
// Make newform 200.3.e.b in Magma, downloaded from the LMFDB on 10 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_200_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_200_e_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_3_e_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_200_3_e_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 := [9, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 6, 0, 2], [0, 6, 0, -2]];
    Rf_basisdens := [1, 3, 3, 3];
    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_e();"
function MakeCharacter_200_e()
    N := 200;
    order := 2;
    char_gens := [151, 101, 177];
    v := [1, 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_e_Hecke();"
function MakeCharacter_200_e_Hecke( : Kf := false)
    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 := 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 := 3;
    raw_aps := [[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-7, 0, 0, 3], [0, 0, 0, 0], [0, 1, 6, 0], [-17, 0, 0, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [23, 0, 0, -12], [0, 14, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [82, 0, 0, 0], [0, 0, 0, 0], [0, 31, 21, 0], [0, 0, 0, 0], [0, 71, 6, 0], [0, 0, 0, 0], [0, -79, -9, 0], [73, 0, 0, 18], [0, 94, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -89, 21, 0], [0, 0, 0, 0], [0, -49, 36, 0], [0, 0, 0, 0], [62, 0, 0, 0], [0, -119, -24, 0], [103, 0, 0, 33], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 161, -9, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-17, 0, 0, 63], [0, 0, 0, 0], [0, 0, 0, 0], [0, -49, 66, 0], [0, 0, 0, 0], [0, 0, 0, 0], [113, 0, 0, 63], [0, 0, 0, 0], [0, -446, 0, 0], [0, 0, 0, 0], [0, 434, 0, 0], [0, 0, 0, 0], [-97, 0, 0, 78], [233, 0, 0, 33], [0, -386, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-238, 0, 0, 0], [0, 41, -99, 0], [0, 0, 0, 0], [0, 271, 51, 0], [0, 0, 0, 0], [0, -526, 0, 0], [0, 0, 0, 0], [-7, 0, 0, -117], [0, -239, -84, 0], [0, -329, -39, 0], [0, 0, 0, 0], [0, 194, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [343, 0, 0, -57], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [383, 0, 0, -42], [-167, 0, 0, -132], [-257, 0, 0, -117], [0, 0, 0, 0], [0, 0, 0, 0], [0, -289, -114, 0], [0, 0, 0, 0], [0, -439, 21, 0], [433, 0, 0, -42], [0, -119, 156, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 34, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [782, 0, 0, 0], [802, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [503, 0, 0, 48], [0, -199, 171, 0], [0, 0, 0, 0], [0, 511, -69, 0], [0, 0, 0, 0], [0, -226, 0, 0], [313, 0, 0, 168], [-658, 0, 0, 0], [0, 1, -204, 0], [0, -569, 51, 0], [0, 431, -144, 0], [0, 0, 0, 0], [-457, 0, 0, 138], [0, 0, 0, 0], [0, 0, 0, 0], [0, 334, 0, 0], [562, 0, 0, 0], [0, 0, 0, 0], [482, 0, 0, 0], [0, 1214, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-497, 0, 0, 153], [0, 0, 0, 0], [0, -1246, 0, 0], [0, 0, 0, 0], [0, -199, 231, 0], [-367, 0, 0, -207], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [322, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-697, 0, 0, 108], [-527, 0, 0, 198], [0, 0, 0, 0], [0, -926, 0, 0], [0, 0, 0, 0], [1582, 0, 0, 0], [-178, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 631, -189, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 601, 216, 0], [823, 0, 0, -87], [0, 0, 0, 0], [0, 0, 0, 0], [-1438, 0, 0, 0], [0, 881, 21, 0], [0, 0, 0, 0], [0, 1714, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1058, 0, 0, 0], [0, -359, 306, 0], [0, 0, 0, 0], [0, 994, 0, 0], [0, 71, 336, 0], [0, 0, 0, 0], [-487, 0, 0, -297], [0, -959, 66, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_200_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_200_3_e_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_200_3_e_b(:prec:=4)
    chi := MakeCharacter_200_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_e_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_200_3_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_200_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![529, 0, 50, 0, 1]>],Snew);
    return Vf;
end function;