262098855635138457850497/8832718501756625526042892800080414024434974720000
    [4 4 4 2 2 2]32 -1262715114805259643875969/8832-2418597082530849040128000132670456717639680000
    [2 2 12 1 1 1]34 -13606280155810240299047/46-3536276046830606081315174400-64106085646761984000
    [2 4 6 1 1 2]34 -13606280155810240299047/46-19792901099269903704422400-186629662309515264000
    [2 2 10 0 1 1]36 -5185812177996539381692273/8832384919278269967053489923200-22520128284315648000
    [2 4 6 0 1 2]36 -5185831694074282788955219/883216792281894126082326998400331288102904500224000
    [2 2 12 1 0 0]36 -2592910968017705542661873/4416-7019804619260868046155667200-301084091153620992000
    [4 4 4 2 2 1]36 -1459224757677652837748353/2484-4938434884050880278528000329329675781406720000
    [2 2 10 0 1 0]38 -206754778225003769780383/18473710771508330234988371200082867508323246080000
    [2 4 6 1 1 1]38 -206754778225003769780383/184-73697104834996273912512000297283767483064320000
    [2 2 10 0 0 0]40 -9182977268776052985912233/44161363856595317463137159808000-133574163222528000000
    [2 4 6 0 0 2]40 -5739361430153347475765831/27604643111558206756435046400037797128204451840000
    [2 4 6 1 0 1]40 -1913587547118829103948653/920-14268929907393030694809600013983669075640320000
    [2 2 14 1 1 1]40 -5737960218950207639685703/2760-24854808454840528017106176000394555089134223360000
    [2 4 6 0 1 1]42 -3607480573675664262987505/966100667328001669462328064000-252704876866928640000
    [2 2 14 1 0 0]42 -9276343709143338187892569/2484-4464658699707953225755584000095721544267776000000
    [2 4 6 1 0 0]42 -9276343709143338187892569/2484-257578271766238000862592000-319793193979822080000
    [2 2 12 0 1 1]44 -3603394275562343543574551/5524281700698047756309534208000-291595965577297920000
    [2 4 6 0 1 0]44 -28813081881253991075660233/4416173429367343244335133395200-774952902416867328000
    [4 4 4 2 1 1]44 -3603394275562343543574551/552-2549554596289561832294400030694312166031360000
    [2 4 6 0 0 1]46 -3070701739252478272547749/276263326678999074769886438400425533196851789824000
    [2 2 12 0 1 0]46 -3070701739252478272547749/2767296292970461747125107942400180486043526283264000
    [2 2 16 1 1 1]46 -3070701739252478272547749/276-133012363424413055189381913600-464094511960375296000
    [2 2 12 0 0 0]48 -92110723001655684133156553/496812159221338771959810226176000782324452231741440000
    [2 4 6 0 0 0]48 -27292168783638150841626323/1472441800980139948444193024000651239445522677760000
    [2 2 16 1 0 0]48 -81856521849700909484620921/4416-221609031010695707789884320000-124140735671500800000
    [2 4 8 1 1 2]48 -92133216461365107423086281/4968-1246565441246692249622016000359199553942978560000
    [2 6 6 1 1 3]48 -81856521849700909484620921/441634492838316242621537376000-955170212166696960000
    [4 4 4 2 0 0]48 -30718569973691510237672003/1656-72918830432730046786560000-3036453856424755200000
    [4 4 4 1 1 -1]50 -41759348180270110612772959/1380155411364310752660929280002818811206497484800000
    [2 2 14 0 1 1]52 -213893565100772633227990873/441631767367977871244237600928000946684090995793920000
    [2 4 8 0 1 2]52 -347746800086424702930794017/71761226200822016125522000896000-3262634154648207360000
    [2 2 18 1 1 1]52 -213893565100772633227990873/4416-579071127174820679976524640000-508949270878003200000
    [2 4 8 1 1 1]52 -347746800086424702930794017/7176-32542531034770552796098560004217297022812160000
    [2 6 6 0 0 3]54 -378565713667422643593730987/4968-3307048653615966027990144004197048448027631616000
    [2 2 14 0 1 0]54 -189282500665292504614316729/248449977550274463935555764147200-2500465528375492608000
    [2 2 18 1 0 0]54 -378565713667422643593730987/4968-9110089024622980532295341184003070364253933182976000
    [2 4 8 1 0 1]54 -189282500665292504614316729/2484-51341655695258961075149568001227181113788055552000
    [4 4 4 1 1 1]54 -378565713667422643593730987/496811862073289556629128281600-5080117150154317824000
    [2 2 14 0 0 0]56 -227770798724380485393651385/1932773229851116280140294914048001748003027055280128000
    [2 4 8 0 0 2]56 -227770798724380485393651385/193229622688784443356077887488004965578692459757568000
    [2 4 8 1 0 0]56 -75942135608741101230372755/644-7930128362093684863558041600-3730842909380837376000
    [2 6 6 1 1 2]56 -227715190622537667096184505/1932394826950596338636550451200-5851211067250311168000
    [4 4 4 1 1 0]56 -75942135608741101230372755/64445030681274452521918054400189911543827267584000
    [4 4 6 2 2 2]56 -520618906525482966517751441/4416-499727810310157462771584000-8497520556464701440000
    [2 4 8 0 1 1]58 -99153848018933269904039881/55245182249181518116904691840001044252684523560960000
    [2 6 6 1 0 2]58 -99153848018933269904039881/552578627142976137624082560004296454578115338240000
    [2 2 20 1 1 1]58 -99153848018933269904039881/552-2147502624962029107780047040000-519254695933747200000
    [2 2 16 0 1 1]60 -670357340620651867971593009/24841769930036805874326948956160002149870211629056000000
    [2 4 8 0 1 0]60 -372422143291676027672459791/138068204428920812669364633600003410302968446976000000
    [2 2 20 1 0 0]60 -6203982334304236291306770/23-3224823987082066598923803648000-4607761650924257280000
    [2 6 6 1 1 1]60 -595584550362812198064335969/22089134646427418268990119040005480385408394444800000
    [4 4 4 1 0 0]60 -670357340620651867971593009/2484989262285023413556305920001490640098063155200000
    [4 4 6 2 2 1]60 -372422143291676027672459791/1380-1134624429647887575462912000-525101042840371200000
    [2 4 8 0 0 1]62 -27592048036537688332224883/6910083934671770389200680448000-5194536699098234880000
    [2 2 16 0 1 0]62 -27592048036537688332224883/692622630503759822124083665920002934636256874004480000
    [2 4 10 1 1 2]62 -27592048036537688332224883/69-26909839831279493396335104000303042914639216640000
    [2 2 16 0 0 0]64 -5169556912681545484827316865/8832383886089189914364578759449600-9474243376047980544000
    [2 4 8 0 0 0]64 -5169556912681545484827316865/883214672720644514763259205145600-519368055609360384000
    [2 6 6 0 0 2]64 -5169557528790298430001336961/8832-1689325359518984403257395200-2994395038597251072000
    [2 6 6 1 0 1]64 -1722765040626768393476052523/2944919530375739882737370848002942163973514723328000
    [2 2 22 1 1 1]64 -1722765040626768393476052523/2944-69960251338331253007284947232006628525497454952448000
    [4 4 4 0 0 0]64 -5172081726501531983973675649/88322994686921115518608336896004835394221355565056000
    [4 4 6 0 2 2]64 -5169556912681545484827316865/883227472454855455037585794560017512906161930633216000
    [2 2 22 1 0 0]66 -233706988112448496153813343/276-10123387409958467573698197120000-6768880630364160000000
    [2 6 6 1 0 0]66 -233706988112448496153813343/276224300463259207819929984000-7120470024265973760000
    [2 4 10 1 1 1]66 -2103354977331541348523197009/2484-57072131438594927412746419200-7099228165051072512000
    [4 4 6 1 2 2]66 -233706988112448496153813343/276224300463259207819929984000-7120470024265973760000
    [2 2 18 0 1 1]68 -5348900854852140027833037793/4416794403645532715272438869408000-5466302648693268480000
    [2 4 10 0 1 2]68 -5348900854852140027833037793/441630261797709179451830398368000-2842699974501703680000
    [2 4 10 1 0 1]68 -55744929798354554505195559/46-81440259352998913402979328000-3700630574017413120000
    [4 4 6 2 1 1]68 -55744929798354554505195559/46-52998936838726158470615040007018850605966295040000
    [2 6 6 0 0 1]70 -2367500817055473065460636343/1380-4803450231118698489052800000-10800997092174643200000
    [2 2 18 0 1 0]70 -1657250585275563079567663265/96611251711348689478269723563520004012092228202168320000
    [2 4 10 1 0 0]70 -2367500817055473065460636343/1380-11531511689401339145294400000029085754223580364800000
    [2 2 24 1 1 1]70 -2367500817055473065460636343/1380-20510405856104722872553540224000-3669325863598080000000
    [2 6 8 1 1 3]70 -1657250585275563079567663265/9663033181187537959216813824000-2606844467600916480000