LEVEL 349
Existence discussion Integrality Congruence Weight 8

EXISTENCE OF NONLIFT
Define L, L' ∈ S2(K(349))+ and Q,Q' ∈ S4(K(349))+ 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(349))= 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 34572 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 34572 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) = 25G1 − 2G10 + 4G11 − 24G2 + 5G3 + 11G4 − 16G5 − 12G6 − 9G7 + 10G9

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 13, which proves that any nontrivial congruence relation involving f and the wt 2 Gritsenko lifts must be modulo a factor of 13.
After solving for all possible congruences modulo 13, we find that the only possible congruence relation is
     f ≡ Grit(φ) mod 13

Continuing to assume that the nonlift exists and is integral, we can prove that
     f ≡ Grit(φ) mod 13
because
     f − Grit(φ) = ( − 13(1488053040930805684238449794152216406789 G12 + 256251512504121507556104040646801088400 G1 G10 − 2104987092190891889015682587143905301 G102 − 249540377227241584811166666698572240777 G1 G11 + 21060579611604257027999368875636610291 G10 G11 − 1535327558000184802343484336105186368 G112 − 3674709184243319558566759280881620903412 G1 G2 − 365101932389753881839671554713311456322 G10 G2 + 90636865136792660119522982652727613039 G11 G2 + 1526763232202908783797604297276614230094 G22 + 367292301616182687251290868568741729449 G1 G3 + 62995326298965987029401008817842864311 G10 G3 − 108508979402652050926084825303211184896 G11 G3 + 713365778530033397539535558058668637550 G2 G3 − 411000915104062920412411535030151468520 G32 + 1900634750740714748825802130555978544025 G1 G4 − 84474726457188532603758399558832293157 G10 G4 + 33166070286556836919054855965105702989 G11 G4 − 1401908429371255592234734323417738773105 G2 G4 − 533415710016686600789590681611882546028 G3 G4 + 370932684829311369676165162185809300642 G42 − 3601969750974885727812461345858602613201 G1 G5 − 153122499358052344770794952693456218071 G10 G5 + 76861880373777071380480358103050903687 G11 G5 + 4618978224926591620274469422781484678105 G2 G5 − 436223910146074497742966750688238386260 G3 G5 − 2226173512178897596161746600474998515517 G4 G5 + 2021239264225696666391255484955193015017 G52 − 1878767054966315674033266392550713693674 G1 G6 + 63440426524460491140245314293302201201 G10 G6 − 6715989265556277079626080080315253019 G11 G6 + 1593398152664822746380275615311209502547 G2 G6 − 71330142156379942810939118110944725832 G3 G6 − 1038794768896189939713519270680481865470 G4 G6 + 2295615148001595276038354302890359344253 G5 G6 + 511317534040718207509718684827839340048 G62 − 397057426714348278592406716712410605539 G1 G7 − 25826354414712832946402518909682088893 G10 G7 + 36162312005208054696936734291305699405 G11 G7 + 196567859215535141722521831305327187233 G2 G7 + 339256694186122456795869367989891725224 G3 G7 − 96789931290318692260102574546366648523 G4 G7 + 533928379929896942893962005248390033598 G5 G7 + 157671728771986806088090028673960108360 G6 G7 − 5240104244397068898859651187396440027 G72 + 533405201051761669520032653501910798950 G1 G8 + 3009830094564460621647858054122154292 G10 G8 + 6846693129448983435741293118392656620 G11 G8 − 199309138154837215773884972911168312876 G2 G8 − 294433611974492694696928857543949690139 G3 G8 + 70206883564815676876765624787877196073 G4 G8 − 832710434910459836260683047799617665327 G5 G8 − 219631058688155082685050222111791229974 G6 G8 − 8971882881971260390899404903064994793 G7 G8 + 3216588329644476169759252306659241301 G82 + 927724744056796859109181369953584555730 G1 G9 + 29342239878618157008465051027610986704 G10 G9 − 41472927730327857132631987024045886318 G11 G9 − 786636060224807139829778690276001669187 G2 G9 − 334316054388116356377320178168846909307 G3 G9 + 312474251216185758634444088416349470087 G4 G9 − 1259784349573632358121485540644429466194 G5 G9 − 359502386028072894599632510656346299506 G6 G9 + 21700957544754308625233253951518287543 G7 G9 + 17116732823622458589353232795981641550 G8 G9 − 13671813847206583572951882792976268470 G92 − 120470454559469826449996112113743746496 C1 − 4303633859298238596800747945137701763113 C10 − 1964006807298236214392585909731444632827 C11 + 4379076679747725879836217466958310140480 C12 + 2864701016217613218430564317762615863932 C13 + 9253120609458083877463477155477570306103 C14 − 7950936178858127588773961768595511202223 C15 + 5022182939840364110967837483222887469091 C16 − 831205111754748737182968937614947474255 C17 + 4019831577021256459648043428525899116283 C18 + 8271159745565991967475355265361299459944 C19 − 1164126422456033958764682687467897367391 C2 + 6061606193418197605077667999811853122417 C20 + 5235615060423899454037576386876293607187 C21 + 8406986900321308474704173670987293064425 C22 + 2682654842867462789360184593055069623889 C23 − 3057932742401206159292342714506882990795 C24 − 839458804742385018659655545436368003687 C25 − 109305026372971133467798474198778178938 C26 + 4208391765567763181809976168500851166479 C27 + 387941685196464533362148273448468578929 C28 + 1666414216578681403222656849915291680314 C29 − 3683358337175275578425115035444696330747 C3 + 1314512812694654113052841176575142323974 C30 − 113978331429768372407567001332879115822 C31 − 31714763859611858916352599894892713102 C32 − 88333859267424995714890449558721967898 C33 + 658641957476363232607275318652742286049 C34 + 501385483530526506920968727893100636918 C35 + 150713462578069665921384238259097537073 C36 + 988337543107641397922855711840764963802 C37 − 125620099003580605288775710053702308112 C38 + 1269690905582997821842542701217120009954 C39 + 2771564933306059681116074869768223766361 C4 + 126708259975824172775069871078256430283 C40 + 775334918610924955047017995755357440594 C41 + 472053005655641015870335282741040257386 C42 + 214571958583160540025362167834428818134 C43 + 14354994749526217873384791149011288827 C44 − 255924631685135201776048708681432380275 C45 − 624480765339381834447430897527861780026 C46 + 141038129303094425415569435342120788162 C47 − 100495968155982168487268430930488916892 C48 − 410774762871164271493121968132239119007 C49 + 220637437182426439080115020346972892545 C5 + 233522466360580335169217557893043524654 C50 + 18107217336517086890017577845876923763 C51 + 19680995378790442862125118578945163214 C52 − 137705387192874912841315552928618417222 C53 − 173735215828688865736049498101064731024 C54 − 218921006451008750344999075044481980 C55 − 2637902533122569728424182015579953027 C56 + 16899499708557219030782623514700829459 C57 + 55153671266109522859810002067603268868 C58 − 15864745259436658454019777090195331912 C59 + 1233353084407402908214673100760447095826 C6 − 98171300477337607695535456828193635559 C60 − 47320290278978282875573235727311613909 C61 − 89275141184243392216020161965093601130 C62 − 14780407797841196133777564498131762653 C63 + 62774430769536359349830640496904319071 C64 + 19249780491839626291918042729741763738 C65 − 5967023012472692487114908213880555171 C66 − 17620220398135687843564615242535128088 C67 − 487523666607090249854427363219799707717 C68 + 15130253188482853012612058737803935478 C69 + 2974639096525499072770382256311334667371 C7 + 1491579427288270185290035729901658767366 C8 + 119182607980811722276575089047857314277 C9 + 17620220398135687843564615242535128088 T2(G10G10) − 35240440796271375687129230485070256176 T2(G10G11) + 12166279559913782055856211184302818464 T2(G11G11) + 43700539218341781603686800233754295518 T2(G1G1) + 65785147848852839519838976848323947226 T2(G1G10) − 47176576729306351025877561113711691524 T2(G1G11) − 289998611433869955836146835313650150098 T2(G1G2) + 191123835031325030950310958983018590810 T2(G1G3) + 196659161934498584981287495912028282352 T2(G1G4) − 141913641107613323426962655971863275698 T2(G1G5) − 5947447913710974794264845345495529884 T2(G1G6) − 109211881024885731274201223145913531446 T2(G1G7) + 84640902997412374217701427716603003398 T2(G1G8) + 129462413055064103203965326773076392902 T2(G1G9) − 59923788285468907146006791088881392860 T2(G2G10) + 73578541124512116498125204965556851126 T2(G2G11) + 300692335725182289165627253227199597105 T2(G2G2) − 317232329312460599365892119948383483188 T2(G2G3) − 390448610330800395394688006110210423473 T2(G2G4) + 384407973358090694666848080198035763016 T2(G2G5) + 138812478296317478780955034111940584296 T2(G2G6) + 125093470289914374412783132203815042352 T2(G2G7) − 194946253877497130172618491669862561766 T2(G2G8) − 268491876913060961227875449076984911209 T2(G2G9) − 3280405913599771443894239801471047205 T2(G3G10) − 21707699970597548243573983225151411547 T2(G3G11) + 57359236021955235609029151669049861026 T2(G3G3) + 207138067736796960872034405092939939350 T2(G3G4) − 246271939043966370838816280343821380053 T2(G3G5) − 117167514697284373890521448848076200378 T2(G3G6) − 22297785952263738876887691530269670274 T2(G3G7) + 98579183116094813949361153810890313433 T2(G3G8) + 145445660545231953762215904776276677675 T2(G3G9) + 39572521173156813088208039781084256957 T2(G4G10) − 16477554636888351530434605156908503517 T2(G4G11) + 132879332006284506232222629909189248865 T2(G4G4) − 287538784191948650313316251886496228306 T2(G4G5) − 90065452275892020237131851015579695879 T2(G4G6) − 109584219380082905241429492734258715587 T2(G4G7) + 152642975502424992185545982669311372642 T2(G4G8) + 215295749323701433099732602402667712444 T2(G4G9) − 67805692220564008863854489203533800019 T2(G5G10) + 57400318522483171440714599242171204930 T2(G5G11) + 98432022895722550573620854813153462160 T2(G5G5) + 41678966158519700237788279959160686595 T2(G5G6) + 109353215318481039861374751171488067210 T2(G5G7) − 114789180446135013316317025301443661458 T2(G5G8) − 150104120721139431573214442437448850639 T2(G5G9) − 69228759926663070370365030875916110925 T2(G6G10) + 51155694809000076364639136241862616650 T2(G6G11) − 7564660179712564894182792302215722013 T2(G6G6) + 103703296574320155269076128631839360574 T2(G6G7) − 54063792386330178236184828858421059209 T2(G6G8) − 131958785541377261424877083665886966604 T2(G6G9) − 3563657671053949060956727035028549415 T2(G7G10) + 48548421272965781738947069463651050876 T2(G7G11) − 20281042271850300598363895640648428099 T2(G7G7) − 23857124962521740115061615167457393757 T2(G7G8) − 15509250909831469171353639469556326765 T2(G7G9) + 31960034882671604243234990683599208971 T2(G8G10) − 27473639465949631179422971794035711233 T2(G8G11) + 6065478599265899054752852512544074411 T2(G8G8) + 97350068859208309590373625364470436940 T2(G8G9) + 10847486434143475761378194519138968136 T2(G9G10) − 47597205809455146671270043019990179842 T2(G9G11) + 50552973655302883136941366282158675869 T2(G9G9) − 487011075881495109634715517539208370868 D1 D2 + 16001812049814742436744383149972415 D1 D3 + 559495894046227888761712241858997670006 D2 D3 − 213859141648282789172110827265331796755 D1 D4 − 9724575697783812636326762211386562797 D2 D4 + 134293828157656897337602788207675745496 D3 D4 + 79585446045616919209891499393169314669 D1 D5 − 421819600325356507638944226753240088213 D2 D5 + 70705858356986650416292200063139880005 D3 D5 − 214693073617636668531436186194858931939 D4 D5 + 101531691792288406775837111415957482221 D1 D6 − 49060600316906405067917124683240868488 D2 D6 − 210710551206943489958742984658105900761 D3 D6 + 108227038383683312174894471877266430923 D4 D6 + 144793302580755163873561280680538560140 D5 D6 + 31811920017109552371649122566338718903 D1 D7 + 513374846561455475330103593569393482814 D2 D7 − 80969803351639755258633236810089279107 D3 D7 + 181591701932252637535848831794143068017 D4 D7 + 56110265083762961762231461545949474862 D5 D7 − 232072110705686818042803049896009145968 D6 D7 − 603239110008264192781058393587750925177 D1 D8 − 720726045453481914830796927315014151072 D2 D8 + 594059623222309767117010967060514374049 D3 D8 − 56490197954685098093463639371393517541 D4 D8 − 579697784723410692264988245163578175327 D5 D8 + 221930077093285780863300721821719380282 D6 D8 + 507015811851942533775122324685647485444 D7 D8 + 445132651075407306151036926589893780165 D1 D9 + 649435644581562301357847904728770545330 D2 D9 − 414475471394319632949743276908899487333 D3 D9 − 52844943062318408591671723040365333044 D4 D9 + 580612891070760048107928777173418160849 D5 D9 − 259526168299476023848546494795175874402 D6 D9 − 353172494332184834772662844528643231356 D7 D9 + 482654463552076588917517938330098062234 D8 D9))/(331444425222207640837965455657748443004 G1 + 34590240241929892382037076809720452913 G10 − 46805462848878790814486604078911992609 G11 − 305653067868741218001528312490413584343 G2 − 45342624812973743355761612889638471176 G3 + 117704875929131648638922772684786889288 G4 − 475656991239502493573245015133850811782 G5 − 163047500742105391994684385574425360464 G6 + 12215222606948898432449527269191539696 G7 + 22251752670709196166159202634776134015 G8 − 17773358001368558644837447630869149011 G9)
and note that this is a multiple of 13 because the content of the denominator is 1, and because the numerator is obviously a multiple of 13.

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(349)) = 3980
We attempt to find cusp forms in the plus and minus parts separately and hope that the dimensions add up to the above 3980.

We use the following wt 8 plus forms in an attempt to fill out the wt 8 plus space of cusp forms. We computed the initial 10014 (or fewer, in the case of multiplying a theta trace with a wt 4 form) Fourier coefficients of these forms (which is up through determinant 2448/4). (This of course required much longer expansions of the Fourier series before computing any Hecke operators.)
and the rank of these 7484 truncated series was computed mod 541 to be 2786.
So these forms span at least 2786 linearly independent plus forms. Hence dim S8(K(349)) ≥ 2786.

For the wt 8 minus space, we use the following forms: and this proved dim S8(K(349))- ≥ 1167.

Because the plus and minus spanned dimensions do not add up the the dimension of the whole space, we cannot yet deduce anything about a set of determining Fourier coefficients.

Here is some evidence that a form in the plus space S8(K(349))+ should be determined by about 7000 initial Fourier coefficients.
         
This is a graph where the horizontal axis is the dimension of forms in the plus space determined by the number of initial Fourier coefficients indicated on the vertical axis.

Dimensions of subspaces of S8(K(349))+ as we Hecke smear.
Just for curiosity, here is a table that shows how the dimensions of the subspaces progress as we increase the number of Hecke smears. Denote W=span of the products of wt 4 Gritsenko lifts.
Subspace of S8(K(349))+ dimension
W2279
Above along with T2(W)2708
Above along with T3(W)2715

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

Weight 4 Theta Blocks (Number of wt 4 Gritsenko lifts: 69)
     C1 = Grit(THBK4(1,1,1,1,1,1,4,26))
     C2 = Grit(THBK4(1,1,1,1,1,2,8,25))
     C3 = Grit(THBK4(1,1,1,1,1,2,17,20))
     C4 = Grit(THBK4(1,1,1,1,1,6,9,24))
     C5 = Grit(THBK4(1,1,1,1,1,8,10,23))
     C6 = Grit(THBK4(1,1,1,1,1,12,15,18))
     C7 = Grit(THBK4(1,1,1,1,2,4,7,25))
     C8 = Grit(THBK4(1,1,1,1,2,11,13,20))
     C9 = Grit(THBK4(1,1,1,1,3,3,10,24))
     C10 = Grit(THBK4(1,1,1,1,4,5,13,22))
     C11 = Grit(THBK4(1,1,1,1,4,7,10,23))
     C12 = Grit(THBK4(1,1,1,1,4,10,17,17))
     C13 = Grit(THBK4(1,1,1,1,4,11,14,19))
     C14 = Grit(THBK4(1,1,1,1,6,12,15,17))
     C15 = Grit(THBK4(1,1,1,1,7,10,16,17))
     C16 = Grit(THBK4(1,1,1,1,10,12,15,15))
     C17 = Grit(THBK4(1,1,1,1,10,13,13,16))
     C18 = Grit(THBK4(1,1,1,2,3,3,12,23))
     C19 = Grit(THBK4(1,1,1,2,3,4,15,21))
     C20 = Grit(THBK4(1,1,1,2,4,5,5,25))
     C21 = Grit(THBK4(1,1,1,2,4,5,17,19))
     C22 = Grit(THBK4(1,1,1,2,4,15,15,15))
     C23 = Grit(THBK4(1,1,1,2,7,7,8,23))
     C24 = Grit(THBK4(1,1,1,2,7,8,17,17))
     C25 = Grit(THBK4(1,1,1,2,7,11,11,20))
     C26 = Grit(THBK4(1,1,1,3,3,9,14,20))
     C27 = Grit(THBK4(1,1,1,3,11,12,14,15))
     C28 = Grit(THBK4(1,1,1,5,8,11,14,17))
     C29 = Grit(THBK4(1,1,1,6,11,12,13,15))
     C30 = Grit(THBK4(1,1,1,7,7,7,8,22))
     C31 = Grit(THBK4(1,1,2,2,2,2,2,26))
     C32 = Grit(THBK4(1,1,2,2,2,2,14,22))
     C33 = Grit(THBK4(1,1,2,2,2,10,10,22))
     C34 = Grit(THBK4(1,1,2,3,3,7,7,24))
     C35 = Grit(THBK4(1,1,2,3,3,12,13,19))
     C36 = Grit(THBK4(1,1,2,3,7,7,12,21))
     C37 = Grit(THBK4(1,1,2,4,7,7,17,17))
     C38 = Grit(THBK4(1,1,2,4,13,13,13,13))
     C39 = Grit(THBK4(1,1,2,8,11,13,13,13))
     C40 = Grit(THBK4(1,1,3,4,5,14,15,15))
     C41 = Grit(THBK4(1,1,4,7,7,7,7,22))
     C42 = Grit(THBK4(1,1,4,10,11,11,13,13))
     C43 = Grit(THBK4(1,1,8,10,10,12,12,12))
     C44 = Grit(THBK4(1,2,2,2,2,5,16,20))
     C45 = Grit(THBK4(1,2,2,2,2,14,14,17))
     C46 = Grit(THBK4(1,2,2,2,6,10,15,18))
     C47 = Grit(THBK4(1,2,2,2,8,8,14,19))
     C48 = Grit(THBK4(1,2,2,3,4,4,18,18))
     C49 = Grit(THBK4(1,2,2,10,10,10,10,17))
     C50 = Grit(THBK4(1,2,4,5,5,7,17,17))
     C51 = Grit(THBK4(1,3,6,7,7,7,8,21))
     C52 = Grit(THBK4(1,3,9,10,11,11,11,12))
     C53 = Grit(THBK4(1,7,8,10,11,11,11,11))
     C54 = Grit(THBK4(2,2,2,2,2,2,7,25))
     C55 = Grit(THBK4(2,2,2,2,2,7,10,23))
     C56 = Grit(THBK4(2,2,2,2,2,10,17,17))
     C57 = Grit(THBK4(2,2,3,4,4,12,12,19))
     C58 = Grit(THBK4(2,3,3,3,8,9,9,21))
     C59 = Grit(THBK4(2,3,3,4,4,4,12,22))
     C60 = Grit(THBK4(2,3,3,5,5,5,5,24))
     C61 = Grit(THBK4(2,4,5,5,5,5,17,17))
     C62 = Grit(THBK4(2,5,5,5,5,12,15,15))
     C63 = Grit(THBK4(2,5,8,11,11,11,11,11))
     C64 = Grit(THBK4(2,7,7,7,7,7,7,20))
     C65 = Grit(THBK4(3,3,3,3,4,9,9,22))
     C66 = Grit(THBK4(3,8,9,9,10,11,11,11))
     C67 = Grit(THBK4(4,9,9,9,9,9,9,14))
     C68 = Grit(THBK4(6,9,9,10,10,10,10,10))
     C69 = Grit(THBK4(7,7,10,10,10,10,10,10))

Weight 2 "Tweak" Theta Blocks that yield Gritsenko lifts with Characters
     D1 = Grit(THBK2(2,5,8,16))
     D2 = Grit(THBK2(2,7,10,14))
     D3 = Grit(THBK2(4,4,11,14))
     D4 = Grit(THBK2(4,8,10,13))
     D5 = Grit(THBK2(5,6,12,12))
     D6 = Grit(THBK2(5,8,8,14))
     D7 = Grit(THBK2(6,6,9,14))
     D8 = Grit(THBK2(7,10,10,10))
     D9 = Grit(THBK2(8,8,10,11))

The set A4 of 4x4 matrices used in theta tracing. Here |A4|=16.
     {{2,1,-1,1},{1,2,-1,1},{-1,-1,2,-1},{1,1,-1,88}}
     {{2,0,0,-1},{0,2,-1,1},{0,-1,4,-1},{-1,1,-1,26}}
     {{2,0,0,-1},{0,2,-1,-1},{0,-1,6,3},{-1,-1,3,18}}
     {{2,0,1,0},{0,2,-1,1},{1,-1,6,-1},{0,1,-1,18}}
     {{2,1,0,-1},{1,2,0,-1},{0,0,4,1},{-1,-1,1,30}}
     {{2,1,1,0},{1,2,1,0},{1,1,6,-1},{0,0,-1,22}}
     {{2,1,1,0},{1,2,1,0},{1,1,8,-1},{0,0,-1,16}}
     {{2,0,-1,1},{0,4,1,0},{-1,1,4,-1},{1,0,-1,14}}
     {{2,0,-1,-1},{0,4,1,2},{-1,1,6,2},{-1,2,2,10}}
     {{2,0,-1,-1},{0,4,1,0},{-1,1,6,3},{-1,0,3,10}}
     {{2,0,-1,0},{0,4,-2,-1},{-1,-2,6,1},{0,-1,1,10}}
     {{2,0,1,-1},{0,6,0,-1},{1,0,6,-1},{-1,-1,-1,6}}
     {{2,-1,1,0},{-1,6,-3,-1},{1,-3,6,2},{0,-1,2,8}}
     {{2,-1,1,-1},{-1,6,0,2},{1,0,6,-3},{-1,2,-3,8}}
     {{2,-1,-1,-1},{-1,6,2,2},{-1,2,6,3},{-1,2,3,8}}
     {{4,1,0,2},{1,4,-1,2},{0,-1,6,1},{2,2,1,6}}