LEVEL 389
Existence discussion Integrality Congruence Weight 8


EXISTENCE OF NONLIFT
Define L, L' ∈ S2(K(389))+ and Q,Q' ∈ S4(K(389))+ as follows. (See bottom of page for definitions of the Theta Blocks Gritsenko lifts Gi, etc.) If we can prove that the weight 8 plus form
     F = Q2 + L Q L' + L2Q'
is identically zero, then by Theorem (see paper), it would follow that the form
     f = Q/L
would be a holomorphic cusp form. And because we know that there is at most one nonlift (see here), then it would follow that
     dim S2(K(389))= 12
and we can compute the action of the Hecke operators to see that actually this f is an eigenform. Its Fourier coefficients can be found here.

Conjecture: The above weight 8 cusp form F is zero.
Evidence: We have checked that the first 29745 coefficients are zero. By the discussion below on Weight 8 cusp forms, it is very likely that this is way more than sufficiently many vanishing Fourier coefficients to show that F=0.

Theorem: If the first 29745 Fourier coefficients determine a weight 8 cusp form, then the above weight 8 cusp form F is zero.


INTEGRALITY
Theorem: If the above f is a holomorphic cusp form, then it is integral.
Proof: This follows because f=Q/L where both Q and L are integral and because L can be checked to have content 1 by looking at its Fourier coefficients..


CONGRUENCES
Assuming that the nonlift f exists, then its first Fourier-Jacobi coefficient is φ where
     Grit(φ) = 3G1 − 5G10 + G11 + 11G2 − 3G3 + G4 − 2G5 − 2G6 + 6G7 − 8G8 − G9

Assuming that the nonlift exists and is integral, then by considering the maximal minors of the matrix of Fourier coefficients of f and the wt 2 Gritsenko lifts given by the listed theta blocks, we find that the GCD of the maximal minors must be a factor of 10, which proves that any nontrivial congruence relation involving f and the wt 2 Gritsenko lifts must be modulo a factor of 10.
After solving for all possible congruences modulo 10, we find that the only possible congruence relation is
     f ≡ Grit(φ) mod 10

Continuing to assume that the nonlift exists and is integral, we can prove that
     f Grit(φ) mod 10
because
     f Grit(φ) = ( − 5(1402499946042738752904567251123759404302849389368 G12 − 593842834335567165051643924931344665668487127102 G1 G10 − 169042944233242990582727465056793524104630782180 G102 − 1880931822133695331170350133688603063408833165454 G1 G11 + 791824722761566361329678904562071712700337431990 G10 G11 + 264778516771589069841298357119041401422851753572 G112 + 2655050778649131624283305477387105832208292139388 G1 G2 − 13569162050221545865548649639774899311183815230 G10 G2 − 2939453258142252464571025798022332841643517084342 G11 G2 + 3042550997343193004713891636329983639658678640342 G22 − 2329053787835319870671663832619669722766307814362 G1 G3 + 865470993169984843762841150160578058645435942986 G10 G3 + 1930612051431848818393555790590693997677295412270 G11 G3 − 3798069772523788623969275955787059716477114286268 G2 G3 + 977168897504599060654316073783054254626561350362 G32 − 767525101244215849998169332724077066341173366408 G1 G4 − 77385686750936596747193312429793965660761269528 G10 G4 + 304759143514749322827305801075096061913075384994 G11 G4 + 575706094453179339765397847380791713622484564100 G2 G4 + 83426925927586560522193155659694893760171660212 G3 G4 + 107761429317617844597315174166681307615621618232 G42 − 1662790712033452479980487304566316085776604143386 G1 G5 + 50941234695830515540840598496559557039889832124 G10 G5 + 1436858910333268082841743599463509685931185584474 G11 G5 − 2033692281974972264476809523690126040142131860956 G2 G5 + 1548618488069538781080984578977015472613544407176 G3 G5 + 192490498983577572977876654869412520941498620326 G4 G5 + 457009257107707373784327723815185653511559280012 G52 − 1603520747495592917599094606502039382451972924976 G1 G6 + 943057007999401217728467617173455055771200487946 G10 G6 + 824879226103761162779186791231001126322954793832 G11 G6 − 3509118793563508814453999875106197996079951633660 G2 G6 + 2202176135943753366046267397089876422722623349546 G3 G6 + 169994821594782547241118975815699037346411799092 G4 G6 + 1056553748597230332900222717213563094837134722534 G5 G6 + 397924088574269208971781407809277431455033602602 G62 + 930870405882820211076118636887997629314449008720 G1 G7 + 337399918654985712466907111890564719124225965484 G10 G7 − 1409068044695565045545714330972860523129528472226 G11 G7 + 1524173775574321119012563473701563329546310078342 G2 G7 − 1375596811994449388211318305579424209873716913374 G3 G7 + 236513207080234517445271365425778765189779389576 G4 G7 − 1328034432580876553709791816435060266250117200 G5 G7 − 2020779713650878212460512629610017177243908325238 G6 G7 − 344667128135729871153687033094041952173418750062 G72 − 1829975348419238289484275222267391567069332055272 G1 G8 + 547453968275897583675346027554881305638489173514 G10 G8 + 1347807164840330592084553035594638908455888909586 G11 G8 − 4048223158249576567134231455426345069030993023074 G2 G8 + 2384876051200827200339910943458982928871538863778 G3 G8 + 62733958666856781188740115668236905062393367664 G4 G8 + 1503360321045307707038149617061173674729578598742 G5 G8 + 2045651452065581423647322757827803571189859818462 G6 G8 − 1578776911107578424596843990938593329895379674212 G7 G8 + 1329769275670681802661655158260207648905162674034 G82 + 946173499966370967280900141581300674656846621604 G1 G9 − 669503991971797431595271524502273565982910844920 G10 G9 + 73403365381100793181915704780234832603424065238 G11 G9 + 1949949866224580448597230681339379145025683377002 G2 G9 − 1146359946218062220215626324755696935413374967298 G3 G9 − 428599219436048146691976786646626261133966009416 G4 G9 − 1150197631546781409153423374038725415290561850118 G5 G9 + 64760817004090362981895132294326899232167884626 G6 G9 + 1320743444339315067137689081392758188581705646762 G7 G9 − 1032505894511555394341071177775482101541895871536 G8 G9 + 293191974040651209121602045631852406182319551416 G92 − 10050107546040087804366708241480559612259932678678 C1 + 8069574289037926758123952963498925515713797386791 C10 − 7034091451257599610916237557970282073255551649385 C11 + 3325850796244666826618586349219639111262294950190 C12 − 63871205788646955780896651534021384596833672859761 C13 − 14482864422365980877509965430839360744052131240929 C14 − 40208337661070823935092131814022177782072832398349 C15 − 10041413610050517529011153413595549142797843688571 C16 + 26586328291548423220348070696691750938553961582159 C17 − 44343132622107226048861849163712466523906447512010 C18 − 36034460987691650669802480549631353523411169481100 C19 − 4317068375220492467058008075900118997862760584732 C2 − 10557267863099233973129046922437825027961022914 C20 − 6811409552540477856408716258617593275972592664507 C21 − 15074174283359657023303103183687608208859526244146 C22 + 2330632375603991951382195966205698067973010841220 C23 − 19636142973528004486944986209148672053066929208798 C24 + 3741798588682644721309657122148081072243430284348 C25 + 1937389921246166918177068770928497794077355769277 C26 − 22781648795825604906167384163677818385369466817528 C27 − 3771362422271335709009847665353807903738193001208 C28 + 23611351780262146910413270847379663959953742338150 C29 − 22057096273092096332203104043642534055063508376638 C3 − 31622245071784561354463305091281873822668212292266 C30 − 4840200481985714701562804473461540400376012433131 C31 − 10277143235610459729993202590388832130239656897171 C32 − 25585088504964359326926355197939287810517175340112 C33 − 8867663765543198868109493404255943110923230367523 C34 − 196298717777044175977595574269825261769448000016 C35 + 795494948734034625262914787462803710912044468197 C36 − 215453168178257875854475376632849961147590409161 C37 + 2071973876607153422186601841147442313542921421071 C38 − 1637221835825809751724666295460766775609015503815 C39 − 8953952336087873970317243258549551505815074059167 C4 − 7544653224655116760300781025249739238463664101326 C40 − 6861437040228529358923569967382939296911259058301 C41 − 3181799558635066380829437288817256338478411713374 C42 + 1094381407152944886534203836645947633287339547896 C43 + 1013266372145838729807307377215219720802367036085 C44 + 2087945099888406410554271832966537416579900682867 C45 − 5005728499303027916521927926198971728062971787176 C46 + 5500776691816399717101527641355679649490978854335 C47 − 419460626747141451727109270757869069028227547197 C48 + 332968886135593469901414881784484526389533686852 C49 − 35310670904449128851081761990640626858835175944 C5 + 6422835622254686536959341603306848519944122332307 C50 − 8745676413427797971991330553683627984395331060 C51 + 76994691725994350317299169699742479157529861054 C52 − 53022268413720149405494212213389704915714097635 C53 − 53779055685212710926133172362802306543563388511 C54 + 208910834170235076910797354978653766801671248965 C55 − 504286098789011936690802911952166801320716875754 C56 − 4534981613691298069495438529155660696421410558268 C57 + 284512987756815315952377075813885273843421378798 C58 + 797421251404245651288720052602348605788072348782 C59 + 6448404575721593559079938845251836732455118814879 C6 − 1161673366668970240519756038481272750948073484386 C60 + 576042514263617155951158912225660111949573934282 C61 + 347380418891112405371568257539149688303891215957 C62 + 1643424107579333279544911624221278909342308586716 C63 − 12407778191788676414540941410614267791252429592 C64 + 111864720843576667988892137461282106574027671048 C65 + 3700988367534890738731843509907755748400336375 C66 + 51176465010214285178553667314847501046125390201 C68 − 211525603709286694311569494276600919692523921520 C69 − 1032622172327500915770354742555630846291280604418 C7 + 207929748911482346570285284482670926504565244267 C70 + 15077925293254578187704055913164534203926943227 C71 − 115186755384182271087887787926852052895889441088 C73 − 105015915556989977068537935455077292864207081353 C74 + 171158565441989820984783394088545800491420220241 C75 − 50972140903110393775198554032462182876723264846575 C8 − 29573521547819606942867708973052842013185066559800 C9 + 631405227623535220485326135033426131826719701968 T2(G10G10) + 729982928817571607983843477048778110063921749220 T2(G10G11) − 9008594025089191204287701968986465067446604680 T2(G11G11) − 122449823349943975752634033614178640097452342844 T2(G1G1) − 569566812173638264582413238374612330054336508350 T2(G1G10) − 611759582382187672204830348075266721781675438488 T2(G1G11) − 192056936342327398103628545954853286348498243576 T2(G1G2) + 673922326303726597959835310260708561049496670512 T2(G1G3) + 802978022056075165849612095879985659806892176126 T2(G1G4) + 709629265225253965667538464834039889486693379120 T2(G1G5) − 509466811429470839344484354629738863892024406946 T2(G1G6) + 430605240314484054650848338156547718563300966278 T2(G1G7) − 711615295532741770133716133135240129786086733980 T2(G1G8) − 561443301883914198958344011715522528091456103058 T2(G1G9) − 328989875143670145677060805739740943714761092686 T2(G2G10) + 282789846308109959625277015542151754602772902066 T2(G2G11) − 282778005143237236819861232381906127981157470684 T2(G2G2) + 426946683388184218753429345552129909086684170854 T2(G2G3) + 188449935388402967920883525045247917490842839884 T2(G2G4) + 210385117204858533839506623715666232773806321830 T2(G2G5) + 411388359712759613467584872627638293335860165984 T2(G2G6) − 61333926935184249162888427416277522264183583014 T2(G2G7) − 85660910800711444512619646314106696950980326894 T2(G2G8) − 287971308433325367599997645816529983886586251224 T2(G2G9) − 505506417133385803084710461754431636537278289846 T2(G3G10) − 299966443636756169473294233274377538839848290174 T2(G3G11) − 69291290816235570939881435164937753714362084854 T2(G3G3) + 23982875135467687173200661393540832072422315288 T2(G3G4) − 61417748237627043576565340678602855727915249822 T2(G3G5) − 1028963836537110895183044913234115512369782209468 T2(G3G6) + 820898342018856490742759142834605781562916040290 T2(G3G7) − 39790438347460032228693159648211912705479952638 T2(G3G8) − 537628608657326116012281229433691062361344452130 T2(G3G9) − 688336110808765495824899541619516704402421717190 T2(G4G10) − 150467633386722951674948354411660034996798118388 T2(G4G11) + 22622726751633083699858110513948839487590090348 T2(G4G4) + 24560805334356617406389174567412994825679743536 T2(G4G5) − 882297089630009893285586413948813537805765898492 T2(G4G6) + 831779233223042538724688354156382021382071449550 T2(G4G7) − 137277421884157528256301337255026861350151376484 T2(G4G8) − 752443558402022127375389522099360177002119952050 T2(G4G9) − 559278676555327428130382624439378399518591003234 T2(G5G10) − 238268108295631484134654998187706137110389081990 T2(G5G11) + 4172554211073636624584250976427142238046498657 T2(G5G5) − 940779263505951160223942350818506083865357876292 T2(G5G6) + 746179373430771847284924055674464305729140106760 T2(G5G7) + 48802170482542185651771348142602135069479520324 T2(G5G8) − 545136982900746013460159267719143052568817973732 T2(G5G9) + 1327009834480312688849301307224219097290830011952 T2(G6G10) + 752881313319491258223604874585358949588382869092 T2(G6G11) + 665740379958809689048807318088251819029652620246 T2(G6G6) − 1256762918890570031198297747303226738987407491198 T2(G6G7) + 639534496664127911904176129866447874061658934156 T2(G6G8) + 1375211660168343110140249274845967239498459777372 T2(G6G9) − 1276922070841505309551914743971523595166586027074 T2(G7G10) − 611796888211538741109863386110764457040985667012 T2(G7G11) + 533652122374393421077696471476815356765838654058 T2(G7G7) − 779650632161265588556823901151217803213183204114 T2(G7G8) − 1195179932367038459473507216678873600237322130580 T2(G7G9) + 834077438983651312386041651619774264922759106016 T2(G8G10) + 19271405391288007235255259581727132260487904344 T2(G8G11) + 153487795073952435249373476984626215905653827130 T2(G8G8) + 776572599389253766717257007179352095574550209752 T2(G8G9) + 1123970743466620325709254609411523229548775493008 T2(G9G10) + 829207244714281632905782953743639559594983968884 T2(G9G11) + 534644143539716672977951127001905923051568611776 T2(G9G9) + 509075190072809650857353885056917710540572129204 T2(D1D2) − 6103961062912988402234054842618116465372766796 T2(D1D3) − 504362812753032178981479568024743858094676404836 T2(D1D4) − 516426220304403153837070711418653734352400649184 T2(D1D5) − 42395476189103029401105901306425949591744685082 T2(D1D6) − 5756083963845265937024733106258680638405047006 T2(D1D7) − 38839911928093931736859710786318029778534971506 T2(D2D3) − 362876277820374697667876174901494754425368933042 T2(D2D4) − 418261354163623912848231796167803725828712553470 T2(D2D5) − 227740998597556065945267420509374457382810407998 T2(D2D6) − 32257475802884520459931477058996513729965472778 T2(D2D7) + 91826379268430450321770387870930960556385578872 T2(D3D4) + 32102778377202380258794908676931511529068207128 T2(D3D5) + 22520793176889308988921743192652778502508738394 T2(D3D6) − 23525714163129271069245385717434472466721839976 T2(D3D7) + 440115164550031127892491187129629631790720537678 T2(D4D5) + 176952507988162443899290988140416093569010586366 T2(D4D6) + 95810698433795101961636836556858651954794081070 T2(D4D7) + 181270089299481743069889441051800979789983299586 T2(D5D6) + 19527858940231261672624619208560621675662049890 T2(D5D7) + 2983768812548002999695730315988399167512891756 T2(D6D7) + 284361526068047466282466561502409360492715374096 D1 D2 + 172151977341737532792382844921188263684066403098 D1 D3 + 71042005653776947833028255067384459540823723894 D2 D3 − 410834274512141196002877990659258988891667211744 D1 D4 − 807767947183981488960863554651483612515248472382 D2 D4 − 89482900294796197354968043442699423454134146266 D3 D4 − 344562463562310123503938182435392848629576780600 D1 D5 − 1041537934896920380380132035044900828507730321674 D2 D5 − 170724636596442656441637903686283467772992978260 D3 D5 + 1070959993696683482569233962608112189787442175796 D4 D5 + 184941993403659243857079069412488993451751444584 D1 D6 + 536026533099713588768191480179636031205962117320 D2 D6 + 502604059901932962599950630749774437569902486024 D3 D6 − 642095464671998978592788583785855023996383376898 D4 D6 − 454608666128108574112699321986904436155317300966 D5 D6 + 128882047093443071076108034216006201254338322218 D1 D7 + 70362609917705748954803375618636986789183562112 D2 D7 − 30596525656186875424513472502661381076754434298 D3 D7 − 234658676624399157576216178618063677154833839788 D4 D7 − 430326221689600548036164719142566514982222436152 D5 D7 + 45272743767302171796086017524912737564743554362 D6 D7))/(94111263497182857051795596379216566786170678731 G1 + 169042944233242990582727465056793524104630782180 G10 − 786535431418655614068196770415725610824530043272 G11 + 1117097434826162547864922373879391811711055753118 G2 − 806296610612820513959694105301689749056441782630 G3 + 360958592045395292079217124837179885652335563017 G4 − 354650925852705016277676880763722718078217570982 G5 − 299212234167864135453129737155536684305710881276 G6 + 409734935996570972835488852078965485075798387757 G7 − 731098591433955421517595819866988610518910689810 G8 + 108088399305998592660053892488181706677287485701 G9)
and note that this is a multiple of 10 because the content of the denominator is 1, and because the numerator is clearly a multiple of 5, and we can prove that it is a multiple of 2 as follows. The numerator mod 2 is
C10 + C11 + C13 + C14 + C15 + C16 + C17 + C21 + C26 + C31 + C32 + C34 + C36 + C37 + C38 + C39 + C4 + C41 + C44 + C45 + C47 + C48 + C50 + C53 + C54 + C55 + C6 + C62 + C66 + C68 + C70 + C71 + C74 + C75 + C8 + T2(G5G5).
We apply the Lemma (see paper) that for a Gritsenko lift g, we have that T2(g2) is congruent mod 2 to the Gritsenko lift of its first Fourier − Jacobi coefficient.


WEIGHT 8 CUSP FORMS
In an attempt to find manageable set of determining coefficients for the weight 8 space of cusp forms, we will attempt to find a spanning set for it. The weight 8 space of cusp forms has dimension
     dim S8(K(389)) = 4885
We attempt to find cusp forms in the plus and minus parts separately and hope that the dimensions add up to the above 4885.

There has not yet been an attempt to span S8(K(389))+.


Weight 2 Theta Blocks (Number of wt 2 Gritsenko lifts: 11)
     G1 = Grit(THBK2(2,3,5,5,7,8,9,11,12,16))
     G2 = Grit(THBK2(2,3,5,5,6,8,10,11,13,15))
     G3 = Grit(THBK2(2,3,4,6,6,8,10,11,14,14))
     G4 = Grit(THBK2(2,3,3,5,5,7,8,10,13,18))
     G5 = Grit(THBK2(2,2,5,6,7,9,9,11,11,16))
     G6 = Grit(THBK2(2,2,5,6,7,8,9,11,13,15))
     G7 = Grit(THBK2(1,4,5,6,7,8,9,9,13,16))
     G8 = Grit(THBK2(1,4,5,6,6,7,10,11,13,15))
     G9 = Grit(THBK2(1,4,4,6,8,8,9,10,12,16))
     G10 = Grit(THBK2(1,4,4,5,6,8,9,9,13,17))
     G11 = Grit(THBK2(1,3,4,5,5,8,8,9,13,18))


Weight 4 Theta Blocks (Number of wt 4 Gritsenko lifts: 75)
     C1 = Grit(THBK4(1,1,1,1,1,1,14,24))
     C2 = Grit(THBK4(1,1,1,1,1,2,12,25))
     C3 = Grit(THBK4(1,1,1,1,1,4,9,26))
     C4 = Grit(THBK4(1,1,1,1,1,7,18,20))
     C5 = Grit(THBK4(1,1,1,1,1,8,15,22))
     C6 = Grit(THBK4(1,1,1,1,1,10,12,23))
     C7 = Grit(THBK4(1,1,1,1,2,3,19,20))
     C8 = Grit(THBK4(1,1,1,1,2,4,5,27))
     C9 = Grit(THBK4(1,1,1,1,2,4,15,23))
     C10 = Grit(THBK4(1,1,1,1,2,5,13,24))
     C11 = Grit(THBK4(1,1,1,1,2,8,9,25))
     C12 = Grit(THBK4(1,1,1,1,2,9,17,20))
     C13 = Grit(THBK4(1,1,1,1,2,15,16,17))
     C14 = Grit(THBK4(1,1,1,1,3,5,8,26))
     C15 = Grit(THBK4(1,1,1,1,3,13,14,20))
     C16 = Grit(THBK4(1,1,1,1,4,6,19,19))
     C17 = Grit(THBK4(1,1,1,1,4,11,14,21))
     C18 = Grit(THBK4(1,1,1,1,5,13,16,18))
     C19 = Grit(THBK4(1,1,1,1,6,7,8,25))
     C20 = Grit(THBK4(1,1,1,1,6,13,13,20))
     C21 = Grit(THBK4(1,1,1,1,7,7,10,24))
     C22 = Grit(THBK4(1,1,1,1,8,14,15,17))
     C23 = Grit(THBK4(1,1,1,2,3,4,11,25))
     C24 = Grit(THBK4(1,1,1,2,3,8,13,23))
     C25 = Grit(THBK4(1,1,1,2,7,7,12,23))
     C26 = Grit(THBK4(1,1,1,2,7,12,17,17))
     C27 = Grit(THBK4(1,1,1,2,11,13,15,16))
     C28 = Grit(THBK4(1,1,1,3,3,9,10,24))
     C29 = Grit(THBK4(1,1,1,3,4,5,7,26))
     C30 = Grit(THBK4(1,1,1,3,4,5,14,23))
     C31 = Grit(THBK4(1,1,1,4,5,7,18,19))
     C32 = Grit(THBK4(1,1,1,4,6,7,7,25))
     C33 = Grit(THBK4(1,1,1,4,13,13,14,15))
     C34 = Grit(THBK4(1,1,1,8,11,13,14,15))
     C35 = Grit(THBK4(1,1,2,2,8,8,8,24))
     C36 = Grit(THBK4(1,1,2,3,3,3,4,27))
     C37 = Grit(THBK4(1,1,2,3,3,3,13,24))
     C38 = Grit(THBK4(1,1,2,3,3,12,13,21))
     C39 = Grit(THBK4(1,1,2,3,4,5,19,19))
     C40 = Grit(THBK4(1,1,2,3,11,11,11,20))
     C41 = Grit(THBK4(1,1,2,7,7,7,7,24))
     C42 = Grit(THBK4(1,1,2,11,12,13,13,13))
     C43 = Grit(THBK4(1,1,3,3,3,3,8,26))
     C44 = Grit(THBK4(1,1,3,3,3,3,16,22))
     C45 = Grit(THBK4(1,1,3,4,5,5,5,26))
     C46 = Grit(THBK4(1,1,4,5,5,5,18,19))
     C47 = Grit(THBK4(1,1,5,10,12,13,13,13))
     C48 = Grit(THBK4(1,1,6,7,7,7,8,23))
     C49 = Grit(THBK4(1,1,6,11,11,11,11,16))
     C50 = Grit(THBK4(1,1,10,10,12,12,12,12))
     C51 = Grit(THBK4(1,2,2,2,2,2,9,26))
     C52 = Grit(THBK4(1,2,2,2,2,6,7,26))
     C53 = Grit(THBK4(1,2,2,2,2,6,10,25))
     C54 = Grit(THBK4(1,2,2,2,2,6,14,23))
     C55 = Grit(THBK4(1,2,2,2,2,8,16,21))
     C56 = Grit(THBK4(1,2,2,2,2,9,14,22))
     C57 = Grit(THBK4(1,2,3,4,5,5,13,23))
     C58 = Grit(THBK4(1,3,3,3,3,4,10,25))
     C59 = Grit(THBK4(1,4,4,4,4,12,13,20))
     C60 = Grit(THBK4(1,4,4,5,6,6,18,18))
     C61 = Grit(THBK4(1,4,5,5,5,5,6,25))
     C62 = Grit(THBK4(1,4,6,7,7,7,7,23))
     C63 = Grit(THBK4(1,4,8,11,12,12,12,12))
     C64 = Grit(THBK4(1,7,10,11,11,11,11,12))
     C65 = Grit(THBK4(2,2,2,2,2,2,5,27))
     C66 = Grit(THBK4(2,2,2,2,2,2,15,23))
     C67 = Grit(THBK4(2,2,2,2,3,8,8,25))
     C68 = Grit(THBK4(2,2,3,4,4,10,10,23))
     C69 = Grit(THBK4(2,3,3,3,3,4,19,19))
     C70 = Grit(THBK4(3,4,4,4,4,5,14,22))
     C71 = Grit(THBK4(3,5,7,7,7,7,8,22))
     C72 = Grit(THBK4(3,8,10,11,11,11,11,11))
     C73 = Grit(THBK4(4,4,5,6,6,6,17,18))
     C74 = Grit(THBK4(5,5,5,5,5,5,12,22))
     C75 = Grit(THBK4(6,10,10,10,10,10,11,11))


Weight 2 "Tweak" Theta Blocks that yield Gritsenko lifts with Characters
     D1 = Grit(THBK2(1,10,12,12))
     D2 = Grit(THBK2(2,4,12,15))
     D3 = Grit(THBK2(2,5,6,18))
     D4 = Grit(THBK2(4,6,9,16))
     D5 = Grit(THBK2(6,6,11,14))
     D6 = Grit(THBK2(6,8,8,15))
     D7 = Grit(THBK2(8,9,10,12))


The set A4 of 4x4 matrices used in theta tracing. Here |A4|=14.
     {{14,0,-1,1},{0,16,3,6},{-1,3,18,8},{1,6,8,44}}
     {{12,1,1,-4},{1,18,2,-7},{1,2,22,8},{-4,-7,8,40}}
     {{12,1,3,-2},{1,16,4,-7},{3,4,22,4},{-2,-7,4,44}}
     {{12,1,-2,-4},{1,16,7,0},{-2,7,28,11},{-4,0,11,38}}
     {{12,1,-3,-3},{1,14,4,5},{-3,4,30,14},{-3,5,14,40}}
     {{16,5,6,4},{5,18,2,4},{6,2,20,3},{4,4,3,34}}
     {{16,0,-4,1},{0,18,5,-2},{-4,5,24,10},{1,-2,10,30}}
     {{14,3,5,0},{3,20,9,-5},{5,9,28,8},{0,-5,8,30}}
     {{16,5,6,-1},{5,18,2,-3},{6,2,20,7},{-1,-3,7,36}}
     {{12,1,-1,-4},{1,16,5,5},{-1,5,18,0},{-4,5,0,52}}
     {{14,0,-6,-1},{0,16,1,-6},{-6,1,20,7},{-1,-6,7,44}}
     {{12,1,-3,4},{1,18,5,7},{-3,5,24,10},{4,7,10,40}}
     {{12,2,3,1},{2,16,7,-8},{3,7,20,1},{1,-8,1,54}}
     {{14,1,1,-6},{1,20,1,-3},{1,1,28,12},{-6,-3,12,28}}