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.
 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 cusp forms | all forms | hide more coefficients

We give the positive definite coefficients of the cusp eigenforms up to determinant 70 (103 coefficients).
 2u det(2u) a(u; f3) [2 2 2 1 1 1] 4 -396362502144000 [2 2 2 1 0 0] 6 2378175012864000 [2 2 2 0 0 0] 8 -7927250042880000 [2 2 4 1 1 1] 10 -91163375493120000 [2 2 4 0 1 1] 12 380508002058240000 [2 2 4 1 0 0] 12 535089377894400000 [2 2 4 0 1 0] 14 -410631552221184000 [2 2 4 0 0 0] 16 133177800720384000 [2 2 6 1 1 1] 16 91956100497408000 [2 2 6 1 0 0] 18 -3210536267366400000 [2 2 6 0 1 1] 20 3975515896504320000 [2 4 4 1 1 2] 20 6500345035161600000 [2 2 6 0 1 0] 22 -9635572427120640000 [2 2 8 1 1 1] 22 5013985652121600000 [2 2 6 0 0 0] 24 25417934537490432000 [2 4 4 0 0 2] 24 -41237554723061760000 [2 2 8 1 0 0] 24 -10292741455675392000 [2 4 4 1 1 1] 24 -26768737944797184000 [2 4 4 1 0 1] 26 30125928337956864000 [2 2 8 0 1 1] 28 -8878520048025600000 [2 4 4 1 0 0] 28 5771038031216640000 [2 2 10 1 1 1] 28 -15953590711296000000 [2 4 4 0 0 1] 30 -76696144164864000000 [2 2 8 0 1 0] 30 29418024909127680000 [2 2 10 1 0 0] 30 123070556915712000000 [2 2 8 0 0 0] 32 -108698452587970560000 [2 4 4 0 0 0] 32 80414024434974720000 [4 4 4 2 2 2] 32 132670456717639680000 [2 2 12 1 1 1] 34 -64106085646761984000 [2 4 6 1 1 2] 34 -186629662309515264000 [2 2 10 0 1 1] 36 -22520128284315648000 [2 4 6 0 1 2] 36 331288102904500224000 [2 2 12 1 0 0] 36 -301084091153620992000 [4 4 4 2 2 1] 36 329329675781406720000 [2 2 10 0 1 0] 38 82867508323246080000 [2 4 6 1 1 1] 38 297283767483064320000 [2 2 10 0 0 0] 40 -133574163222528000000 [2 4 6 0 0 2] 40 37797128204451840000 [2 4 6 1 0 1] 40 13983669075640320000 [2 2 14 1 1 1] 40 394555089134223360000 [2 4 6 0 1 1] 42 -252704876866928640000 [2 2 14 1 0 0] 42 95721544267776000000 [2 4 6 1 0 0] 42 -319793193979822080000 [2 2 12 0 1 1] 44 -291595965577297920000 [2 4 6 0 1 0] 44 -774952902416867328000 [4 4 4 2 1 1] 44 30694312166031360000 [2 4 6 0 0 1] 46 425533196851789824000 [2 2 12 0 1 0] 46 180486043526283264000 [2 2 16 1 1 1] 46 -464094511960375296000 [2 2 12 0 0 0] 48 782324452231741440000 [2 4 6 0 0 0] 48 651239445522677760000 [2 2 16 1 0 0] 48 -124140735671500800000 [2 4 8 1 1 2] 48 359199553942978560000 [2 6 6 1 1 3] 48 -955170212166696960000 [4 4 4 2 0 0] 48 -3036453856424755200000 [4 4 4 1 1 -1] 50 2818811206497484800000 [2 2 14 0 1 1] 52 946684090995793920000 [2 4 8 0 1 2] 52 -3262634154648207360000 [2 2 18 1 1 1] 52 -508949270878003200000 [2 4 8 1 1 1] 52 4217297022812160000 [2 6 6 0 0 3] 54 4197048448027631616000 [2 2 14 0 1 0] 54 -2500465528375492608000 [2 2 18 1 0 0] 54 3070364253933182976000 [2 4 8 1 0 1] 54 1227181113788055552000 [4 4 4 1 1 1] 54 -5080117150154317824000 [2 2 14 0 0 0] 56 1748003027055280128000 [2 4 8 0 0 2] 56 4965578692459757568000 [2 4 8 1 0 0] 56 -3730842909380837376000 [2 6 6 1 1 2] 56 -5851211067250311168000 [4 4 4 1 1 0] 56 189911543827267584000 [4 4 6 2 2 2] 56 -8497520556464701440000 [2 4 8 0 1 1] 58 1044252684523560960000 [2 6 6 1 0 2] 58 4296454578115338240000 [2 2 20 1 1 1] 58 -519254695933747200000 [2 2 16 0 1 1] 60 2149870211629056000000 [2 4 8 0 1 0] 60 3410302968446976000000 [2 2 20 1 0 0] 60 -4607761650924257280000 [2 6 6 1 1 1] 60 5480385408394444800000 [4 4 4 1 0 0] 60 1490640098063155200000 [4 4 6 2 2 1] 60 -525101042840371200000 [2 4 8 0 0 1] 62 -5194536699098234880000 [2 2 16 0 1 0] 62 2934636256874004480000 [2 4 10 1 1 2] 62 303042914639216640000 [2 2 16 0 0 0] 64 -9474243376047980544000 [2 4 8 0 0 0] 64 -519368055609360384000 [2 6 6 0 0 2] 64 -2994395038597251072000 [2 6 6 1 0 1] 64 2942163973514723328000 [2 2 22 1 1 1] 64 6628525497454952448000 [4 4 4 0 0 0] 64 4835394221355565056000 [4 4 6 0 2 2] 64 17512906161930633216000 [2 2 22 1 0 0] 66 -6768880630364160000000 [2 6 6 1 0 0] 66 -7120470024265973760000 [2 4 10 1 1 1] 66 -7099228165051072512000 [4 4 6 1 2 2] 66 -7120470024265973760000 [2 2 18 0 1 1] 68 -5466302648693268480000 [2 4 10 0 1 2] 68 -2842699974501703680000 [2 4 10 1 0 1] 68 -3700630574017413120000 [4 4 6 2 1 1] 68 7018850605966295040000 [2 6 6 0 0 1] 70 -10800997092174643200000 [2 2 18 0 1 0] 70 4012092228202168320000 [2 4 10 1 0 0] 70 29085754223580364800000 [2 2 24 1 1 1] 70 -3669325863598080000000 [2 6 8 1 1 3] 70 -2606844467600916480000