
// Make newform 25.34.b.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_25_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_25_b_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_25_34_b_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_25_34_b_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [39700341273686227371481766034965422500156905479886340096, 0, 544606882049987247786598582768552479276511789056, 0, 862054500541443268723145801151954288640, 0, 529361026408998369955851761920, 0, 151877130254993985540, 0, 20265063301, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [785111750092814602956289868633733557315025731867482052503163514126336, 0, 81459543577587949471936977571247424351350986792107360911360, 0, 381336324335695060763529782512521527645565887970560, 0, 421129850774205585968397622485070495258260, 0, 109213289539878176731567001176505, 0, 7848608243207638774181, 0], [-105845966553333815167366271812502087039371044695831940423983755689984, 0, -182331702486824967493652637302002413650216267250347718737920, 0, 381336324335695060763529782512521527645565887970560, 0, 421129850774205585968397622485070495258260, 0, 109213289539878176731567001176505, 0, 7848608243207638774181, 0], [-15603268507081392682905408953736073373661942030099639487442616777552989000249408225280, 0, 109537866708333871775635719606437171305047505021772092327545119908522069327872, 0, 206127504938446746433228654810638600563046039495876637084303269261568, 0, 91421103788768827293549552555160022965071709572377777478372, 0, 14672818909053308825667435805646958110478587041053, 0, 779752462997331626941642252141933671097, 0], [2688093683288572499035759923067300517762486156350574577888951867806370937897356915376128, 0, 61901518503950409976851528633913451342380819934807932614755613333109919113543680, 0, 77945238850822526911332130188006417390399749054184804469668844032638720, 0, 33820802615823330154452308433429840116220207365086042601427420, 0, 5930321400364670713635104696727227370104247784262355, 0, 350121954757571366830303302940430164875383, 0], [-71127929854873306295743175898375108398234865175452447802103135771770073552607610637647872, 0, -398838530929058664273405850353152822901361248471364156585016941696421113960071168, 0, -388426630249828903567866623629991497994373241922559938227693430213231360, 0, -144980767243686270440910960213045026975993319693153437596133484, 0, -22900200286228827686473732776956860686313477667476743, 0, -1260157163036734973356299516482954226011675, 0], [0, 126440810607153327159583966138456550631963166294708474985847342207541548652777439232, 0, 347366823341471024896056257591738384815291360423060076094233818300426485760, 0, 254393319261217910034937430305239497400556441348296313134824600320, 0, 70435390443738777895157996530061258130201029617557170220, 0, 7920651499707618803034482542423463816736535735, 0, 305741908930145219113367453699319307], [0, 184410304855167094315329602634795845312568168489133897461041620761090338089045327872, 0, 347366823341471024896056257591738384815291360423060076094233818300426485760, 0, 254393319261217910034937430305239497400556441348296313134824600320, 0, 70435390443738777895157996530061258130201029617557170220, 0, 7920651499707618803034482542423463816736535735, 0, 305741908930145219113367453699319307], [0, -23160563136970997671185406011512713506497212271658669090427004997906583447764946575358088504672256, 0, 314454056120339222704179649703689557610574047325056910118020291732667983303867040898482176, 0, 426996580431878775852548048539618306269770758194704895103780003606967593161731328, 0, 187681215299007891017685709688015161841719903400351707676356292227648596, 0, 32960723459820308450012945104979951568213799778958510174781673, 0, 1944450224141509551646454723003976476225546494028565], [0, 9585586557448277882274465049333961979288519782045263988028056796561546129445741823590741489882518213623808, 0, 6905313259195267793725755994721382248055408174753843865541750908567224804073601690047726579875840, 0, 124813727667233807799714203982445992703314268528943905333768057067475900317954570904320, 0, -637303592126158236865521164476403065951016187043209510897132788146124023724820, 0, -121373811008439054423450250271726674607060180144251347121559580895065, 0, -6343611233696978283098560680796068308541609661585446402757], [0, -20416816640190752728293140786868464918158989530157054613009497945788202282542071062781429621786467631104, 0, -59567574832231103765739698377827666064309240603592276593252070386748690280785267855720560721920, 0, -50377594766547976156348043325392292201729856167246626801392494745385489272613291961600, 0, -18013762616940056908809752282121265425121517228350851288641445526596862237140, 0, -2820099803700849041789140124949210320756688238282603768291873832905, 0, -155479391572602641159066185080512461386936113732306168949], [0, 3553330477262402028584406411820723859185320385886689150489178644288955366498124987951864359133570744188928, 0, 6515405891501136657153380794369972299257743663039682141407907077885141573840317940390838686187520, 0, 7844104704451327292269870346705240370343437520565065059217592697922718221930313011815680, 0, 3595078317848008527051769911037813537149561352082169062563665554483368021192500, 0, 644228774523659181338624890170824298614243573573226217177381744137345, 0, 38189949680671456751671975463594993303659794271529556137933]];
    Rf_basisdens := [1, 65947811516103229241397403718312459500391813510613769912320, 65947811516103229241397403718312459500391813510613769912320, 499618883920760122135852497252976754875576191094881939072616363254783934464, 312261802450475076334907810783110471797235119434301211920385227034239959040, 499618883920760122135852497252976754875576191094881939072616363254783934464, 28984747124006883577872818248169647340302501097212711237597139276774394718133944320, 28984747124006883577872818248169647340302501097212711237597139276774394718133944320, 11212606268710592700633296739775011272084898784570714085095512817837091028289890033680353591296, 1906143065680800759107660445761751916254432793377021394466237179032305474809281305725660110520320, 10359473183047830212541632857400825631817569529222942361229549886045138450050441878943804948480, 1906143065680800759107660445761751916254432793377021394466237179032305474809281305725660110520320];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_25_b();"
function MakeCharacter_25_b()
    N := 25;
    order := 2;
    char_gens := [2];
    v := [1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_25_b_Hecke();"
function MakeCharacter_25_b_Hecke(Kf)
    N := 25;
    order := 2;
    char_gens := [2];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 34;
    raw_aps := [[0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 158, -77, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -81027079, -97559503, 216966, 1, 3, 2], [25126816090199447, -14413046, -7346728, 577206, -1350, 1323, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8387074752769, 4227611318652, 13181999140, -33220, -113336, 2572], [0, 0, 0, 0, 0, 0, -731985235456705, 71979343991268, 1206195272012, 7165684, -6713080, 2320148], [-285264060547553437715, -34918200554, -342670207344, 5328047433, -9006490, 12206789, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 45885434488382249, -42933048238174583, 639665020750, -1006032495, -788124893, 31884690], [-217537234337621831392874, -9766658160152, 192624564103984, 2475437599560, -136047966712, 9668691956, 0, 0, 0, 0, 0, 0], [1804220232784683688838838, 123262136993740, -46308791692232, 3329891184987, 33609440748, 38296391322, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1513908060285215715699, 176883503558777976856, 65271889433181280, -2041651136528, 2224952922280, 1717470790328], [74668096588566883570901118, 17945598000906768, -59056610166152808, 2609528006825850, 672376427456, -4908195482260, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 11494219637176081894384, 2377231320609283250275, 1224143130911256203, -5317073654818, -20864986451446, -10524453770708], [0, 0, 0, 0, 0, 0, -24279425445185965814877, 17975787291846864043269, 6187862865866354348, -5602146991521, 71852550727037, -121583657907426], [0, 0, 0, 0, 0, 0, 259798428324436968254611, -41747544953250159396364, 67930793307109529004, -177214407399756, 2946295216711704, 309915317090916], [29719192702546499531079704603, -5620821608272942550, -2867801225814653632, -1134982169053807539, -11623024768669862, 59309055537403, 0, 0, 0, 0, 0, 0], [-47052948168343613011540995810, 9352186873198468576, 25248306766384150432, -38414614471709496, 14347001668460576, -485676071454784, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -17024935161227846269788120, -4111334884584395131241137, 6996178539994758134127, 2060362376056926, -46044127358382198, -33999454406991060], [1408536338538163739059610629740, 40756229004286498688, -382328459315970193512, -20871986387770004487, -64648932742573312, 12974099236071008, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -14718257451683951733098595, 25906659657416708556784844, -24376522837754793654108, 83403997468707356, 39246665920146360, -24174987547283300], [3675417559094817199400270756408, 720951491640157040096, 1583933805409419110224, 34831780714467703974, -672035536007085472, 12106380026426672, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -283863398515904932288617774, 152890408833893802249037683, 63935233224572146100901, -94592695921901700, 3274640421804628836, 338664555292237800], [56081893066373650302459671570862, -1578104921946746169600, 28524299170066203018192, 267739219285912756764, 11503355727335767200, 407407482369246744, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -13126682738414442551232482385, -2692238672614581131241177940, -1289035869686423272448596, 1951017452115648516, -31743670890463421248, -279181356653221956], [163791961400329966812982504679098, 8474607884977501334504, 188596107983593738222960, -56382311255936870280, 40495727235139968136, 13100009887482087892, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -12299832598678633040884673855, 5750345922234720780446917253, -2529033900979781713121374, 4363323670473569713, -67807193853596825821, 41195614362924215666], [0, 0, 0, 0, 0, 0, 930789814437792794261808150, -1303519705691054845239602657, 12993990498344198673304357, -63757116435387343168, -210098407608274142912, -1141766395703371136], [1108617831717509275762255285328786, 70541304665124970319952, -76146293896049672741552, 12222996050224090589316, 1187154351010013296, -54986182253773702736, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -27583529422402886512937245239, -58802306038057870226136097128, -38199897509716477278792560, 164472034253566677408, 225445423687912672792, 75513750980175490680], [0, 0, 0, 0, 0, 0, -426375325861906760749232903457, 41663895957092159556092897957, -128622901688854527930698504, -617736297603875149677, 4090663499834186595817, 865922601791881685334], [45333691300133934456444003358826501, -3542453590386371381676418, -7284381838362950140008472, 208724313697798535382270, -1661051382604631456242, 388094735141645517041, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -706124310905852596820821070741, 727547476184318606219562793712, 364654088556928169075252456, -381228389885019465240, -9701451134187697999336, 548221854307177574448], [16195440986390577075351401384276735, 13888191736051166525774242, 20308110443549957907478912, 1371227013892503528088065, 10931151876518639845138, 1545443257745203120255, 0, 0, 0, 0, 0, 0], [-126564833461844976128266148846270534, -23269087824475552312202096, -103474519417962191817484880, 1694810263815736978069452, -5958997230910670560912, -7587243210846958627360, 0, 0, 0, 0, 0, 0], [566070621142696234365635329956884864, -6427673008758456979841352, -57821481103577802463340040, -2115931520433957390348735, -15448885453536455480840, -4496743594053789096284, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -44006095152317032149993933544911, -7677497698582947274206327005384, -5779327526054453345460720320, -3347264143816164426800, 7736385992001376016104, 13788523801131906915608], [0, 0, 0, 0, 0, 0, 21951021326543000009551090656162, 4271089645318377777271558349191, -649456186586696454533748791, -31663371199885982187036, 179833494288997464997324, -42185787535787368713240], [0, 0, 0, 0, 0, 0, -81039479930616944429811799593483, 16693312949970828151904369555741, 1048750389475344335190444350, -36716992052530417394675, 191030624490984483147863, -53795582389802770839478], [0, 0, 0, 0, 0, 0, 21388225751230051369463276360983, -61758955652202562121271673065272, 51998078965620703805715839296, 123789378959550571600240, -106670118712507141023464, -50910952645536078598744], 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-240127274097058515247782056876453644], [0, 0, 0, 0, 0, 0, 4786255696401666207558643777229348157824785, -85984199200635946130321945055612154307856495, 18276827493852260156570858906860934773278, 448616874792799363466651076486268169, 439752950483798472791008414009396731, 271876579359430195714124939868950770], [-2099347297496794214414870882517245068411594315682, -2884536483263186988845825495025413903972, -2678102313669689711819125873335414225448, -31882207276710649818188996022840228225, -294471972821070962241010684608152260, -55170557510460607082811571981384654, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 339118695882616844058540877316958816936993893, -27609299031223217725105552592736063241995036, -21505477134662094764406402524966563383092, -145724824795969066023508732890737292, -529479533784794635973531228503207544, 612183507445579908694626565644675348]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_25_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_25_34_b_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_25_34_b_c(:prec:=12)
    chi := MakeCharacter_25_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 34));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_25_34_b_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_25_34_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_25_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,34,sign)));
    Vf := Kernel([<2,R![162612597857018787313589313679218370560642684845614449033216, 0, 557677447219186941733476948754997738779148071993344, 0, 220685952138609476793125325094900297891840, 0, 33879105690175895677174512762880, 0, 2430034084079903768640, 0, 81060253204, 0, 1]>],Snew);
    return Vf;
end function;