
// Make newform 200.4.f.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_200_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_200_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_200_4_f_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_200_4_f_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 := [4, 0, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [0, -1, 0, 1], [-6, 0, 4, 0]];
    Rf_basisdens := [1, 1, 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_200_f();"
function MakeCharacter_200_f()
    N := 200;
    order := 2;
    char_gens := [151, 101, 177];
    v := [2, 1, 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_200_f_Hecke();"
function MakeCharacter_200_f_Hecke(Kf)
    N := 200;
    order := 2;
    char_gens := [151, 101, 177];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [-1, 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 := 4;
    raw_aps := [[0, -1, 0, 0], [0, -2, 1, 0], [0, 0, 0, 0], [0, 0, -4, 0], [0, 0, 0, 3], [0, -20, 10, 0], [0, 0, -7, 0], [0, 0, 0, -7], [0, 0, 76, 0], [0, 0, 0, -30], [224, 0, 0, 0], [0, 92, -46, 0], [-70, 0, 0, 0], [0, 166, -83, 0], [0, 0, 168, 0], [0, -12, 6, 0], [0, 0, 0, 101], [0, 0, 0, -18], [0, 66, -33, 0], [-72, 0, 0, 0], [0, 0, 147, 0], [464, 0, 0, 0], [0, 206, -103, 0], [-266, 0, 0, 0], [0, 0, 497, 0], [0, 0, 0, -142], [0, 0, -588, 0], [0, -102, 51, 0], [0, 0, 0, -358], [0, 0, 855, 0], [0, 0, -832, 0], [0, 0, 0, 127], [0, 0, -531, 0], [0, 0, 0, 509], [0, 0, 0, 150], [744, 0, 0, 0], [0, 68, -34, 0], [0, 670, -335, 0], [0, 0, -980, 0], [0, -756, 378, 0], [0, 0, 0, 617], [0, 0, 0, 442], [3904, 0, 0, 0], [0, 0, -1665, 0], [0, -452, 226, 0], [1736, 0, 0, 0], [0, 0, 0, 551], [0, 0, -784, 0], [0, -478, 239, 0], [0, 0, 0, 974], [0, 0, 419, 0], [-6288, 0, 0, 0], [-2926, 0, 0, 0], [0, 0, 0, -1029], [0, 0, 1281, 0], [0, 0, 2948, 0], [0, 0, 0, -1014], [-1680, 0, 0, 0], [0, -596, 298, 0], [-2742, 0, 0, 0], [0, -1130, 565, 0], [0, 3492, -1746, 0], [0, 978, -489, 0], [-2744, 0, 0, 0], [0, 0, -1141, 0], [0, 3620, -1810, 0], [0, 0, 0, 803], [0, 0, 3065, 0], [0, 938, -469, 0], [0, 0, 0, -62], [0, 0, 5103, 0], [3176, 0, 0, 0], [0, 0, -5880, 0], [0, 4148, -2074, 0], [0, 0, 0, 581], [0, 0, -1344, 0], [0, 0, 0, 1982], [0, -2156, 1078, 0], [12402, 0, 0, 0], [12278, 0, 0, 0], [0, 0, 0, -1551], [0, 0, 0, 274], [7632, 0, 0, 0], [0, 0, -1897, 0], [1848, 0, 0, 0], [0, 4662, -2331, 0], [3582, 0, 0, 0], [0, 0, 1357, 0], [0, 0, 0, -1578], [0, 0, -1112, 0], [0, 3890, -1945, 0], [-17696, 0, 0, 0], [0, 0, 652, 0], [0, 0, 0, -3149], [0, 0, 0, 585], [0, 0, 9660, 0], [0, 0, 0, -846], [-2982, 0, 0, 0], [0, -762, 381, 0], [0, 0, 0, -2914], [0, -4766, 2383, 0], [0, -2700, 1350, 0], [0, 2862, -1431, 0], [-15594, 0, 0, 0], [0, 0, 0, 3163], [0, 0, 3297, 0], [0, -8774, 4387, 0], [0, 0, -7161, 0], [16088, 0, 0, 0], [-21238, 0, 0, 0], [0, 0, -6832, 0], [0, -7708, 3854, 0], [0, 0, -1891, 0], [0, 0, 0, 1101], [2056, 0, 0, 0], [11842, 0, 0, 0], [0, 6142, -3071, 0], [0, 0, 9660, 0], [0, 876, -438, 0], [0, 0, 0, 5241], [0, 0, 0, -4246], [0, 0, 5039, 0], [0, 6108, -3054, 0], [0, 6182, -3091, 0], [0, 0, 0, 5527], [0, 0, 0, -466], [0, 0, 0, -5986], [9296, 0, 0, 0], [0, 0, 10836, 0], [0, 3580, -1790, 0], [0, 0, 0, 1297], [0, 0, -8716, 0], [-11632, 0, 0, 0], [0, -6324, 3162, 0], [39466, 0, 0, 0], [-35266, 0, 0, 0], [0, -6140, 3070, 0], [0, 13170, -6585, 0], [0, 964, -482, 0], [24390, 0, 0, 0], [0, 0, 0, 1811], [0, 0, 0, 1658], [0, 0, 10844, 0], [0, -7350, 3675, 0], [0, 0, 0, -3694], [18760, 0, 0, 0], [0, -10860, 5430, 0], [0, 0, 4389, 0], [0, 0, 0, -1067], [0, 0, 4656, 0], [0, 52, -26, 0], [-31150, 0, 0, 0], [0, -4754, 2377, 0], [0, 0, 18564, 0], [0, 13306, -6653, 0], [-10512, 0, 0, 0], [46104, 0, 0, 0], [5726, 0, 0, 0], [0, 0, 637, 0], [0, 0, 0, 4998], [0, 9042, -4521, 0], [0, 0, -19125, 0], [0, 0, 2332, 0], [0, 0, 0, -5853], [0, 0, -2407, 0], [0, 0, 6188, 0], [45344, 0, 0, 0], [0, -9828, 4914, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_200_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_200_4_f_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_200_4_f_a(:prec:=4)
    chi := MakeCharacter_200_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_200_4_f_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_200_4_f_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_200_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-28, 0, 1]>],Snew);
    return Vf;
end function;