
// Make newform 50.10.b.d in Magma, downloaded from the LMFDB on 03 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_50_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_50_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_50_10_b_d();".
// 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_50_10_b_d();".
// 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 := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    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_50_b();"
function MakeCharacter_50_b()
    N := 50;
    order := 2;
    char_gens := [27];
    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_50_b_Hecke();"
function MakeCharacter_50_b_Hecke(Kf)
    N := 50;
    order := 2;
    char_gens := [27];
    char_values := [[-1, 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]];
    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 := 10;
    raw_aps := [[0, -8], [0, 87], [0, 0], [0, -2329], [28992, 0], [0, -82223], [0, 297411], [295780, 0], [0, 1272267], [3722970, 0], [2335772, 0], [0, -5420209], [21593862, 0], [0, 5416147], [0, -2586069], [0, 49089837], [-16162860, 0], [-43928158, 0], [0, 40778711], [161307732, 0], [0, -123573983], [583345720, 0], [0, -7285893], [-470133690, 0], [0, 58919231], [-86092698, 0], [0, 964360687], [0, -698422629], [604327390, 0], [0, -842901123], [0, -1853579449], [2540801472, 0], [0, 576947991], [5627209060, 0], [2136877350, 0], [-9675148, 0], [0, 344243771], [0, -7154080973], [0, -4993673889], [0, 1756978317], [-1195024980, 0], [-9120526378, 0], [-9374306988, 0], [0, 12000019597], [0, 278061831], [25125494200, 0], [-16391501008, 0], [0, 23262842467], [0, 957073851], [14582465170, 0], [0, 20639499357], [-36550217640, 0], [-86606962738, 0], [-72076909248, 0], [0, 56364562371], [0, 27761252127], [-27972555570, 0], [33288431732, 0], [0, -27312017149], [-83781811578, 0], [0, -41804291393], [0, -114273505503], [0, -97531815529], [215637432492, 0], [0, 95877624277], [0, 169442852211], [-178426590328, 0], [0, -132346732609], [0, 40595585031], [226688041450, 0], [0, 16436646537], [312096166440, 0], [0, -314104618189], [0, -49238506883], [-289023741380, 0], [0, 100674769707], [-393769497690, 0], [0, 257812445231], [-843309264798, 0], [-564548742110, 0], [-692474998020, 0], [408125314682, 0], [441054560772, 0], [0, -35769481343], [-474967117640, 0], [0, -98535886053], [620156168250, 0], [0, -394002976429], [15349559742, 0], [0, 97246909327], [0, -53251345689], [831146028120, 0], [0, 692832220091], [392804354112, 0], [752029324900, 0], [0, -917145785313], [619863696090, 0], [525682625082, 0], [0, 84212874367], [2206122543662, 0], [0, 1758196809131], [0, -314997801729], [0, 201309496527], [-2554817360670, 0], [1162432881032, 0], [0, -2331660893749], [0, -1942107615909], [0, -1006705835703], [-2105782224000, 0], [4740576995702, 0], [0, 2007380798771], [0, 651992603377], [0, -2550821085789], [450629987980, 0], [79710463172, 0], [-3126484893138, 0], [0, -2974908033953], [0, -1780936518669], [0, -68941460763], [4166544473340, 0], [-3290827458358, 0], [0, 2873658075817], [0, -4656226176249], [0, 4599835936707], [10312547234312, 0], [-9525655030398, 0], [8385367901890, 0], [-6940126686120, 0], [0, -3874919326849], [0, 2062774962757], [-3339308538740, 0], [0, -6323566038453], [1565515642652, 0], [0, -477390662629], [5273296939242, 0], [6969226252030, 0], [0, -1896913578783], [0, 10296960130091], [0, -727533256869], [-13174886289210, 0], [9978331382192, 0], [-18091349708118, 0], [0, 699005995867], [0, -11310052781049], [-17264635520930, 0], [1766066867760, 0], [0, -11892209406263], [0, -3809704665129], [12769504611940, 0], [0, 13151905717227], [0, -238416005149], [17483879355822, 0], [0, -3241603996193], [0, 4962250026591], [0, -703840106929], [6913431077292, 0], [8501894648080, 0], [-8906389927830, 0], [0, -3700320282709], [-8138673844338, 0], [0, 5848750631631], [0, -12520478825763], [0, -13445212601389], [32878001398632, 0], [0, 406752668151], [0, -20661994671393], [-36122726805988, 0], [0, 15358284912131]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_50_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_50_10_b_d();".
// 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_50_10_b_d(:prec:=2)
    chi := MakeCharacter_50_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 10));
    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_50_10_b_d();".
// 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_50_10_b_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_50_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<3,R![30276, 0, 1]>],Snew);
    return Vf;
end function;