DEGREE 3 WEIGHT 20

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


DIMENSIONS
     dim M203)= 11
     dim S203) = 6


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 [4 4 0 2 0 0]
t6½ [2 2 2 1 1 1]
t7½ [2 2 2 1 0 0]
t8½ [2 2 2 0 0 0]
t9½ [2 2 4 0 1 1]
t10½ [4 4 4 2 2 2]
t11½ [4 4 4 2 0 0]


PULLBACK OF EISENSTEIN SERIES
Let E(6)20 denote the monic Eisenstein series of degree 6 and weight 20.
The rescaled Eisenstein series
     E(6)20 = 107441973609722531026066150430595653959970701892533/525311474688000 E(6)20
has rational Fourier coefficients that are near-integers in that their denominator prime factors are at most 39.
Consider the Witt map W3,3 and the pullback W*3,3E(6)20 ∈ M203) ⊗ M203).
The columns of the 11 x 11 matrix [a(ti x tj; W*3,3E(6)20)] are determining truncations of an M203)-basis gt1,...,gt11. Each gt is the cofficient function of e(tr(t,z)) in the Fourier expansion of W*3,3E(6)20(w,z). For convenience let gj = gtj for j=1,...,11. The basis elements g1,...,g11 are not Hecke eigenforms. The matrix [a(ti x tj; W*3,3E(6)20)] is as follows (switch to nonfactored form).
t1t2t3t4t5t6t7t8t9t10t11
t1 283*617*151628697551*154210205991661*26315271553053477373/(2^13*3^6*5^3*7*11*13*19*37)151628697551*154210205991661*26315271553053477373/(2^9*3^5*5*7*13*19*37)7691*8609*151628697551*26315271553053477373/(2^6*3^5*5*13*37)79*349*87224971*151628697551*26315271553053477373/(2^8*3^4*5*7*13*19*37)109*7691*8609*174763*151628697551*26315271553053477373/(2^6*3^4)73*151628697551*26315271553053477373/(2^6*3*5*13*37)757*151628697551*26315271553053477373/(2^3*3^4*5)73*109*151628697551*26315271553053477373/(2^6*3)13*37*109*757*151628697551*26315271553053477373/(2^3*3^4)5^2*13^2*37^2*73*109^3*151628697551*26315271553053477373/(2^6*3)13*37*109*757*87211*262657*151628697551*26315271553053477373/(2^3*3^3)
t2 151628697551*154210205991661*26315271553053477373/(2^9*3^5*5*7*13*19*37)5*11*97*1033*244199*151628697551*26315271553053477373/(2^5*3^4*7*13*19*37)5*11*19*1619*151628697551*26315271553053477373/(2^2*3^4*13*37)5*11*97*7732427*151628697551*26315271553053477373/(2^4*3^3*7*13*19*37)5^2*11*109*1229*57679*151628697551*26315271553053477373/(2^2*3^3)5*11*23*151628697551*77164768567975057/(2^2*13*37)2*5*11^2*347*151628697551*3227008665665987/(3^3)5^2*11*29*151628697551*3360281728589778199/(2^2)2*5^2*11*13*1277*244530679*80378574721*151628697551/(3^3)5^2*11*13^2*53*149*151628697551*70348105062766120689563/(2^2)2*5^5*11*13*114974938229*151628697551*801345191012719049/(3^2)
t3 7691*8609*151628697551*26315271553053477373/(2^6*3^5*5*13*37)5*11*19*1619*151628697551*26315271553053477373/(2^2*3^4*13*37)2*5*7*101*14177*119869*185527*9773653*151628697551/(3^4*13*37)5*1031*76564469*151628697551*4926945230257/(2*3^3*13*37)2*5^2*7*23*2903*16657*105619*151628697551*178394708671/(3^3)2*5*7*43*151628697551*31913318658756547/(13*37)2^4*5*7*41*181*151628697551*479797579043969/(3^3)2*5^2*7*11^2*100391*4198439699*1516286975512^4*5^3*7*13*1459*151628697551*1081747135708691/(3^3)2*5^2*7*13*151628697551*8539017812759567398210092^4*5^2*7*11*13*3590759*6081091*151628697551*5132139491977/(3^2)
t4 79*349*87224971*151628697551*26315271553053477373/(2^8*3^4*5*7*13*19*37)5*11*97*7732427*151628697551*26315271553053477373/(2^4*3^3*7*13*19*37)5*1031*76564469*151628697551*4926945230257/(2*3^3*13*37)5*263*151628697551*531910085533*561790921818923/(2^3*3^2*7*13*19*37)5^2*151628697551*102732600734329*362302504512373/(2*3^2)3*5*2364353*151628697551*3241595697617/(2*13*37)2^2*5*151628697551*73616523910143654589/(3^2)3*5^2*11*137*21345001*23254046503*151628697551/(2)2^2*5^2*13*53*151628697551*1895021128514743395401/(3^2)3*5^2*13*2011*1513957*670167301*687631579*151628697551/(2)2^2*5^3*11*13*149*151628697551*5541929722507*47417671579679/(3)
t5 109*7691*8609*174763*151628697551*26315271553053477373/(2^6*3^4)5^2*11*109*1229*57679*151628697551*26315271553053477373/(2^2*3^3)2*5^2*7*23*2903*16657*105619*151628697551*178394708671/(3^3)5^2*151628697551*102732600734329*362302504512373/(2*3^2)2*5^3*7*13*151628697551*6018707792291*1997908115676524071109/(3^2)2*3*5^6*7*101*151628697551*4474832990903392^4*5^2*7*13*43*1634632981*133912524937*151628697551/(3^2)2*3*5^2*7*11*13*6301*41227*151628697551*1612992256906392^4*5^2*7*13*641*17477*151628697551*192779626948725368243/(3^2)2*3*5^2*7*13*5179*304163*151628697551*165506005206129753384966192^4*5^2*7*11*13^2*31*509*6089*26267*127112977*151628697551*944777886859907/(3)
t6 73*151628697551*26315271553053477373/(2^6*3*5*13*37)5*11*23*151628697551*77164768567975057/(2^2*13*37)2*5*7*43*151628697551*31913318658756547/(13*37)3*5*2364353*151628697551*3241595697617/(2*13*37)2*3*5^6*7*101*151628697551*4474832990903392*3^4*5*7*2569166306826890960364216917/(13*37)2^4*3*5*7*23799483193*1478077897982812*3^4*5^2*7*11*1277*1107845023*224012521612^4*3*5^2*7*13*982095143683*367318816606672*3^4*5^2*7*13*53*4373*26783*342324982875187903872^4*3^2*5^3*7*11*13*19^3*503*5653*1851132293*133593750569
t7 757*151628697551*26315271553053477373/(2^3*3^4*5)2*5*11^2*347*151628697551*3227008665665987/(3^3)2^4*5*7*41*181*151628697551*479797579043969/(3^3)2^2*5*151628697551*73616523910143654589/(3^2)2^4*5^2*7*13*43*1634632981*133912524937*151628697551/(3^2)2^4*3*5*7*23799483193*1478077897982812^7*5*7*13*367*727*14030679921089513878121/(3^2)2^4*3*5^2*7*11*13*5297*4246907537*4216248381792^7*5^2*7*13^3*5119*127114052405372502455387/(3^2)2^4*3*5^2*7*13*43*463163297*399844249410432599200992^7*5^2*7*11*13*29*43*56489*18669155464301*77071518266279/(3)
t8 73*109*151628697551*26315271553053477373/(2^6*3)5^2*11*29*151628697551*3360281728589778199/(2^2)2*5^2*7*11^2*100391*4198439699*1516286975513*5^2*11*137*21345001*23254046503*151628697551/(2)2*3*5^2*7*11*13*6301*41227*151628697551*1612992256906392*3^4*5^2*7*11*1277*1107845023*224012521612^4*3*5^2*7*11*13*5297*4246907537*4216248381792*3^4*5^3*7*11*13^2*1123*4244508419*76993985321512^4*3*5^2*7*11*13*29*65587*41634822169*80746309876932*3^4*5^4*7*11*13*661*1993425854724098022754288560112^4*3^2*5^2*7*11*13*677*750311*9609653887629820893232096427
t9 13*37*109*757*151628697551*26315271553053477373/(2^3*3^4)2*5^2*11*13*1277*244530679*80378574721*151628697551/(3^3)2^4*5^3*7*13*1459*151628697551*1081747135708691/(3^3)2^2*5^2*13*53*151628697551*1895021128514743395401/(3^2)2^4*5^2*7*13*641*17477*151628697551*192779626948725368243/(3^2)2^4*3*5^2*7*13*982095143683*367318816606672^7*5^2*7*13^3*5119*127114052405372502455387/(3^2)2^4*3*5^2*7*11*13*29*65587*41634822169*80746309876932^7*5^2*7*13*541*21752469677135459388316187334049/(3^2)2^4*3*5^2*7*13*41*39153511909838347571586057953238598012^7*5^2*7*11*13*29*53*14569624925983066098711824837303838413/(3)
t10 5^2*13^2*37^2*73*109^3*151628697551*26315271553053477373/(2^6*3)5^2*11*13^2*53*149*151628697551*70348105062766120689563/(2^2)2*5^2*7*13*151628697551*8539017812759567398210093*5^2*13*2011*1513957*670167301*687631579*151628697551/(2)2*3*5^2*7*13*5179*304163*151628697551*165506005206129753384966192*3^4*5^2*7*13*53*4373*26783*342324982875187903872^4*3*5^2*7*13*43*463163297*399844249410432599200992*3^4*5^4*7*11*13*661*1993425854724098022754288560112^4*3*5^2*7*13*41*39153511909838347571586057953238598012*3^4*5^3*7*13*33749*48827566973*2081142652328573375463409278412^4*3^2*5^2*7*11*13*1223*32441*6529869521178337668852992542777553913743
t11 13*37*109*757*87211*262657*151628697551*26315271553053477373/(2^3*3^3)2*5^5*11*13*114974938229*151628697551*801345191012719049/(3^2)2^4*5^2*7*11*13*3590759*6081091*151628697551*5132139491977/(3^2)2^2*5^3*11*13*149*151628697551*5541929722507*47417671579679/(3)2^4*5^2*7*11*13^2*31*509*6089*26267*127112977*151628697551*944777886859907/(3)2^4*3^2*5^3*7*11*13*19^3*503*5653*1851132293*1335937505692^7*5^2*7*11*13*29*43*56489*18669155464301*77071518266279/(3)2^4*3^2*5^2*7*11*13*677*750311*96096538876298208932320964272^7*5^2*7*11*13*29*53*14569624925983066098711824837303838413/(3)2^4*3^2*5^2*7*11*13*1223*32441*65298695211783376688529925427775539137432^7*5^2*7*11*13*199*120851*66461167*5202334619*51791942157367425355627127


EIGENFORM NUMBER FIELD GENERATORS
  • a = Sqrt[63737521]
  • b1 = Root[-4382089113600+893191104 #1-54971 #1^2+#1^3&,1]
  • b2 = Root[-4382089113600+893191104 #1-54971 #1^2+#1^3&,2]
  • b3 = Root[-4382089113600+893191104 #1-54971 #1^2+#1^3&,3]

  • 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,...,t11 and also the action of Ti(4) for i=0,1,2,3 at t6.
    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 6 0 0 0 0]
    u6½ [2 2 0 1 0 0]
    u7½ [4 4 0 0 0 0]
    u8½ [4 4 0 2 0 0]
    u9½ [8 8 0 4 0 0]
    u10½ [2 2 2 1 1 1]
    u11½ [2 2 2 1 0 0]
    u12½ [2 2 2 0 0 0]
    u13½ [2 2 4 0 1 1]
    u14½ [2 2 4 0 0 0]
    u15½ [2 2 6 1 1 1]
    u16½ [2 2 6 0 0 0]
    u17½ [2 2 8 1 0 0]
    u18½ [2 4 4 1 1 1]
    u19½ [4 4 4 2 2 2]
    u20½ [4 4 4 2 0 0]
    u21½ [2 6 6 0 0 2]
    u22½ [4 4 4 0 0 0]
    u23½ [2 6 8 0 0 0]
    u24½ [2 8 8 0 0 4]
    u25½ [4 4 8 0 2 2]
    u26½ [8 8 8 4 4 4]
    u27½ [8 8 8 4 0 0]


    EIGENFORM BASIS FROM PULLBACK-GENUS BASIS
    The M203) eigenform basis f1,...,f11 comprises algebraic integer linear combinations of the rational near-integer non-eigenform basis g1,...,g11 from above. Specifically fi = Σj=111 bi,jgj for each i, where the matrix (bi,j) is as follows.
    g1g2g3g4g5g6g7g8g9g10g11
    f1 10000000000
    f2 -13200174611000000000
    f3 -123673711558895794707809982652851940872000-1684161178678974339285015858117048000*a-3514453979357044959995394537031737687168+64888441581477656256500830853570688*a39446746505262288436494712064467640258784+3013639463435544100367619440701671456*a-4370710298809734921051925209258991792536-678822249162982050400754408283392424*a52235315139895711454259444509683393/2+(709547342997564447214140524087*a)/2000000
    f4 -123673711558895794707809982652851940872000+1684161178678974339285015858117048000*a-3514453979357044959995394537031737687168-64888441581477656256500830853570688*a39446746505262288436494712064467640258784-3013639463435544100367619440701671456*a-4370710298809734921051925209258991792536+678822249162982050400754408283392424*a52235315139895711454259444509683393/2-(709547342997564447214140524087*a)/2000000
    f5 -23866128000-796883712-4207339584-1241295845041000000
    f6 -17322788530107964020847050048607119494201376000000/7+(2494737286123631219652841680724386774166304000*b1)/7-(62941307888519510737340208439260447328000*b1^2)/7-7209504483781321748230647083576907870198593694720/7+(14915314021651995300498498459200167478900194048*b1)/105-(376363897422967065700870162954448511585536*b1^2)/105-18111349196077947466123946010572480999215134720+(12622668293719106853653232358667246123279936*b1)/5-(324441937229963787741505966746623157952*b1^2)/5-88342218144971179197317895707632340840856080640/7+(216187359982584724688343795855680090447977456*b1)/105-(5833654550930426311887852037066420912592*b1^2)/105-129289399627877107404698835387751964851005250/7+(12782774284235136726026384491627717419563401*b1)/5040-(322558898458894675661590410108933048107*b1^2)/5040-231732632703937702038087621618407270326330229760+52665769223058988834105855852354625449956824*b1-1411910689822835542175559811448993630728*b1^2-85685407856894449856516425862255506803265751040+(34247555044204763963046279895370112655500352*b1)/3-(834859640175228289337827163573426389184*b1^2)/39553703799314240967232222299768151413929984000-(11944781940427015147689923527392385111632616*b1)/9+(302905951095237942472385624799762733112*b1^2)/9-162550708066993218979278099824542952493077760+(307206088446135907353496247579149647343277*b1)/15-(7440883847033196415091300874693960439*b1^2)/15-62968691402300612457937370945196367106030+(6303801954995561690909585062765129276007*b1)/720-(159062442666676234898247664265603749*b1^2)/7201141367625173365906204906017360670206655-(45217993480208769217049267536299493327*b1)/288+(1141017908499434811863904881016029*b1^2)/288
    f7 -17322788530107964020847050048607119494201376000000/7+(2494737286123631219652841680724386774166304000*b2)/7-(62941307888519510737340208439260447328000*b2^2)/7-7209504483781321748230647083576907870198593694720/7+(14915314021651995300498498459200167478900194048*b2)/105-(376363897422967065700870162954448511585536*b2^2)/105-18111349196077947466123946010572480999215134720+(12622668293719106853653232358667246123279936*b2)/5-(324441937229963787741505966746623157952*b2^2)/5-88342218144971179197317895707632340840856080640/7+(216187359982584724688343795855680090447977456*b2)/105-(5833654550930426311887852037066420912592*b2^2)/105-129289399627877107404698835387751964851005250/7+(12782774284235136726026384491627717419563401*b2)/5040-(322558898458894675661590410108933048107*b2^2)/5040-231732632703937702038087621618407270326330229760+52665769223058988834105855852354625449956824*b2-1411910689822835542175559811448993630728*b2^2-85685407856894449856516425862255506803265751040+(34247555044204763963046279895370112655500352*b2)/3-(834859640175228289337827163573426389184*b2^2)/39553703799314240967232222299768151413929984000-(11944781940427015147689923527392385111632616*b2)/9+(302905951095237942472385624799762733112*b2^2)/9-162550708066993218979278099824542952493077760+(307206088446135907353496247579149647343277*b2)/15-(7440883847033196415091300874693960439*b2^2)/15-62968691402300612457937370945196367106030+(6303801954995561690909585062765129276007*b2)/720-(159062442666676234898247664265603749*b2^2)/7201141367625173365906204906017360670206655-(45217993480208769217049267536299493327*b2)/288+(1141017908499434811863904881016029*b2^2)/288
    f8 -17322788530107964020847050048607119494201376000000/7+(2494737286123631219652841680724386774166304000*b3)/7-(62941307888519510737340208439260447328000*b3^2)/7-7209504483781321748230647083576907870198593694720/7+(14915314021651995300498498459200167478900194048*b3)/105-(376363897422967065700870162954448511585536*b3^2)/105-18111349196077947466123946010572480999215134720+(12622668293719106853653232358667246123279936*b3)/5-(324441937229963787741505966746623157952*b3^2)/5-88342218144971179197317895707632340840856080640/7+(216187359982584724688343795855680090447977456*b3)/105-(5833654550930426311887852037066420912592*b3^2)/105-129289399627877107404698835387751964851005250/7+(12782774284235136726026384491627717419563401*b3)/5040-(322558898458894675661590410108933048107*b3^2)/5040-231732632703937702038087621618407270326330229760+52665769223058988834105855852354625449956824*b3-1411910689822835542175559811448993630728*b3^2-85685407856894449856516425862255506803265751040+(34247555044204763963046279895370112655500352*b3)/3-(834859640175228289337827163573426389184*b3^2)/39553703799314240967232222299768151413929984000-(11944781940427015147689923527392385111632616*b3)/9+(302905951095237942472385624799762733112*b3^2)/9-162550708066993218979278099824542952493077760+(307206088446135907353496247579149647343277*b3)/15-(7440883847033196415091300874693960439*b3^2)/15-62968691402300612457937370945196367106030+(6303801954995561690909585062765129276007*b3)/720-(159062442666676234898247664265603749*b3^2)/7201141367625173365906204906017360670206655-(45217993480208769217049267536299493327*b3)/288+(1141017908499434811863904881016029*b3^2)/288
    f9 28369879025955120448572981760128000+1713849775081040077642541952000*a4526421489183642268309022718853632+763658765671805407915273317888*a143457790759471388817479871396864+16544972284515548764900402176*a312160523551867374085436811583104+64631386062717706229231742336*a68168383177532163354609150819+13777736332849233230982021*a-30894320849506804162257483082210560-3263180234312655865731542711040*a-178033897931463945711219807014400+39900942817408889878133030400*a40321067744016359516487412115712+5891347726486051065622564608*a-1061237048746119890597528381664-229608336474035843967545376*a358351010051811385680832563+46111876479909187416117*a-9065389213552223437338611/2-(1698427560025511583349*a)/2
    f10 28369879025955120448572981760128000-1713849775081040077642541952000*a4526421489183642268309022718853632-763658765671805407915273317888*a143457790759471388817479871396864-16544972284515548764900402176*a312160523551867374085436811583104-64631386062717706229231742336*a68168383177532163354609150819-13777736332849233230982021*a-30894320849506804162257483082210560+3263180234312655865731542711040*a-178033897931463945711219807014400-39900942817408889878133030400*a40321067744016359516487412115712-5891347726486051065622564608*a-1061237048746119890597528381664+229608336474035843967545376*a358351010051811385680832563-46111876479909187416117*a-9065389213552223437338611/2+(1698427560025511583349*a)/2
    f11 190165307904000-1847079788350464-82320894931968-32587262742528-36536803518-7007474440350720217025197240320-12962395367040-204454185600-912741902173195


    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 1801463902856806532452072009481628538123930822246521664012669578812292002611202066037306610371526656281474976710656
    f2 15668220002760-682966125587237653199770560309481987324332198462750720885442075367357546496281474976710656
    f3 90339705936-6291504*a5570371796636818395648-163954714123279872*a228849019758468661248-324260417085898752*a295148623108906156032-618475290624*a281474976710656
    f4 90339705936+6291504*a5570371796636818395648+163954714123279872*a228849019758468661248+324260417085898752*a295148623108906156032+618475290624*a281474976710656
    f5 -11022715008020935108578662852198401876399758014278533120295151765581857816576281474976710656
    f6 10944*b1-125204137833922560+11899911536640*b1-333213696*b1^287934816820920320-7059013632000*b1+150994944*b1^2-11773295632318464+618475290624*b1281474976710656
    f7 10944*b2-125204137833922560+11899911536640*b2-333213696*b2^287934816820920320-7059013632000*b2+150994944*b2^2-11773295632318464+618475290624*b2281474976710656
    f8 10944*b3-125204137833922560+11899911536640*b3-333213696*b3^287934816820920320-7059013632000*b3+150994944*b3^2-11773295632318464+618475290624*b3281474976710656
    f9 -363914496+25344*a29263117797163008-4495071117312*a30351506075025408-1557965832192*a1003785396682752-618475290624*a281474976710656
    f10 -363914496-25344*a29263117797163008+4495071117312*a30351506075025408+1557965832192*a1003785396682752+618475290624*a281474976710656
    f11 4716288013608666228326407864145852497920-4048539252424704281474976710656


    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)a(u; f9)a(u; f10)a(u; f11)
    u1 107441973609722531026066150430595653959970701892533/5253114746880000000000000
    u2 615321907610187966543151063968453613804231703/397963238403831758538274897591784149796692536273085110/703000000000
    u3 107535502539679279596980709392318858909268945111389/1326544128091962204918597542202819595120620870554042640/37000000000
    u4 9595831481772000639515348360087185449988643/6632720640717690288336064027488596775323234598695460/703-973284976155142030303384857817435121208727668573991294217784360960000000000/19-(152304955239236756143508567176463034208658156670003940986224640000000000*a)/19-973284976155142030303384857817435121208727668573991294217784360960000000000/19+(152304955239236756143508567176463034208658156670003940986224640000000000*a)/19-1414396037808558912447212627011357900800000000000
    u5 36310769078907701071518090877839053529563790738929/3740256012663016014225149063345722725122597676281052949680/37-223341756488886127906803131440595837438074135667640668403703291576320000000000-118767673195675114137451951955506132174509186521053000253445242880000000000*a-223341756488886127906803131440595837438074135667640668403703291576320000000000+118767673195675114137451951955506132174509186521053000253445242880000000000*a-144562590232337189323404708181576868324966400000000000
    u6 264195112842425041846621723344923428337/37402560276406812376685548811321692787689042440/373344876432667102570653690336573262785008625885505555426707005440000000000+253336863252654290267758658696348345534877631094272360488960000000000*a3344876432667102570653690336573262785008625885505555426707005440000000000-253336863252654290267758658696348345534877631094272360488960000000000*a-353599009452139728111803156752839475200000000000
    u7 1318846067537306558767147498754370544674796877301068323/663272064085603209607005452877193209247131522236369318268960/70350283599686168513457888270920773989685914164327703186552343090576752640000000000/19-(18329939867768330253256313763024358536088466724327061025471059722240000000000*a)/1950283599686168513457888270920773989685914164327703186552343090576752640000000000/19+(18329939867768330253256313763024358536088466724327061025471059722240000000000*a)/191560225926890772570556170037706736945804083200000000000
    u8 5032696825119199307118285593825244988604045279/51848930040731541006036776656721907542507534544001530336818641283744656078867740726961588528312592295975153659176550400000000000+14053804165332498398450699695084949209280736930100867082590617600000000000*a1530336818641283744656078867740726961588528312592295975153659176550400000000000-14053804165332498398450699695084949209280736930100867082590617600000000000*a297362622988871425752901982702867885064192000000000000
    u9 4990533139341652135412026366707943061308958560568501721794033/37402560-6008961961046342722487734171828551960093508016892157440/37727687671396260877910298955356930520733152102890401075898315026991740354560000000000-5185490337003979426597864233332054342846995247677452289927879285800960000000000*a727687671396260877910298955356930520733152102890401075898315026991740354560000000000+5185490337003979426597864233332054342846995247677452289927879285800960000000000*a-87783146659285993424375795434173689677145230540800000000000
    u10 291280975644197723291719047987179/46176049392241099834973985483265826756280/371127708274939220414352823727839775760774142234008550834610176000000000+79988729877618676479524300636556806427614204405631442944000000000*a1127708274939220414352823727839775760774142234008550834610176000000000-79988729877618676479524300636556806427614204405631442944000000000*a-128167524348404380989325857128448000000-7138232385576367464624841473651551596101068959775745739262045611622400000000+1608829883330774682344591920524154132865152165889278394156816793600000000*b1-43069115455769722584363389030317503749195520049488426801561600000000*b1^2-7138232385576367464624841473651551596101068959775745739262045611622400000000+1608829883330774682344591920524154132865152165889278394156816793600000000*b2-43069115455769722584363389030317503749195520049488426801561600000000*b2^2-7138232385576367464624841473651551596101068959775745739262045611622400000000+1608829883330774682344591920524154132865152165889278394156816793600000000*b3-43069115455769722584363389030317503749195520049488426801561600000000*b3^2-937028397471318220949967613962893847106554534297600000000000000-99034803247766498521580875853970982528509542400000000000000*a-937028397471318220949967613962893847106554534297600000000000000+99034803247766498521580875853970982528509542400000000000000*a-211651687003412582437755439068217344000000000
    u11 3020543815926817486737415333236911/32401316324577062887149180743530575805440342014144772462293091827563492889340695295878090196636994830336000000000+17878765009319115886520927689122585846350052523993686933504000000000*a342014144772462293091827563492889340695295878090196636994830336000000000-17878765009319115886520927689122585846350052523993686933504000000000*a-49406207163636770271737167444180992000000-1776252521006238466739524387666011177179936634174650791529918272792166400000000+237098907014152702552643529787945180939770790387011251981925325209600000000*b1-5773412156097984529295802323198492331664321652334580932909465600000000*b1^2-1776252521006238466739524387666011177179936634174650791529918272792166400000000+237098907014152702552643529787945180939770790387011251981925325209600000000*b2-5773412156097984529295802323198492331664321652334580932909465600000000*b2^2-1776252521006238466739524387666011177179936634174650791529918272792166400000000+237098907014152702552643529787945180939770790387011251981925325209600000000*b3-5773412156097984529295802323198492331664321652334580932909465600000000*b3^2-3453577660492055302968510664493419835147662078771200000000000000+1030955938338132276531267257206882749577691136000000000000000*a-3453577660492055302968510664493419835147662078771200000000000000-1030955938338132276531267257206882749577691136000000000000000*a5079640488081901978506130537637216256000000000
    u12 31749626345217551838797376230602511/192175195066649255933659966307985966702000-17588896806489825607327918627389871924477453634382096775475036160000000000-2753183077602036594765466772237796438318286363149134811955200000000000*a-17588896806489825607327918627389871924477453634382096775475036160000000000+2753183077602036594765466772237796438318286363149134811955200000000000*a-48650493464515808143829734982890291200000073929684548040709169599838233062333084813005927839804641348542225448960000000000-10361507137389617245545395103911414984912823089065783104896118030336000000000*b1+263784194854850597286893503361796924277291362702515970785673216000000000*b1^273929684548040709169599838233062333084813005927839804641348542225448960000000000-10361507137389617245545395103911414984912823089065783104896118030336000000000*b2+263784194854850597286893503361796924277291362702515970785673216000000000*b2^273929684548040709169599838233062333084813005927839804641348542225448960000000000-10361507137389617245545395103911414984912823089065783104896118030336000000000*b3+263784194854850597286893503361796924277291362702515970785673216000000000*b3^2273933779155093656653891172012457974509790816947404800000000000000+36397290284439853013566658638208987879975996620800000000000000*a273933779155093656653891172012457974509790816947404800000000000000-36397290284439853013566658638208987879975996620800000000000000*a-110058877241774542867632828315473018880000000000
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    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(1-524288*x)*(1-262144*x)*(1-131072*x)*(1-x/131072)*(1-x/262144)*(1-x/524288)
    f2(1-262144*x)*(1-131072*x)*(1-x/131072)*(1-x/262144)*(1+(13135*x)/8192+x^2)
    f3(1-131072*x)*(1-x/131072)*(1+(6075/32768+(3*a)/32768)*x+x^2/2)*(1+(6075/16384+(3*a)/16384)*x+2*x^2)
    f4(1-131072*x)*(1-x/131072)*(1+(6075/32768-(3*a)/32768)*x+x^2/2)*(1+(6075/16384-(3*a)/16384)*x+2*x^2)
    f5(1-131072*x)*(1-x/131072)*(1-(1719*x)/2048+(3076409*x^2)/2097152-(1719*x^3)/2048+x^4)
    f6(1+(3/2-(3*b1)/32768)*x+x^2/2)*(1+(13135*x)/8192+x^2)*(1+(3-(3*b1)/16384)*x+2*x^2)
    f7(1+(3/2-(3*b2)/32768)*x+x^2/2)*(1+(13135*x)/8192+x^2)*(1+(3-(3*b2)/16384)*x+2*x^2)
    f8(1+(3/2-(3*b3)/32768)*x+x^2/2)*(1+(13135*x)/8192+x^2)*(1+(3-(3*b3)/16384)*x+2*x^2)
    f9(1+(6075/32768+(3*a)/32768)*x+x^2/2)*(1-(65*x)/512+x^2)*(1+(6075/16384+(3*a)/16384)*x+2*x^2)
    f10(1+(6075/32768-(3*a)/32768)*x+x^2/2)*(1-(65*x)/512+x^2)*(1+(6075/16384-(3*a)/16384)*x+2*x^2)
    f111+(43793*x)/16384+(503689609*x^2)/134217728+(277369629719*x^3)/68719476736+(503689609*x^4)/134217728+(43793*x^5)/16384+x^6


    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(1-18014398509481984*x)*(1-137438953472*x)*(1-68719476736*x)*(1-34359738368*x)*(1-524288*x)*(1-262144*x)*(1-131072*x)*(1-x)
    f2(1-456*x+524288*x^2)*(1-59768832*x+9007199254740992*x^2)*(1-119537664*x+36028797018963968*x^2)*(1-15668040695808*x+618970019642690137449562112*x^2)
    f3(1-68719476736*x)*(1-34359738368*x)*(1-524288*x)*(1-262144*x)*(1+(97200+48*a)*x+137438953472*x^2)*(1+(12740198400+6291456*a)*x+2361183241434822606848*x^2)
    f4(1-68719476736*x)*(1-34359738368*x)*(1-524288*x)*(1-262144*x)*(1+(97200-48*a)*x+137438953472*x^2)*(1+(12740198400-6291456*a)*x+2361183241434822606848*x^2)
    f5(1+840960*x+390238044160*x^2+115580662311813120*x^3+18889465931478580854784*x^4)*(1+110226309120*x+6704238549288815165440*x^2+260264513859358348994791876853760*x^3+5575186299632655785383929568162090376495104*x^4)
    f6(1-59768832*x+9007199254740992*x^2)*(1-119537664*x+36028797018963968*x^2)*(1+(179306496-10944*b1)*x+(52180622830993408-9895604649984*b1+301989888*b1^2)*x^2+(3230098674282837326168064-197149577287770832896*b1)*x^3+324518553658426726783156020576256*x^4)
    f7(1-59768832*x+9007199254740992*x^2)*(1-119537664*x+36028797018963968*x^2)*(1+(179306496-10944*b2)*x+(52180622830993408-9895604649984*b2+301989888*b2^2)*x^2+(3230098674282837326168064-197149577287770832896*b2)*x^3+324518553658426726783156020576256*x^4)
    f8(1-59768832*x+9007199254740992*x^2)*(1-119537664*x+36028797018963968*x^2)*(1+(179306496-10944*b3)*x+(52180622830993408-9895604649984*b3+301989888*b3^2)*x^2+(3230098674282837326168064-197149577287770832896*b3)*x^3+324518553658426726783156020576256*x^4)
    f9(1+138412032*x+9007199254740992*x^2)*(1+276824064*x+36028797018963968*x^2)*(1+(-51321600-25344*a)*x+(22773418298441728+1223059046400*a)*x^2+(-924527754544230590054400-456556915824311402496*a)*x^3+324518553658426726783156020576256*x^4)
    f10(1+138412032*x+9007199254740992*x^2)*(1+276824064*x+36028797018963968*x^2)*(1+(-51321600+25344*a)*x+(22773418298441728-1223059046400*a)*x^2+(-924527754544230590054400+456556915824311402496*a)*x^3+324518553658426726783156020576256*x^4)
    f111-47162880*x-10683351983718400*x^2+1421324803102308481105920*x^3-77113000621438799588267195891712*x^4+25604311414496000290391002970030635745280*x^5-3466945933980179205187242843942752625497630310400*x^6-275714505364963578491560813292872520408281551249887723520*x^7+105312291668557186697918027683670432318895095400549111254310977536*x^8


    IDENTIFIED EIGENFORMS
    Based on matching standard and spinor 2-Euler factors we identify the following eigenforms.
  • f1 = E 20 : Basic Siegel Eisenstein series
  • f2 = K1 20 1 : Klingen lift from degree 1
  • f3 = K2 20 1 li : Klingen lift of degree 2 lift
  • f4 = K2 20 2 li : Klingen lift of degree 2 lift
  • f5 = K2 20 3 nl : Klingen lift of degree 2 nonlift
  • f6 = M1 20 1 1 : Miyawaki lift of type 1
  • f7 = M1 20 2 1 : Miyawaki lift of type 1
  • f8 = M1 20 3 1 : Miyawaki lift of type 1
  • f9 = M2 20 1 1 : Conjectural Miyawaki lift of type 2
  • f10 = M2 20 2 1 : Conjectural Miyawaki lift of type 2

  • UNIMODULARITY AT 2 OF APPARENT NON-LIFT EIGENFORMS
  • f11 is unimodular at 2

  • SPECIAL VALUES
    The special values ci such that W*3,3E(6)20i=111 ci fi⊗fi are given in the following table (switch to factored denominator form).
    c1525311474688000/107441973609722531026066150430595653959970701892533
    c2703/669067190126718143399022180150280451179664142210
    c3175053632000065463924740728245787213231004139677/22388294951949851321884020502866209737329675483989735497166613895516693782971850481604507046429030430852641314724321528284936675861246194483200000000000000000000-(116335318729093499464013747732593622010254026524673*a)/129724947241281603597769137851442327574859515916725539094100226696762461440248887498193398324219845645100843063882641147255749578306942946094822195200000000000000000000
    c4175053632000065463924740728245787213231004139677/22388294951949851321884020502866209737329675483989735497166613895516693782971850481604507046429030430852641314724321528284936675861246194483200000000000000000000+(116335318729093499464013747732593622010254026524673*a)/129724947241281603597769137851442327574859515916725539094100226696762461440248887498193398324219845645100843063882641147255749578306942946094822195200000000000000000000
    c51/3162284483602503178245730819178002456516755456000000
    c6126477737178266613203928656355421856950525152340227499/60368244453423384350375945234491626605064085819177495058552507362623686994828163800712017975158053717189325707183796451941069315597975911506096569284834413772800000000000000000-(3772959381919010895682673447384327323978133904766447321*b1)/16516751682456637958262858616156909039145533880126962648019966014413840761784985615874808118003243497022999513485486709251076564747606209388068021356330695608238080000000000000000000+(6343918399425229909411817495383129749695318323274573*b1^2)/1057072107677224829328822951434042178505314168328125609473277824922485808754239079415987719552207583809471968863071149392068900143846797400836353366805164518927237120000000000000000000
    c7126477737178266613203928656355421856950525152340227499/60368244453423384350375945234491626605064085819177495058552507362623686994828163800712017975158053717189325707183796451941069315597975911506096569284834413772800000000000000000-(3772959381919010895682673447384327323978133904766447321*b2)/16516751682456637958262858616156909039145533880126962648019966014413840761784985615874808118003243497022999513485486709251076564747606209388068021356330695608238080000000000000000000+(6343918399425229909411817495383129749695318323274573*b2^2)/1057072107677224829328822951434042178505314168328125609473277824922485808754239079415987719552207583809471968863071149392068900143846797400836353366805164518927237120000000000000000000
    c8126477737178266613203928656355421856950525152340227499/60368244453423384350375945234491626605064085819177495058552507362623686994828163800712017975158053717189325707183796451941069315597975911506096569284834413772800000000000000000-(3772959381919010895682673447384327323978133904766447321*b3)/16516751682456637958262858616156909039145533880126962648019966014413840761784985615874808118003243497022999513485486709251076564747606209388068021356330695608238080000000000000000000+(6343918399425229909411817495383129749695318323274573*b3^2)/1057072107677224829328822951434042178505314168328125609473277824922485808754239079415987719552207583809471968863071149392068900143846797400836353366805164518927237120000000000000000000
    c9483432187771016063/18149461138353634637502786044119654845666662594922789160036864425169060087484694185796131880960000000000000000000000-(257744445542717860189*a)/88984743111115284087243170234275648249110051241831212882032462082566607375101348680442989037035192320000000000000000000000
    c10483432187771016063/18149461138353634637502786044119654845666662594922789160036864425169060087484694185796131880960000000000000000000000+(257744445542717860189*a)/88984743111115284087243170234275648249110051241831212882032462082566607375101348680442989037035192320000000000000000000000
    c111/6628601160119271086457327965472426463084561406361600000000000


    CONGRUENCES
    From Garrett's formula we can prove pairwise congruences of the eigenforms. We list congruences of cusp forms, omitting the trivial congruences within each Galois orbit and giving only one representative congruence for any pair of Galois orbits.
  • Eigenforms f6, f11 are congruent modulo a prime ideal P lying over 157. The ideal is
      P = <157,122+49y+105y^2>.
    Here y = Root[-4382089113600+893191104#1-54971#1^2+#1^3&,1]. We may need to rescale the eigenforms by algebraic integers relatively prime to P to obtain this congruence.

  • MORE COEFFICIENTS
    Show more coefficients for all forms | cusp forms