
// Make newform 190.2.i.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_190_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_190_i_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_190_2_i_a();".
// 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_190_2_i_a();".
// 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 := [10000, 0, -34500, 0, 85425, 0, -89920, 0, 66046, 0, -29980, 0, 9835, 0, -1928, 0, 270, 0, -20, 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, 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, 25455372239291500, 0, -62032454787180475, 0, 69104511948824710, 0, -41552708308853972, 0, 16414690458001860, 0, -3709100447264955, 0, 583092465086296, 0, -52636278341640, 0, 3129986432010, 0, -67521462807], [1502788932814500, 0, -3736670324512425, 0, 3686209593721840, 0, -2989442526333086, 0, 1395350556245180, 0, -485110638968595, 0, 97142278802248, 0, -14058915653070, 0, 1052563531840, 0, -54067767641, 0], [-30857811000, 0, 72562221150, 0, -87629121165, 0, 31400754873, 0, -4819884240, 0, -4864178705, 0, 1511616336, 0, -323563365, 0, 26942635, 0, -1689612, 0], [1319197705722000, 0, -5573461833504850, 0, 9564408458723185, 0, -7714416724559674, 0, 3925919355491645, 0, -1220448373480805, 0, 236572159532307, 0, -30750494419705, 0, 2229773264110, 0, -102026243919, 0], [14548369045450000, -36463203971865500, -82674556465981000, 89077189670499575, 105765732867681625, -100088150548870545, -85775748439887650, 61102611369853779, 40174551627595125, -24769624245471020, -13920603313595625, 6085604333806910, 2787911259052075, -1020126168778872, -403507493841375, 107681048716605, 30222412702750, -7051030576145, -1553004285275, 244351455749], [-2039151855515000, 153417896376500, 3250501619144750, 1829435517540000, -7249999966229525, -4450950438561000, -9043065764527855, 6465395719130639, 8518486506772400, -2934356933309220, -5790156943291675, 1138391333481600, 1400052307666640, -230227796742002, -248406186805725, 35828758140480, 19810551139225, -2721738955500, -1149846030380, 149747246509], [-20418794431595000, 27137894626863000, 51637617199916750, -6091324597145000, -49624022863167125, 14819972405224250, 46855437547688585, -12707879205563957, -23916420959174800, 9770292745110985, 9667163825331875, -3790410246400800, -2049268929759280, 828828557891851, 318379497312075, -119296140068240, -24408237958125, 9062355730875, 1317043616260, -446324990017], [0, -507053781808000, 0, 3736670324512425, 0, -3686209593721840, 0, 2989442526333086, 0, -1395350556245180, 0, 485110638968595, 0, -97142278802248, 0, 14058915653070, 0, -1052563531840, 0, 54067767641], [14548369045450000, 36463203971865500, -82674556465981000, -89077189670499575, 105765732867681625, 100088150548870545, -85775748439887650, -61102611369853779, 40174551627595125, 24769624245471020, -13920603313595625, -6085604333806910, 2787911259052075, 1020126168778872, -403507493841375, -107681048716605, 30222412702750, 7051030576145, -1553004285275, -244351455749], [6025736267915000, -1688006956684000, -21007325474025000, 6388839899846325, 51110062986716250, -12287062395469070, -55933105556144665, 9989686502483928, 33695088219276825, -4533820841544040, -13072096299636000, 1639801849219460, 2776792115716845, -333029982590554, -411420012550800, 50221730714535, 31253605576875, -3807391628420, -1607322513115, 203038749068], [-20418794431595000, -27137894626863000, 51637617199916750, 6091324597145000, -49624022863167125, -14819972405224250, 46855437547688585, 12707879205563957, -23916420959174800, -9770292745110985, 9667163825331875, 3790410246400800, -2049268929759280, -828828557891851, 318379497312075, 119296140068240, -24408237958125, -9062355730875, 1317043616260, 446324990017], [0, -338972268000, 0, 1759737293025, 0, -2460726816390, 0, 1989836142906, 0, -973758960830, 0, 318602734545, 0, -62729493258, 0, 8601035320, 0, -634052340, 0, 30857811], [29718570766840000, 15128311906211500, -73530246893056000, 22533817961915000, 83517958451080375, -54823963992404750, -56492909891037245, 64220585227787889, 25507367166390600, -36143534077323595, -8230431307703625, 14021977179381600, 1614060863412160, -2949303034590877, -224419872053775, 441315753410480, 16617984808375, -33524641658625, -824090674470, 1748616823509], [29718570766840000, -15128311906211500, -73530246893056000, -22533817961915000, 83517958451080375, 54823963992404750, -56492909891037245, -64220585227787889, 25507367166390600, 36143534077323595, -8230431307703625, -14021977179381600, 1614060863412160, 2949303034590877, -224419872053775, -441315753410480, 16617984808375, 33524641658625, -824090674470, -1748616823509], [-3545087107550000, 39129533465861000, 41887421567691750, -98123129209653650, -25902149580820875, 102158342714201665, 21216766903863350, -83798444377726023, -8277414347445375, 41068854338096240, 3599970911967875, -15521091246493545, -759984778383425, 3226296089478864, 127425840962875, -487754598990885, -9902404316250, 37105178622365, 571224133225, -1964304297788], [3545087107550000, 39129533465861000, -41887421567691750, -98123129209653650, 25902149580820875, 102158342714201665, -21216766903863350, -83798444377726023, 8277414347445375, 41068854338096240, -3599970911967875, -15521091246493545, 759984778383425, 3226296089478864, -127425840962875, -487754598990885, 9902404316250, 37105178622365, -571224133225, -1964304297788], [33796874477870000, 15435147698964500, -80031250131345500, 26192688996995000, 98017958383539425, -63725864869526750, -38406778361981535, 77151376666049167, 8470394152845800, -42012247943942035, 3349882578879725, 16298759846344800, -1186043751921120, -3409758628074881, 272392501557675, 512973269691440, -23003117470075, -38968119569625, 1475601386290, 2048111316527], [3118729185560000, -24710905979022000, 51158960520975000, 124388687432107225, -124467900390033750, -179486239332213435, 141149049661139735, 145302569602609524, -82057360899758175, -69980800005882695, 31834364605164000, 23387185211627430, -6727434627073605, -4635489168550857, 1001927646889200, 649440263556155, -76111639048125, -48190455491610, 3943558637035, 2399045777394]];
    Rf_basisdens := [1, 1, 9957351510065000, 995735151006500, 22157483500, 995735151006500, 9957351510065000, 4978675755032500, 9957351510065000, 995735151006500, 9957351510065000, 4978675755032500, 9957351510065000, 221574835000, 9957351510065000, 9957351510065000, 9957351510065000, 9957351510065000, 9957351510065000, 9957351510065000];
    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_190_i();"
function MakeCharacter_190_i()
    N := 190;
    order := 6;
    char_gens := [77, 21];
    v := [3, 4];
    // 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_190_i_Hecke();"
function MakeCharacter_190_i_Hecke(Kf)
    N := 190;
    order := 6;
    char_gens := [77, 21];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 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, 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, -1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0], [-2, 0, 0, 1, -1, -1, -1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1, -1, 0, 0], [0, -1, 3, 0, 0, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -2, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 1, 0, -1, 0, -1, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 1, 1, 2, 1, -1, 0, 2, 2, 0, 1], [0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0], [0, 2, 2, 0, 0, 0, -1, -1, 0, -2, 2, 1, 0, 0, 0, 0, 2, 2, -1, 1], [0, 1, 1, 0, 0, 0, 1, 1, 0, -1, -2, -1, 0, 0, 0, 0, -2, -2, 1, -1], [0, 0, 0, -4, -1, 0, 0, 0, 1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -2, 0, 0, 0, 1, 1, 0, -1, -2, -1, 0, 0, 0, 0, 1, 1, 1, -1], [0, 0, 0, -4, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0], [0, -1, -4, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, -4, 1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 4, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, -2, -2, 0, 0, -2, -2, 0, 0], [1, 0, 0, -1, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, -5, 2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0], [4, 0, 0, -4, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 1, 0, -1, 0, -1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 4, -4, 0, 0, 0, 2, 0, 0, 0, 2, 0, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, -3, -3, 2, 0, 3, 0, -2, 2, 3, 3, 0, 0], [0, 1, -3, 0, 0, 0, 3, -1, 0, -1, -2, 1, 0, 0, 0, 0, -2, -2, -1, 1], [0, 0, 0, -5, -2, 0, 0, 0, -3, 0, 0, 0, -3, 0, 2, 2, 0, 0, 0, 0], [0, -1, -6, 0, 0, 0, 2, 0, 0, 1, -2, 0, 0, 0, 0, 0, -1, -1, 0, 0], [-4, 0, 0, 4, 0, 3, -3, 2, 0, 0, -1, -2, 0, 0, 0, 0, -1, 1, -2, 2], [0, 0, 0, -6, -4, 0, 0, 2, 2, 0, 0, 0, 2, 0, 1, -1, 0, 0, -2, 0], [2, 0, 0, 0, 2, 2, 2, 0, 5, 0, 1, 1, 5, 0, -1, -2, -5, 5, 0, -1], [0, -3, 7, 0, 0, 0, 0, -1, -2, 0, 0, 0, 2, 7, 0, 1, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, -3, 0, 1, 1, 2, -1, -1, 6, -2, 3, -1, -1, 0, -1], [0, -3, 3, 0, 0, 0, -3, -1, 0, 3, 4, 1, 0, 0, 0, 0, 0, 0, -1, 1], [0, 3, 3, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 3, 2, -1, 0, 0, -1, 0], [1, 0, 0, 0, 4, 4, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0], [-8, 0, 0, 8, 0, -2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -2, 2, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, -4, 0, -2, 0, -4, 0, 2, 2, 4, -4, 0, 0], [0, 6, -10, 0, 0, 0, 0, 1, 2, 0, 0, 0, -2, -10, -1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0, 5, -1, -2, -1, -5, -3, 2, -1, -5, -5, 0, -1], [-4, 0, 0, 4, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0], [0, 0, 0, 11, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, -1, -1, 0, 0, 0, 0], [0, -1, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -2, -2, 0, 2, 2, 0, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 2, 2, 0, 2, 0, -2, 2, 0, -2], [0, -3, 0, 0, 0, 0, 0, -1, 3, 0, 0, 0, -3, 0, -1, 2, 0, 0, -1, 0], [4, 0, 0, 0, -8, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, -6, 0, -3, 0, -2, 0, 0, -2, 2, 0, 0, 0, 0, 3, -3, 2, -2], [-6, 0, 0, 6, 0, 5, -2, 1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 1], [0, 3, -10, 0, 0, 0, 0, -1, 0, -3, 1, 1, 0, 0, 0, 0, 3, 3, -1, 1], [0, 1, -13, 0, 0, 0, 0, 4, -5, 0, 0, 0, 5, -13, -5, 1, 0, 0, 4, 0], [0, 0, 0, 0, 8, 0, 0, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, -2, 0], [0, 0, 0, 2, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -4, 2, 2, 1, -10, -2, 0, 1, 1, 0, 2], [3, 0, 0, -3, 0, 2, -4, 1, 0, 0, -3, -1, 0, 0, 0, 0, 2, -2, -1, 1], [0, -3, -12, 0, 0, 0, 0, 0, 2, 0, 0, 0, -2, -12, -2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 3, 3, -2, -3, -3, 3, 2, 1, -3, -3, 0, -3], [0, -5, 4, 0, 0, 0, 0, 1, -3, 0, 0, 0, 3, 4, -4, 3, 0, 0, 1, 0], [8, 0, 0, 0, -4, -4, -1, 0, -2, 0, 2, -3, -2, 0, -2, 1, 2, -2, 0, 3], [0, 3, 0, 0, 0, 0, 0, -3, 0, -3, 3, 3, 0, 0, 0, 0, 3, 3, -3, 3], [0, -5, -7, 0, 0, 0, -2, 1, 0, 5, 1, -1, 0, 0, 0, 0, 1, 1, 1, -1], [-13, 0, 0, 0, -4, -4, 2, 0, -1, 0, 2, 0, -1, 0, -2, -2, 1, -1, 0, 0], [0, -5, -8, 0, 0, 0, 0, -2, -3, 0, 0, 0, 3, -8, 0, 2, 0, 0, -2, 0], [0, 11, -2, 0, 0, 0, 0, 1, 5, 0, 0, 0, -5, -2, 2, -3, 0, 0, 1, 0], [-10, 0, 0, 0, 0, 0, 3, 0, 1, 0, 5, -2, 1, 0, -5, -3, -1, 1, 0, 2], [0, 0, 0, 0, 0, 0, 6, 0, -1, 5, -4, 2, 1, 0, 4, -6, 1, 1, 0, 2], [0, 0, 0, 10, 2, 0, 0, 1, 4, 0, 0, 0, 4, 0, 2, 1, 0, 0, -1, 0], [0, -2, 12, 0, 0, 0, 4, -1, 0, 2, -3, 1, 0, 0, 0, 0, 1, 1, -1, 1], [0, 0, 0, 0, 0, 0, 4, 0, 1, 1, -5, -1, -1, -3, 5, -4, -1, -1, 0, -1], [6, 0, 0, 0, 8, 8, -1, 0, -3, 0, -2, 1, -3, 0, 2, 1, 3, -3, 0, -1], [0, 6, -2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -2, 8, -5, 0, 0, -3, 0], [-8, 0, 0, 8, 0, 6, 5, -2, 0, 0, 3, 2, 0, 0, 0, 0, 1, -1, 2, -2], [0, 3, 21, 0, 0, 0, 0, -3, 3, 0, 0, 0, -3, 21, 2, 1, 0, 0, -3, 0], [0, 0, 0, 10, -3, 0, 0, -1, 1, 0, 0, 0, 1, 0, -3, -2, 0, 0, 1, 0], [17, 0, 0, -17, 0, 4, -2, 3, 0, 0, 1, -3, 0, 0, 0, 0, 4, -4, -3, 3], [-3, 0, 0, 0, 3, 3, -2, 0, -3, 0, 1, -3, -3, 0, -1, 2, 3, -3, 0, 3], [0, 0, 0, 2, 0, 0, 0, -4, -2, 0, 0, 0, -2, 0, 5, 9, 0, 0, 4, 0], [-4, 0, 0, 4, 0, -10, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0], [0, -2, 0, 0, 0, 0, -2, -1, 0, 2, 3, 1, 0, 0, 0, 0, 2, 2, -1, 1], [0, 0, 0, 8, 8, 0, 0, -1, 3, 0, 0, 0, 3, 0, -3, -2, 0, 0, 1, 0], [0, -1, -6, 0, 0, 0, -2, -2, 0, 1, 4, 2, 0, 0, 0, 0, -4, -4, -2, 2], [3, 0, 0, 0, 2, 2, -2, 0, 8, 0, -1, -1, 8, 0, 1, 2, -8, 8, 0, 1], [0, 0, 0, 0, 0, 0, -4, 0, -5, -5, 3, -1, 5, 0, -3, 4, 5, 5, 0, -1], [0, 0, 0, 0, 4, 0, 0, 4, -5, 0, 0, 0, -5, 0, 5, 1, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, -1, -8, -1, 0, 11, 8, -7, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -5, 0, -5, 7, 6, 1, 5, -8, -6, 5, 5, 5, 0, 1], [10, 0, 0, -10, 0, -2, -4, 0, 0, 0, -4, 0, 0, 0, 0, 0, 2, -2, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, -1, -11, 1, -2, 1, -11, -1, 3, 1, 1, 0, -2], [0, 0, 0, 15, 5, 0, 0, -1, -5, 0, 0, 0, -5, 0, -1, 0, 0, 0, 1, 0], [0, 0, 0, 5, 2, 0, 0, 0, -1, 0, 0, 0, -1, 0, -2, -2, 0, 0, 0, 0], [0, 3, -5, 0, 0, 0, 6, 2, 0, -3, -8, -2, 0, 0, 0, 0, -6, -6, 2, -2], [2, 0, 0, -2, 0, 0, 3, 2, 0, 0, 5, -2, 0, 0, 0, 0, -1, 1, -2, 2], [-2, 0, 0, 0, -1, -1, 0, 0, -1, 0, -1, 1, -1, 0, 1, 0, 1, -1, 0, -1], [0, 2, 0, 0, 0, 0, 4, 4, 0, -2, -8, -4, 0, 0, 0, 0, -10, -10, 4, -4], [8, 0, 0, -8, 0, -2, -5, 0, 0, 0, -5, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 4, 6, 0, 0, 0, -4, 2, 0, -4, 2, -2, 0, 0, 0, 0, 0, 0, 2, -2], [0, -5, -3, 0, 0, 0, -3, 1, 0, 5, 2, -1, 0, 0, 0, 0, -2, -2, 1, -1], [0, 0, 0, 0, 0, 0, 2, 0, -2, -5, -4, -2, 2, 10, 4, -2, 2, 2, 0, -2], [19, 0, 0, 0, -1, -1, 0, 0, -1, 0, -4, 4, -1, 0, 4, 0, 1, -1, 0, -4], [6, 0, 0, 0, 3, 3, 0, 0, 10, 0, 1, -1, 10, 0, -1, 0, -10, 10, 0, 1], [0, 0, 0, 0, 0, 0, 4, 0, -5, -7, 0, 4, 5, -24, 0, -4, 5, 5, 0, 4], [0, 4, 4, 0, 0, 0, 0, 0, -8, 0, 0, 0, 8, 4, -6, 6, 0, 0, 0, 0], [0, -3, 2, 0, 0, 0, 0, 5, 0, 3, -5, -5, 0, 0, 0, 0, -3, -3, 5, -5], [8, 0, 0, -8, 0, 4, 1, 1, 0, 0, 2, -1, 0, 0, 0, 0, -6, 6, -1, 1], [-11, 0, 0, 0, 4, 4, 4, 0, -2, 0, 5, -1, -2, 0, -5, -4, 2, -2, 0, 1], [0, 0, 0, 0, 0, 0, 5, 0, 1, -1, -3, 2, -1, 5, 3, -5, -1, -1, 0, 2], [0, -5, 1, 0, 0, 0, 0, 1, -5, 0, 0, 0, 5, 1, -3, 2, 0, 0, 1, 0], [0, -3, 20, 0, 0, 0, -4, 0, 0, 3, 4, 0, 0, 0, 0, 0, -3, -3, 0, 0], [-13, 0, 0, 0, -9, -9, -6, 0, -7, 0, -7, 1, -7, 0, 7, 6, 7, -7, 0, -1], [-12, 0, 0, 12, 0, 2, -4, 3, 0, 0, -1, -3, 0, 0, 0, 0, -3, 3, -3, 3], [0, 0, 0, 2, -1, 0, 0, 0, -5, 0, 0, 0, -5, 0, -6, -6, 0, 0, 0, 0], [0, 1, -14, 0, 0, 0, 0, -2, 10, 0, 0, 0, -10, -14, 6, -4, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 3, 0, 8, 5, -6, -3, -8, 9, 6, -3, -8, -8, 0, -3], [0, 0, 0, 0, 0, 0, -1, 0, -6, 9, 4, 3, 6, 3, -4, 1, 6, 6, 0, 3], [15, 0, 0, -15, 0, -1, 4, 1, 0, 0, 5, -1, 0, 0, 0, 0, -5, 5, -1, 1], [0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 4, -2, 0, 0, 0, 0, -7, 7, -2, 2], [0, 0, 0, 0, 0, 0, -4, 0, -6, 0, 7, 3, 6, 0, -7, 4, 6, 6, 0, 3], [0, 0, 0, 0, 0, 0, -7, 0, -2, 1, 6, -1, 2, -23, -6, 7, 2, 2, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 4, -2, 0, 0, 0, 0, 2], [3, 0, 0, 0, 1, 1, -2, 0, 3, 0, 1, -3, 3, 0, -1, 2, -3, 3, 0, 3], [0, 0, 0, -6, -8, 0, 0, -2, 5, 0, 0, 0, 5, 0, 0, 2, 0, 0, 2, 0], [-8, 0, 0, 8, 0, -6, 3, -4, 0, 0, -1, 4, 0, 0, 0, 0, 0, 0, 4, -4], [0, 0, 0, -26, -4, 0, 0, -2, -8, 0, 0, 0, -8, 0, 6, 8, 0, 0, 2, 0], [0, 8, 20, 0, 0, 0, 0, 2, -4, 0, 0, 0, 4, 20, 4, -6, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 3, -1, 1, 0, 23, 1, -2, 0, 0, 0, 1], [0, 0, 0, -21, 2, 0, 0, 1, 6, 0, 0, 0, 6, 0, 2, 1, 0, 0, -1, 0], [0, -5, -5, 0, 0, 0, 0, -2, 4, 0, 0, 0, -4, -5, -2, 4, 0, 0, -2, 0], [2, 0, 0, -2, 0, -4, -4, 4, 0, 0, 0, -4, 0, 0, 0, 0, 8, -8, -4, 4], [0, 1, 23, 0, 0, 0, 0, 3, -3, 0, 0, 0, 3, 23, -1, -2, 0, 0, 3, 0], [-7, 0, 0, 0, -2, -2, 4, 0, 0, 0, 2, 2, 0, 0, -2, -4, 0, 0, 0, -2], [-6, 0, 0, 6, 0, 2, -10, 3, 0, 0, -7, -3, 0, 0, 0, 0, 0, 0, -3, 3], [0, 9, 3, 0, 0, 0, 2, -3, 0, -9, 1, 3, 0, 0, 0, 0, 6, 6, -3, 3], [0, 0, 0, 0, 0, 0, 8, 0, 2, 5, -14, -6, -2, -2, 14, -8, -2, -2, 0, -6], [0, 0, 0, 0, 0, 0, 3, 0, -1, 5, 4, 7, 1, -9, -4, -3, 1, 1, 0, 7], [10, 0, 0, 0, 11, 11, 6, 0, 9, 0, 2, 4, 9, 0, -2, -6, -9, 9, 0, -4], [-21, 0, 0, 21, 0, 1, 4, -5, 0, 0, -1, 5, 0, 0, 0, 0, 1, -1, 5, -5], [10, 0, 0, -10, 0, 4, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, -2, 2, 0, 0], [0, -3, 5, 0, 0, 0, 7, 1, 0, 3, -8, -1, 0, 0, 0, 0, 0, 0, 1, -1], [0, -13, -2, 0, 0, 0, 0, 6, 0, 13, -6, -6, 0, 0, 0, 0, -4, -4, 6, -6], [22, 0, 0, 0, 0, 0, -3, 0, -1, 0, 1, -4, -1, 0, -1, 3, 1, -1, 0, 4], [0, 0, 0, 0, -10, 0, 0, -1, 4, 0, 0, 0, 4, 0, -2, -1, 0, 0, 1, 0], [0, -4, -20, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, -20, 1, -1, 0, 0, 0, 0], [0, -11, 14, 0, 0, 0, 0, 0, -5, 0, 0, 0, 5, 14, -8, 8, 0, 0, 0, 0], [-17, 0, 0, 17, 0, 0, -6, 0, 0, 0, -6, 0, 0, 0, 0, 0, 5, -5, 0, 0], [0, 0, 0, 0, 0, 0, -11, 0, -3, 3, 9, -2, 3, -1, -9, 11, 3, 3, 0, -2], [0, -1, -1, 0, 0, 0, 0, -1, 3, 0, 0, 0, -3, -1, -6, 7, 0, 0, -1, 0], [9, 0, 0, 0, 4, 4, 2, 0, 6, 0, 1, 1, 6, 0, -1, -2, -6, 6, 0, -1], [0, 1, -8, 0, 0, 0, -6, -2, 0, -1, 8, 2, 0, 0, 0, 0, 4, 4, -2, 2], [0, 6, -10, 0, 0, 0, 3, -2, 0, -6, -1, 2, 0, 0, 0, 0, 3, 3, -2, 2], [0, -3, -16, 0, 0, 0, 0, 1, -3, 0, 0, 0, 3, -16, -10, 9, 0, 0, 1, 0], [2, 0, 0, 0, 10, 10, 11, 0, 0, 0, 12, -1, 0, 0, -12, -11, 0, 0, 0, 1], [-16, 0, 0, 0, -10, -10, 5, 0, 5, 0, 5, 0, 5, 0, -5, -5, -5, 5, 0, 0], [0, 0, 0, 5, 12, 0, 0, -2, 4, 0, 0, 0, 4, 0, -6, -4, 0, 0, 2, 0], [0, -5, 4, 0, 0, 0, 4, 1, 0, 5, -5, -1, 0, 0, 0, 0, -5, -5, 1, -1], [20, 0, 0, -20, 0, -2, 8, -2, 0, 0, 6, 2, 0, 0, 0, 0, -8, 8, 2, -2], [0, 8, 12, 0, 0, 0, 0, 4, 2, 0, 0, 0, -2, 12, -2, -2, 0, 0, 4, 0], [0, 1, 10, 0, 0, 0, 0, 1, -5, 0, 0, 0, 5, 10, -7, 6, 0, 0, 1, 0], [0, -2, 8, 0, 0, 0, 0, 6, -4, 0, 0, 0, 4, 8, -4, -2, 0, 0, 6, 0], [0, 0, 0, 8, 1, 0, 0, 2, 5, 0, 0, 0, 5, 0, -3, -5, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, -8, 0, 3, 3, 6, -2, -3, 18, -6, 8, -3, -3, 0, -2], [0, 9, -5, 0, 0, 0, 0, 1, 9, 0, 0, 0, -9, -5, 5, -6, 0, 0, 1, 0], [0, 0, 0, 0, 8, 0, 0, 2, -2, 0, 0, 0, -2, 0, 6, 4, 0, 0, -2, 0], [0, 5, 15, 0, 0, 0, -12, -5, 0, -5, 17, 5, 0, 0, 0, 0, 2, 2, -5, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_190_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_190_2_i_a();".
// 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_190_2_i_a(:prec:=20)
    chi := MakeCharacter_190_i();
    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_190_2_i_a();".
// 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_190_2_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_190_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;