DEGREE 3 WEIGHT 14

Scroll through or jump to chosen results (a jump may take some time due to page reload)
 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.
 t1 ½ [0 0 0 0 0 0] t2 ½ [2 2 0 1 0 0] t3 ½ [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).
 t1 t2 t3 t1 -12080581424652674443/20832215040 -11218859197184483/1192320 18361471681153/8832 t2 -11218859197184483/1192320 -2009862675060793/6210 8191964515763/46 t3 18361471681153/8832 8191964515763/46 -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.
 u1 ½ [0 0 0 0 0 0] u2 ½ [2 2 0 1 0 0] u3 ½ [4 4 0 2 0 0] u4 ½ [2 2 2 1 1 1] u5 ½ [2 2 2 0 0 0] u6 ½ [2 2 4 0 0 0] u7 ½ [2 2 6 1 1 1] u8 ½ [4 4 4 2 2 2] u9 ½ [2 6 6 0 0 2] u10 ½ [4 4 4 0 0 0] u11 ½ [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.
 g1 g2 g3 f1 -1 0 0 f2 10675392 -657931 0 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 68778211329 4724386558799742245889 2017982459832966144 123152825843712 1073741824 f2 25079760 209967984380160 51629363527680 17599555436544 1073741824 f3 -293760 -87148707840 46489927680 -7398752256 1073741824

EIGENFORM FOURIER COEFFICIENTS
The eigenform Fourier coefficients at the indices ui are as follows (switch to factored denominator form).
 u a(u; f1) a(u; f2) a(u; f3) u1 12080581424652674443/20832215040 0 0 u2 11218859197184483/1192320 112491331633296000 0 u3 125526772203391842130081/397440 1376893899191543040000 0 u4 -18361471681153/8832 -94974591714710400 -396362502144000 u5 -75226949477683841/8832 17898982533165888000 -7927250042880000 u6 -102709867807354897963/2944 24451529733772038374400 133177800720384000 u7 -308054394834058691201/8832 -416979395062743293827200 91956100497408000 u8 -1262715114805259643875969/8832 -2418597082530849040128000 132670456717639680000 u9 -5169557528790298430001336961/8832 -1689325359518984403257395200 -2994395038597251072000 u10 -5172081726501531983973675649/8832 299468692111551860833689600 4835394221355565056000 u11 -86773152975921590746522484699787905/8832 -27130025762780132634389950464000 8719725433918649794560000

STANDARD EULER FACTORS
The standard 2-Euler factors Q2(f,  x,   st) are given by the following table (switch to expanded form).
 f Q2(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
The spinor 2-Euler factors Q2(f,  x,   spin) are given by the following table (switch to expanded form).
 f Q2(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 c3 1/1369432444907520000

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).
 2u det(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 -2804892398710617600 2378175012864000 [2 2 2 0 0 0] 8 -75226949477683841/8832 17898982533165888000 -7927250042880000 [2 2 4 1 1 1] 10 -342092578891561543/2760 -1497955409786493504000 -91163375493120000 [2 2 4 0 1 1] 12 -5491567311870920393/4968 418750365762279936000 380508002058240000 [2 2 4 1 0 0] 12 -4879028616587656313/4416 -13114263861056746080000 535089377894400000 [2 2 4 0 1 0] 14 -13576196738969310905/1932 4516000746581361945600 -410631552221184000 [2 2 4 0 0 0] 16 -102709867807354897963/2944 24451529733772038374400 133177800720384000 [2 2 6 1 1 1] 16 -308054394834058691201/8832 -416979395062743293827200 91956100497408000 [2 2 6 1 0 0] 18 -356169089010898910849/2484 -1714792964809522308480000 -3210536267366400000 [2 2 6 0 1 1] 20 -2241390595258982910889/4416 336129967028161174176000 3975515896504320000 [2 4 4 1 1 2] 20 -1401553295718727641671/2760 -59823086818923445248000 6500345035161600000 [2 2 6 0 1 0] 22 -879520203944921538583/552 1043707338671383641024000 -9635572427120640000 [2 2 8 1 1 1] 22 -879520203944921538583/552 -19048014300311489908800000 5013985652121600000 [2 2 6 0 0 0] 24 -22487970822886284457673/4968 2955343266846582158131200 25417934537490432000 [2 4 4 0 0 2] 24 -19989380242159627914361/4416 323273171243170963584000 -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 -476605453623621484377600 30125928337956864000 [2 2 8 0 1 1] 28 -55621678039557266777785/1932 18895012391720534952960000 -8878520048025600000 [2 4 4 1 0 0] 28 -55621678039557266777785/1932 -1196709247262338596864000 5771038031216640000 [2 2 10 1 1 1] 28 -127073201495114221751953/4416 -344026311512799699796320000 -15953590711296000000 [2 4 4 0 0 1] 30 -90901182155644624767503/1380 3229300706630316506496000 -76696144164864000000 [2 2 8 0 1 0] 30 -163621513453905752494897/2484 43191578946976884902016000 29418024909127680000 [2 2 10 1 0 0] 30 -90901182155644624767503/1380 -787483797572685433739904000 123070556915712000000 [2 2 8 0 0 0] 32 -1262098855635138457850497/8832 93767587710091128763776000 -108698452587970560000 [2 4 4 0 0 0] 32 -1262098855635138457850497/8832 7185017566255260428928000 80414024434974720000 [4 4 4 2 2 2] 32 -1262715114805259643875969/8832 -2418597082530849040128000 132670456717639680000 [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 -5185812177996539381692273/8832 384919278269967053489923200 -22520128284315648000 [2 4 6 0 1 2] 36 -5185831694074282788955219/8832 16792281894126082326998400 331288102904500224000 [2 2 12 1 0 0] 36 -2592910968017705542661873/4416 -7019804619260868046155667200 -301084091153620992000 [4 4 4 2 2 1] 36 -1459224757677652837748353/2484 -4938434884050880278528000 329329675781406720000 [2 2 10 0 1 0] 38 -206754778225003769780383/184 737107715083302349883712000 82867508323246080000 [2 4 6 1 1 1] 38 -206754778225003769780383/184 -73697104834996273912512000 297283767483064320000 [2 2 10 0 0 0] 40 -9182977268776052985912233/4416 1363856595317463137159808000 -133574163222528000000 [2 4 6 0 0 2] 40 -5739361430153347475765831/2760 46431115582067564350464000 37797128204451840000 [2 4 6 1 0 1] 40 -1913587547118829103948653/920 -142689299073930306948096000 13983669075640320000 [2 2 14 1 1 1] 40 -5737960218950207639685703/2760 -24854808454840528017106176000 394555089134223360000 [2 4 6 0 1 1] 42 -3607480573675664262987505/966 100667328001669462328064000 -252704876866928640000 [2 2 14 1 0 0] 42 -9276343709143338187892569/2484 -44646586997079532257555840000 95721544267776000000 [2 4 6 1 0 0] 42 -9276343709143338187892569/2484 -257578271766238000862592000 -319793193979822080000 [2 2 12 0 1 1] 44 -3603394275562343543574551/552 4281700698047756309534208000 -291595965577297920000 [2 4 6 0 1 0] 44 -28813081881253991075660233/4416 173429367343244335133395200 -774952902416867328000 [4 4 4 2 1 1] 44 -3603394275562343543574551/552 -25495545962895618322944000 30694312166031360000 [2 4 6 0 0 1] 46 -3070701739252478272547749/276 263326678999074769886438400 425533196851789824000 [2 2 12 0 1 0] 46 -3070701739252478272547749/276 7296292970461747125107942400 180486043526283264000 [2 2 16 1 1 1] 46 -3070701739252478272547749/276 -133012363424413055189381913600 -464094511960375296000 [2 2 12 0 0 0] 48 -92110723001655684133156553/4968 12159221338771959810226176000 782324452231741440000 [2 4 6 0 0 0] 48 -27292168783638150841626323/1472 441800980139948444193024000 651239445522677760000 [2 2 16 1 0 0] 48 -81856521849700909484620921/4416 -221609031010695707789884320000 -124140735671500800000 [2 4 8 1 1 2] 48 -92133216461365107423086281/4968 -1246565441246692249622016000 359199553942978560000 [2 6 6 1 1 3] 48 -81856521849700909484620921/4416 34492838316242621537376000 -955170212166696960000 [4 4 4 2 0 0] 48 -30718569973691510237672003/1656 -72918830432730046786560000 -3036453856424755200000 [4 4 4 1 1 -1] 50 -41759348180270110612772959/1380 15541136431075266092928000 2818811206497484800000 [2 2 14 0 1 1] 52 -213893565100772633227990873/4416 31767367977871244237600928000 946684090995793920000 [2 4 8 0 1 2] 52 -347746800086424702930794017/7176 1226200822016125522000896000 -3262634154648207360000 [2 2 18 1 1 1] 52 -213893565100772633227990873/4416 -579071127174820679976524640000 -508949270878003200000 [2 4 8 1 1 1] 52 -347746800086424702930794017/7176 -3254253103477055279609856000 4217297022812160000 [2 6 6 0 0 3] 54 -378565713667422643593730987/4968 -330704865361596602799014400 4197048448027631616000 [2 2 14 0 1 0] 54 -189282500665292504614316729/2484 49977550274463935555764147200 -2500465528375492608000 [2 2 18 1 0 0] 54 -378565713667422643593730987/4968 -911008902462298053229534118400 3070364253933182976000 [2 4 8 1 0 1] 54 -189282500665292504614316729/2484 -5134165569525896107514956800 1227181113788055552000 [4 4 4 1 1 1] 54 -378565713667422643593730987/4968 11862073289556629128281600 -5080117150154317824000 [2 2 14 0 0 0] 56 -227770798724380485393651385/1932 77322985111628014029491404800 1748003027055280128000 [2 4 8 0 0 2] 56 -227770798724380485393651385/1932 2962268878444335607788748800 4965578692459757568000 [2 4 8 1 0 0] 56 -75942135608741101230372755/644 -7930128362093684863558041600 -3730842909380837376000 [2 6 6 1 1 2] 56 -227715190622537667096184505/1932 394826950596338636550451200 -5851211067250311168000 [4 4 4 1 1 0] 56 -75942135608741101230372755/644 45030681274452521918054400 189911543827267584000 [4 4 6 2 2 2] 56 -520618906525482966517751441/4416 -499727810310157462771584000 -8497520556464701440000 [2 4 8 0 1 1] 58 -99153848018933269904039881/552 4518224918151811690469184000 1044252684523560960000 [2 6 6 1 0 2] 58 -99153848018933269904039881/552 57862714297613762408256000 4296454578115338240000 [2 2 20 1 1 1] 58 -99153848018933269904039881/552 -2147502624962029107780047040000 -519254695933747200000 [2 2 16 0 1 1] 60 -670357340620651867971593009/2484 176993003680587432694895616000 2149870211629056000000 [2 4 8 0 1 0] 60 -372422143291676027672459791/1380 6820442892081266936463360000 3410302968446976000000 [2 2 20 1 0 0] 60 -6203982334304236291306770/23 -3224823987082066598923803648000 -4607761650924257280000 [2 6 6 1 1 1] 60 -595584550362812198064335969/2208 913464642741826899011904000 5480385408394444800000 [4 4 4 1 0 0] 60 -670357340620651867971593009/2484 98926228502341355630592000 1490640098063155200000 [4 4 6 2 2 1] 60 -372422143291676027672459791/1380 -1134624429647887575462912000 -525101042840371200000 [2 4 8 0 0 1] 62 -27592048036537688332224883/69 10083934671770389200680448000 -5194536699098234880000 [2 2 16 0 1 0] 62 -27592048036537688332224883/69 262263050375982212408366592000 2934636256874004480000 [2 4 10 1 1 2] 62 -27592048036537688332224883/69 -26909839831279493396335104000 303042914639216640000 [2 2 16 0 0 0] 64 -5169556912681545484827316865/8832 383886089189914364578759449600 -9474243376047980544000 [2 4 8 0 0 0] 64 -5169556912681545484827316865/8832 14672720644514763259205145600 -519368055609360384000 [2 6 6 0 0 2] 64 -5169557528790298430001336961/8832 -1689325359518984403257395200 -2994395038597251072000 [2 6 6 1 0 1] 64 -1722765040626768393476052523/2944 91953037573988273737084800 2942163973514723328000 [2 2 22 1 1 1] 64 -1722765040626768393476052523/2944 -6996025133833125300728494723200 6628525497454952448000 [4 4 4 0 0 0] 64 -5172081726501531983973675649/8832 299468692111551860833689600 4835394221355565056000 [4 4 6 0 2 2] 64 -5169556912681545484827316865/8832 274724548554550375857945600 17512906161930633216000 [2 2 22 1 0 0] 66 -233706988112448496153813343/276 -10123387409958467573698197120000 -6768880630364160000000 [2 6 6 1 0 0] 66 -233706988112448496153813343/276 224300463259207819929984000 -7120470024265973760000 [2 4 10 1 1 1] 66 -2103354977331541348523197009/2484 -57072131438594927412746419200 -7099228165051072512000 [4 4 6 1 2 2] 66 -233706988112448496153813343/276 224300463259207819929984000 -7120470024265973760000 [2 2 18 0 1 1] 68 -5348900854852140027833037793/4416 794403645532715272438869408000 -5466302648693268480000 [2 4 10 0 1 2] 68 -5348900854852140027833037793/4416 30261797709179451830398368000 -2842699974501703680000 [2 4 10 1 0 1] 68 -55744929798354554505195559/46 -81440259352998913402979328000 -3700630574017413120000 [4 4 6 2 1 1] 68 -55744929798354554505195559/46 -5299893683872615847061504000 7018850605966295040000 [2 6 6 0 0 1] 70 -2367500817055473065460636343/1380 -4803450231118698489052800000 -10800997092174643200000 [2 2 18 0 1 0] 70 -1657250585275563079567663265/966 1125171134868947826972356352000 4012092228202168320000 [2 4 10 1 0 0] 70 -2367500817055473065460636343/1380 -115315116894013391452944000000 29085754223580364800000 [2 2 24 1 1 1] 70 -2367500817055473065460636343/1380 -20510405856104722872553540224000 -3669325863598080000000 [2 6 8 1 1 3] 70 -1657250585275563079567663265/966 3033181187537959216813824000 -2606844467600916480000