
// Make newform 93.2.f.b in Magma, downloaded from the LMFDB on 10 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_93_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_93_f_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_93_2_f_b();".
// 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_93_2_f_b();".
// 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 := [81, 729, 2493, -1014, 4006, -4987, 5846, -2812, 1869, -356, 382, -119, 90, -28, 13, -3, 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, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [18367596905865675808987299, 67641903119350020752531222, -2901465258950677118364214, 32489734032064370240389814, 32796799292110852121729335, 18212878108015687881999594, 15561519488218694052210369, 12490682782745364426018319, 3706018372881128476972647, 1793683750426651810121786, 600385702469931835961882, 260509979293514344130927, 33865550375742065906783, 43125610572414463271763, 2330587915797073985660, 6500286224746908751917], [278306355927043401829142687, 13628888248025179328839806, 51436689561055977234002141, 3689824972942688356079624, 6449587415728253780210312, 65213726694923686067539414, -19787795161854740681707188, 33840324352207001462600973, 341647828693391968685446, 6020120268891027992655099, -689425587426667333110508, 1373919763214813977440005, -324515780933707443541603, 215873564668655510960459, -41378110349295350503158, 21831446590037800241411], [129167826948808679389300128, 713682661358917933210669158, -732422546447373268441870533, 2181127958580515078674731236, -2031028846476039463511003291, 1913577279177785207367677953, -912205646810106829049904173, 525332190293440409331753786, -137927709461136820182572209, 113779604063609029850798732, -41448266466980465289179263, 26276610085974128132487894, -8574195438807392835881906, 3599706879955688357045285, -848738624401195914243366, 243065667137128335093899], [84551066914997790754337934, 896786593581035880961681473, -110547679637783198097923259, 1763100297358242740386431029, -2429050502360160178049077037, 2620347205847620634295739087, -1365947791872377293710475826, 736109226591611846048787306, -189131280266274847412673460, 157058474608680776297211353, -59066956335528689748989236, 38341111134141214461376650, -12859298484277755946594031, 5477421101341002829883204, -1360746056802259921002081, 381347425120524238619540], [407480974038167292517920201, -8138524884064812273944305683, 8464663182882423029488873275, -19832024620468671266068629763, 23477936828842639032216227515, -22404715123212579464459559410, 11014164722922012856709260822, -6420206583914782724454968478, 1585996898705272458837719411, -1373441889803962219919452552, 487674332027784276708804548, -320414785169492570354241873, 104114851215872569271925988, -44792011129479790126205989, 10538605265916882799673946, -3131520996851770027938442], [823719471491936374276799520, 292579293863521212636678114, 1063916127241744319117359145, -193798865795456522517896458, -149711935420306102611481638, 1443033288761734766213804529, -542170822777819291493648116, 588021493545825637170289847, -26963499778417821684022434, 107601854428267482723512325, -20516497013407610004324380, 25902555365512798416268279, -7019167160089192657724915, 4072850354928416762317603, -871082679612617407405420, 366392117045781113232526], [-2765152280324584159575378237, -9356573813882069284470928179, 4155780379222843809661417911, -18215610577455541571348622859, 24904245253272382657754486497, -27073034462618173767685692902, 13539387443235020798999405305, -8408089810805919239635413243, 1769607328984029243321739601, -1744208464137120735021154507, 575164377850561073828753006, -414246313006200032372398224, 131959205976701705115216466, -59583872628911579074404268, 14000532126349486874203515, -4470059708110377542441611], [1346483343541080543879095982, 14123763660948787868937656751, -11394353376707915639755687404, 25257764863758769117701021457, -29598507874337388367082283427, 30703843471642747883810769863, -15028586547919210355503111726, 9355067149395389044940381043, -2044440530478823010389094180, 1951830351047049378338376637, -648125067550819897261416098, 456282009479117776471979478, -144750941179372362215595247, 64738132552259618559800530, -15081177773501238270759489, 4783993590696617755159264], [-407456567547809284228184406, 11150184111702752549603896443, -8505759093409582516677854280, 19698887613937622314454101954, -23317901282481175643125217848, 22564109215821412505764549484, -11296165526598158812784803606, 6554957600201864877612293778, -1604064003711880304280864590, 1394258857679713693252179574, -492800171692889487306502862, 326389316980173834558097131, -106131545886127949690215492, 45843245381683327008517180, -10857670809390915401751066, 3249246200779668573937210], [-3156419684299580806768733466, -33014717168964294322342042074, 28145157816899979774898820544, -57557684013820765781754441512, 63891278369739782443492363784, -65038495952310069883746616687, 31546372200522610267509929873, -19657111637380247895185729886, 4363714262452439787845542939, -4111282589004027399993975404, 1369210824801898617955495288, -954599715431153297043540210, 302293617561415381531569275, -134630525027867276604997322, 31243001194154227938074952, -9881921063979475723036181], [3054067375294617395048916093, 32077719095867910411430320567, -19220132855241182324554764165, 58070328882026250903270226540, -69536263154130120834173964646, 72383985676813464880531517021, -36113352754142173808076024562, 21342407508196104978261847353, -4994127770136048121886747420, 4487613819741432179949396253, -1564294356458044204117728644, 1061643146416705867835833842, -343642434477036971562325330, 150492163711691932655800957, -35857583065958115456042189, 10834971721198425418516672], [1582105302989182476074507562, -32622392038373688760814011419, 32880423944385816049351495056, -76942195499390971929103527460, 91169933024483949997494534325, -86983321458550154364981779699, 43217512730814682353182440756, -24961819102988425410169055124, 6210197378077424897588216159, -5337150106844938380601041700, 1903448554236860622740087753, -1248052789076742639846863439, 406554907267264541146823869, -174579099737671295144874094, 41277366187040059916124276, -12228121404828688393781047], [7118839307733660123386095026, 40808511483922265619490675686, -25319016954540565004235017316, 78945914914501366830963044792, -91808797757698588016476472816, 97866172129911145084428682834, -47617732762836931225357350272, 29685087109674267856080970368, -6454904896308155760278210737, 6191773120944446579249786504, -2058802949113978816666776838, 1453316387682703482907798140, -461743994842631789699145890, 206510879560008908575936160, -48257574909563060220741054, 15240938266645313950302968], [-11001544164184014452873695491, -37282214122345803507201255987, 16299961657333204257180934791, -71404580429911190875424520595, 95994760325001993726152230462, -105835556757083153701549584674, 52679703605657684314771388023, -32946244270762171106244072417, 6865301940693674320163188739, -6822644860667525541417185533, 2237871953750566277780224082, -1619414895791813549250577050, 514634723775909086538754912, -233030420238742917937858870, 54682568883878887494340266, -17521613963472077442881437]];
    Rf_basisdens := [1, 1, 69708219777811975359511991, 69708219777811975359511991, 627373978000307778235607919, 627373978000307778235607919, 1882121934000923334706823757, 209124659333435926078535973, 1882121934000923334706823757, 1882121934000923334706823757, 627373978000307778235607919, 1882121934000923334706823757, 1882121934000923334706823757, 1882121934000923334706823757, 1882121934000923334706823757, 1882121934000923334706823757];
    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_93_f();"
function MakeCharacter_93_f()
    N := 93;
    order := 5;
    char_gens := [32, 34];
    v := [5, 3];
    // 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_93_f_Hecke();"
function MakeCharacter_93_f_Hecke( : Kf := false)
    N := 93;
    order := 5;
    char_gens := [32, 34];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := 2;
    raw_aps := [[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1], [-1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1], [-1, 2, -2, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0], [2, 0, 0, -1, 0, 0, -2, -1, 1, -1, -1, 0, 0, 1, -1, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -2, 0, -2, -1, 0, 0, 0, -1, 1, 1, -1, 0], [-1, -2, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, -1, 1, 1], [1, 0, 0, 0, 0, 0, 1, 1, 1, -1, 1, 0, -1, -1, 0, 0], [1, 0, 0, 1, 0, 0, -5, 0, 1, -1, -1, 1, 0, -1, 0, 0], [-2, 2, 0, 0, 0, -2, 0, 1, 4, 1, 0, -1, 0, 1, 1, 0], [-2, 0, -2, 0, 0, -2, 1, -1, -1, 1, -1, -1, 0, 0, 1, -1], [-1, 0, 2, 0, 0, 0, 3, -1, -7, 5, 0, -2, 0, 2, -1, 0], [1, -2, 2, -2, -2, 2, -1, -1, 0, 0, 0, 0, 1, 1, -1, -1], [4, -2, 2, 1, 0, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -2, 0, 0, -1, 1, -1, 1, 1, 0, 0, 2, 1, 0], [-2, 0, 0, 0, 0, 0, 2, 0, -2, 1, 2, -1, -2, 0, 2, 0], [-3, 0, -2, -1, 2, -2, 2, -1, -4, 4, -1, -1, 0, 1, -2, -1], [-6, 2, -2, 1, 2, 0, 5, 0, 1, 6, 1, 1, 0, 1, 0, 0], [0, 2, 0, 1, 2, 0, -1, 0, 0, 1, -1, 2, 2, 0, -1, 1], [3, -2, 0, -1, -2, 0, -1, 0, 3, -6, -1, 1, 1, 0, -1, -1], [2, 0, 0, 1, 2, 0, 0, 0, 1, -1, 0, 2, 0, -1, -1, 2], [2, 2, -2, 1, 2, 0, 0, 1, 2, -2, 0, 1, -1, 0, 0, 0], [5, 0, -2, 1, 0, -2, -5, 0, 0, 0, 0, 0, 0, -1, 0, 0], [1, 0, 4, 0, 0, 0, 2, 0, 3, 1, 0, 1, 0, -1, 0, 3], [1, 0, 0, -1, 0, 2, -2, 0, 1, -1, -2, -1, 1, 0, -2, -1], [-4, 2, -4, 0, 0, -2, -4, 0, 2, 0, -1, -3, 1, 3, 0, 0], [-7, -2, 0, -2, -2, 2, 2, 0, -7, 1, 2, -2, -1, 0, 2, -2], [-1, -2, 0, 0, 0, 2, -3, 3, 3, -5, 2, 0, -2, 0, 3, -4], [-3, -2, 2, 0, 0, 2, 2, 1, 3, 4, -2, 0, 2, 0, 1, -1], [2, 4, -4, 4, 2, -2, 4, -1, -2, 0, 0, 0, -2, -2, 2, 2], [2, -2, 0, -2, -2, 0, -1, 0, 2, -6, -1, 0, 0, 0, -1, -2], [-2, 2, 2, 0, -2, 0, -5, -1, -1, 2, 1, 0, 1, 0, 0, 0], [6, 0, 4, 0, 0, 4, -7, 0, 2, -2, 0, -1, 0, 0, -1, -1], [4, -4, 4, 2, -4, 0, 1, -2, 0, -4, -1, 2, 2, -1, 0, 0], [0, 2, -4, 0, 0, -2, -5, 0, 5, -5, -1, 1, 1, -1, 0, 1], [-2, 0, 0, -2, 0, -2, -1, 0, -2, 6, -1, 2, 1, 0, -1, -2], [-5, 4, -4, -1, 4, -4, 0, -2, -2, 0, 0, 0, 2, 3, -2, -3], [-4, 0, 0, -2, 0, 0, 5, 2, -4, 0, 0, 0, -1, 2, 1, -2], [2, -4, 0, -1, -4, 4, -2, 0, 2, -8, -2, -2, 0, 0, -2, -1], [-8, 2, 0, 0, 2, 4, -2, 0, -8, 6, -2, -2, 2, 0, -2, 0], [-1, 4, 0, 0, 0, -4, -1, -1, 2, 1, -1, -1, 1, 1, -1, 2], [0, 0, 0, 0, -4, 4, -1, -1, -1, 0, 0, 0, 2, -2, -2, 2], [9, -4, 4, -4, -4, 4, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0], [-5, 2, 0, 2, 0, 0, 1, -5, -3, 5, -1, 2, 5, -1, 0, 0], [2, -6, 0, 0, 0, 0, -9, 3, 3, -2, 4, 0, -3, -2, 0, 0], [3, 2, -4, 0, 0, -2, 1, -4, 9, 2, 0, -1, 0, 1, -4, 0], [7, 0, 2, 0, -4, 2, -7, 0, 1, -1, 0, 0, 0, 0, -1, 0], [1, 4, 0, 3, 4, -4, -2, 0, 1, 0, -2, 0, -2, 0, -2, 3], [-3, -4, 4, 0, 0, 4, 0, 4, -1, -1, -1, 2, 1, -2, 4, -2], [7, 0, 6, 4, 0, 6, -4, -3, 8, -8, -3, 3, 0, -4, -2, 3], [2, 8, -4, 0, 0, -8, 5, 0, -5, 3, 3, -1, -3, 1, 0, 1], [-11, -4, 0, -5, 0, 0, 11, 1, -4, 11, 1, -5, -1, 1, 0, 0], [0, 0, -2, 2, 2, -2, 0, -2, -3, 3, -2, 0, 0, -2, -2, 0], [3, 6, -2, 1, 2, 0, -5, 3, 4, -3, -1, 1, -3, 3, 0, 0], [-7, 4, -2, 2, 2, 0, -5, -1, 1, 7, -1, 2, 1, -1, 0, 0], [-9, -8, 0, -1, -8, 0, -4, 0, -9, 2, -4, -2, 5, 0, -4, -1], [8, 0, -6, 2, 6, -6, -8, -3, 4, -4, -3, 0, 0, -2, -4, 0], [0, 0, 0, 0, 0, 0, -3, -1, -9, -2, 0, -1, 0, 1, -1, 3], [2, -4, 4, -2, 0, 0, 2, 3, 1, 0, 0, 0, -3, 1, 3, -1], [6, 0, 4, 0, 0, 0, 11, 0, -4, 5, 2, 4, -2, -4, 0, 3], [5, 0, 0, 1, 0, -4, -2, 0, 5, -9, -2, 3, 5, 0, -2, 1], [-4, -4, 2, -1, -2, 0, -1, 2, 1, 4, 3, -1, -2, -1, 0, 0], [3, -2, 0, 2, -2, 6, 4, 0, 3, 3, 4, 1, -1, 0, 4, 2], [-11, -6, 6, -1, -4, 4, 9, 0, -11, 0, 0, 0, 2, 1, -2, -1], [-16, 0, 4, 2, 4, 4, 16, -2, -13, 13, -2, 0, 0, -2, 2, 0], [-6, 4, 2, 0, -2, 0, 1, 0, 0, 6, 2, 0, 0, 5, 0, 0], [1, 0, -4, 0, 0, 0, 4, 3, -2, 0, 3, 1, -3, -1, 3, 3], [9, 6, -6, -1, 6, -6, -6, -5, 7, 0, 0, 0, -1, 1, 1, -1], [-2, 0, 0, -5, -4, 4, 8, -6, -12, 0, 0, 0, 4, 1, -4, -1], [-3, 0, 0, 2, 0, 4, 3, 0, -3, -1, 3, 3, -4, 0, 3, 2], [4, 0, 0, -3, 0, 0, 1, 0, 4, -10, 1, -3, 1, 0, 1, -3], [-6, 0, -2, 3, 4, -2, 7, 2, 8, -8, 2, 1, 0, -3, 4, 1], [-3, 0, 0, -4, 0, 0, 2, -4, -6, 0, 0, 0, 4, 1, -4, -1], [0, 4, -6, 0, 6, 0, -10, 0, 0, 0, -5, 0, 0, -2, 0, 0], [4, -2, 2, 4, 0, 0, -3, 0, 0, 0, 0, 0, 3, 0, -3, 0], [-4, 2, -2, 0, 0, -2, -10, -3, 8, -3, -6, -1, 6, 1, -3, 0], [-2, 2, 4, 0, 0, -2, -8, 1, 4, -7, 3, -4, -3, 4, 1, -1], [2, 6, -4, 0, 0, -6, 13, 6, 0, 5, 7, 1, -7, -1, 6, -4], [-11, 6, -6, 5, 2, -2, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0], [-1, -4, 4, 7, 0, 0, 10, -5, -12, 0, 0, 0, 2, -7, -2, 7], [-3, 0, -4, 8, 2, -4, 9, -3, 5, -5, -3, 6, 0, -8, 1, 6], [-2, 4, 2, 0, 0, -4, -6, 1, 5, -5, 0, -1, 0, 1, 1, -7], [5, 0, -4, -4, 0, -4, -4, 2, 6, -6, 2, 1, 0, 4, 4, 1], [18, -2, 0, -2, -2, -4, -1, 0, 18, -4, -1, -6, -3, 0, -1, -2], [-1, -2, -2, -3, 2, 0, 3, 0, -3, 1, -1, -3, 0, -2, 0, 0], [11, -2, 0, 2, 0, 0, -7, 5, 7, -11, 2, 2, -5, 1, 0, 0], [20, 0, 0, -2, -2, 0, -20, 0, 12, -12, 0, 0, 0, 2, -6, 0], [6, 0, -6, -6, 0, -6, -13, 3, 14, -14, 3, -7, 0, 6, -7, -7], [6, 0, 0, 6, 0, 0, 2, 0, -2, 0, 0, 0, 0, -2, 0, 2], [3, 4, 2, 0, 0, -4, -8, 2, 8, -13, 2, 3, -2, -3, 2, 1], [-3, 0, 0, 0, 0, 0, 0, 0, -3, -6, 0, -3, 0, 0, 0, 0], [-2, -2, 6, 0, 0, 2, 15, 8, -19, 9, 5, 1, -5, -1, 8, 5], [-8, 2, -2, -4, 2, -2, 13, -3, -16, 0, 0, 0, 3, 2, -3, -2], [7, 6, 0, 0, 6, 0, -3, 0, 7, 4, -3, 5, 0, 0, -3, 0], [-17, 0, 0, -4, 2, 0, 14, 1, -1, 1, 1, -3, 0, 4, 1, -3], [-7, 4, 0, 2, 4, 4, -5, 0, -7, 3, -5, 3, 8, 0, -5, 2], [-4, -2, 2, -2, -2, 2, 1, -2, -3, 0, 0, 0, 2, -3, -2, 3], [-6, 8, -8, -6, 2, -2, 0, 2, 2, 0, 0, 0, -2, 2, 2, -2], [3, 4, 0, 2, 4, -4, 2, 0, 3, 2, 2, 7, -6, 0, 2, 2], [-2, 4, -8, 0, 0, -4, 12, 2, 0, 12, 2, -2, -2, 2, 2, -6], [12, 2, -4, 1, 4, 0, -5, 5, 6, -12, 6, 1, -5, -7, 0, 0], [-1, 2, 0, -4, 0, 0, 10, -1, -5, 1, 3, -4, 1, 2, 0, 0], [5, 2, 0, -2, 2, -4, 5, 0, 5, -5, 5, -2, -4, 0, 5, -2], [-13, 4, 4, -3, -4, 0, 2, -1, -4, 13, 0, -3, 1, 4, 0, 0], [-11, -4, 2, 0, -2, 0, 3, 0, 0, 11, 3, 0, 0, -3, 0, 0], [-1, 2, 8, 0, 0, -2, 3, 0, -1, 4, 1, -2, -1, 2, 0, 0], [8, 0, 0, -5, 0, 8, -2, 0, 8, -6, -2, 2, 2, 0, -2, -5], [4, -4, 12, 0, 0, 4, 21, 5, 4, 12, 5, 4, -5, -4, 5, 6], [11, 0, 0, 0, 2, -2, 7, 0, -2, 0, 0, 0, -5, -1, 5, 1], [11, -8, 0, 4, -8, 4, -1, 0, 11, -13, -1, 1, 0, 0, -1, 4], [-7, 6, -4, -1, 4, 0, 9, 5, 4, 7, 0, -1, -5, 3, 0, 0], [8, 0, 2, -3, -6, 2, -10, -1, 6, -6, -1, -2, 0, 3, 0, -2], [-6, 0, 0, -4, 0, 0, -2, 0, -6, 13, -2, -9, 3, 0, -2, -4], [11, -12, 8, -1, -8, 0, -15, 5, 4, -11, 1, -1, -5, 3, 0, 0], [1, 0, -2, -9, -4, -2, -8, -4, -1, 1, -4, -7, 0, 9, -5, -7], [10, -2, -4, 6, 4, 0, -1, -3, 3, -10, -4, 6, 3, -5, 0, 0], [-6, 0, -8, 8, 8, -8, 10, -2, -1, 1, -2, 4, 0, -8, 2, 4], [-10, -6, 6, 4, -6, 6, -3, -3, 0, 0, 0, 0, 3, -6, -3, 6], [-9, 6, -6, 3, 6, -6, -8, -1, 8, 0, 0, 0, 0, 2, 0, -2], [-2, 0, 2, 2, -4, 2, 2, 0, 11, -11, 0, 0, 0, -2, 1, 0], [12, -6, -2, 1, 2, 0, -13, -6, -5, -12, -4, 1, 6, 1, 0, 0], [-1, -2, 0, 2, -2, 0, 3, 0, -1, -4, 3, -7, -5, 0, 3, 2], [-24, 2, -2, 6, 2, -2, 11, -1, -12, 0, 0, 0, 1, 0, -1, 0], [10, 0, 4, 0, -2, 4, -7, -5, -5, 5, -5, 3, 0, 0, -3, 3], [-14, -2, 0, 2, -2, 10, 6, 0, -14, 3, 6, 5, -1, 0, 6, 2], [14, 2, -2, 3, 8, -8, -9, -7, 9, 0, 0, 0, 0, -2, 0, 2], [2, 8, -8, 4, 12, -12, -5, -3, 5, 0, 0, 0, 0, 5, 0, -5], [3, 2, 0, 2, 2, -2, 4, 0, 3, -9, 4, 4, 1, 0, 4, 2], [1, 0, -4, 0, 0, 0, 8, 0, -9, 7, -2, 3, 2, -3, 0, 4], [0, 0, 10, -5, -10, 10, 1, 2, 6, -6, 2, 1, 0, 5, 0, 1], [-13, 4, -4, 7, 0, 0, 2, 6, -2, 0, 0, 0, 0, -3, 0, 3], [16, -10, 12, 0, -12, 0, -10, 4, 4, -16, 0, 0, -4, -2, 0, 0], [-4, 0, 2, -10, -2, 0, -4, 2, -8, 4, -5, -10, -2, 8, 0, 0], [-2, 0, 0, 8, -2, 0, 6, 6, -10, 10, 6, 4, 0, -8, 4, 4], [-2, 2, -4, 0, 0, -2, 13, -5, -18, 20, -7, 0, 7, 0, -5, -2], [5, -4, 4, -9, -2, 2, -8, 1, 4, 0, 0, 0, 4, 4, -4, -4], [-8, -4, -2, 0, 0, 4, -4, -4, -19, 8, -6, -6, 6, 6, -4, -1], [-2, 0, -4, -8, 2, -4, 0, -1, 11, -11, -1, -2, 0, 8, 2, -2], [-1, 8, -6, 1, 6, 0, -1, -1, 0, 1, 6, 1, 1, 1, 0, 0], [-20, 0, 0, -1, -12, 0, 18, 2, -14, 14, 2, -2, 0, 1, 8, -2], [2, -4, -6, -2, 6, 0, 3, -10, -12, -2, -1, -2, 10, -3, 0, 0], [-3, -2, -10, 0, 0, 2, -4, -2, 5, 1, -3, -2, 3, 2, -2, 0], [10, 0, 0, -8, 0, 4, 5, 0, 10, -8, 5, 0, -1, 0, 5, -8], [15, 0, 6, -7, -8, 6, -18, -3, 12, -12, -3, -3, 0, 7, -5, -3], [2, -2, 2, -2, -12, 12, 10, 3, -10, 0, 0, 0, 0, 2, 0, -2], [-10, 0, 6, 2, 0, 6, 12, 7, -22, 22, 7, 2, 0, -2, 2, 2], [2, 8, -8, 0, 0, -8, -16, -7, -9, -11, -2, -3, 2, 3, -7, -1], [2, 4, 10, 0, 0, -4, 8, 0, -1, 6, 1, 1, -1, -1, 0, 8], [-13, 2, -12, 6, 12, 0, -2, -1, 5, 13, -3, 6, 1, -4, 0, 0], [12, 0, -4, -6, 0, -4, -14, -5, -5, 5, -5, -2, 0, 6, -1, -2], [2, 8, -8, 1, 8, 0, 4, 2, 3, -2, 6, 1, -2, -4, 0, 0], [-1, -4, 0, -3, -4, 6, 4, 0, -1, 8, 4, -9, -2, 0, 4, -3], [-11, -12, 12, -1, -10, 10, 10, 5, -4, 0, 0, 0, -6, 2, 6, -2], [-3, 2, 0, 3, 2, 2, -4, 0, -3, -6, -4, 0, -3, 0, -4, 3], [-25, 2, 0, -2, 2, -6, -3, 0, -25, 34, -3, 1, 7, 0, -3, -2], [9, 0, -10, -1, 6, -10, -19, -3, 12, -12, -3, -10, 0, 1, 0, -10], [12, 0, 16, -1, -8, 16, -8, 2, 1, -1, 2, 4, 0, 1, -1, 4], [-13, 10, -10, 1, 12, -12, 17, 7, -12, 0, 0, 0, -5, -1, 5, 1], [1, 2, -8, 2, 8, 0, 0, -5, -3, -1, -7, 2, 5, 6, 0, 0], [-1, -6, 6, 0, 0, 6, 0, -1, 3, 2, -3, 1, 3, -1, -1, 2], [2, 0, 2, -2, 12, 2, -4, 0, 8, -8, 0, -2, 0, 2, 4, -2], [-1, -6, 6, 0, -6, 6, 22, 0, -22, 0, 0, 0, 0, -3, 0, 3], [12, 2, -2, 0, 0, 0, -5, 6, 8, 0, 0, 0, -3, 1, 3, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_93_f_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_93_2_f_b();".
// 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_93_2_f_b(:prec:=16)
    chi := MakeCharacter_93_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_93_2_f_b();".
// 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_93_2_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_93_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![81, -729, 2493, 1014, 4006, 4987, 5846, 2812, 1869, 356, 382, 119, 90, 28, 13, 3, 1]>],Snew);
    return Vf;
end function;