
// Make newform 456.2.bf.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_456_bf();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_456_bf_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_456_2_bf_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_456_2_bf_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, -3, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 2, 1], [3, 2, 0, -1]];
    Rf_basisdens := [1, 1, 6, 2];
    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_456_bf();"
function MakeCharacter_456_bf()
    N := 456;
    order := 6;
    char_gens := [343, 229, 305, 97];
    v := [6, 6, 3, 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_456_bf_Hecke();"
function MakeCharacter_456_bf_Hecke(Kf)
    N := 456;
    order := 6;
    char_gens := [343, 229, 305, 97];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [0, 0, 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+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, 0, 0], [1, -1, 0, 1], [1, -1, 0, 0], [1, -1, -1, -1], [2, 0, -4, 0], [0, 0, -1, -2], [1, -1, 0, 0], [-3, 0, -2, 0], [-1, 0, 3, 5], [-1, 2, 3, -1], [3, 1, -7, -1], [-5, -1, 11, 1], [3, 1, -4, -2], [-5, -2, 7, 4], [3, 0, -1, 1], [1, -2, -9, 1], [5, -1, -4, 2], [0, 0, 1, 0], [6, 0, -1, 4], [9, 1, -10, -2], [1, -2, 1, 4], [-5, 8, 3, 0], [-2, 0, 4, 0], [1, -2, -5, 1], [3, 3, 6, 0], [1, 0, -1, -1], [1, 3, -5, -3], [-10, -2, -2, -2], [-1, 3, 2, 0], [8, 2, 2, 2], [-9, 0, 1, -7], [7, -7, 0, 0], [17, 0, -9, -1], [4, -8, -7, 4], [-3, 3, 0, 0], [6, -4, -8, 4], [-11, 4, 7, -8], [13, -1, -1, -1], [-3, 6, 11, -3], [3, -5, 2, 10], [-8, -4, -4, -4], [-11, 0, 9, 7], [2, -4, 0, 4], [9, 0, 9, 0], [4, -12, 4, 12], [4, -8, -11, 4], [3, -8, -5, 0], [7, -12, -5, 0], [14, -2, -2, -2], [-11, -3, -3, -3], [-7, 3, -4, 0], [-6, 0, 12, 0], [12, 0, -9, -6], [-17, 0, 11, 5], [-3, 6, -9, -3], [-5, 5, 0, 0], [-1, 5, -4, -10], [3, 5, -8, -10], [-10, 4, 4, 4], [7, -14, -9, 7], [-17, 5, 12, -10], [16, -2, -2, -2], [-11, 15, 4, 0], [-8, -2, 18, 2], [9, -18, -11, 9], [-1, 2, 19, -1], [-9, 5, 13, -5], [-11, 12, 1, 0], [11, -11, 0, 0], [-7, -7, -7, -7], [-14, 14, 14, -14], [-9, 7, -2, 0], [-6, 12, 5, -6], [4, 0, -8, 0], [3, -7, 1, 7], [-7, -7, 14, 14], [3, 0, 7, 17], [5, 6, -11, -12], [3, 9, -12, -18], [13, 0, 1, 15], [18, -8, -28, 8], [-5, 15, 10, 0], [5, -10, -25, 5], [-18, 0, 15, 12], [3, 0, 3, 0], [5, 0, 7, 19], [-2, 12, 12, 12], [33, 1, 1, 1], [-15, 5, -10, 0], [21, -1, -1, -1], [-4, -14, 22, 14], [21, 0, -13, -5], [-2, -4, 8, 4], [-1, 9, 8, 0], [19, -2, -17, 4], [-47, 0, 23, -1], [-3, 6, 3, -3], [-10, 0, 0, 0], [6, 0, -1, 4], [-8, 16, -3, -8], [-36, 0, 15, -6], [-27, 0, 11, -5], [10, 10, 10, 10], [32, 2, 2, 2], [13, -1, -1, -1], [-14, -4, -4, -4], [27, -1, 26, 0], [15, 0, -13, -11], [3, -6, 7, 3], [9, -15, -3, 15], [-1, 1, 1, -1], [-15, -3, 18, 6], [-11, 0, 7, 3], [15, 5, 5, 5], [6, -12, 5, 6], [23, 1, -24, -2], [3, 0, -3, 0], [22, -8, -36, 8], [10, -6, -14, 6], [11, -22, -13, 11], [-19, 0, 13, 7], [9, -7, -11, 7], [20, 2, 2, 2], [-2, 2, 2, 2], [12, -16, -16, -16], [5, 3, 8, 0], [4, -8, 9, 4], [-1, -1, -2, 0], [-29, 2, 27, -4], [2, -8, -8, -8], [-21, 4, 17, -8], [13, 1, -14, -2], [18, 0, -5, 8], [21, -2, -19, 4], [12, -16, -8, 16], [2, -4, 25, 2], [1, -2, -41, 1], [-3, 3, 3, -3], [34, 4, 4, 4], [-2, 2, 2, -2], [15, 0, -3, 9], [-19, 0, -1, -21], [-5, 10, -19, -5], [-5, 10, -17, -5], [-13, 11, 15, -11], [5, 1, -6, -2], [5, -2, -3, 4], [7, 1, -8, -2], [6, -12, 21, 6], [24, -4, -4, -4], [-3, 10, 7, 0], [-6, -2, 14, 2], [2, -4, -15, 2], [-7, 14, -13, -7], [13, 3, 16, 0], [4, -4, -4, -4], [3, 5, 5, 5], [13, -1, 12, 0], [-8, 16, -7, -8], [-7, 14, 39, -7], [3, 1, 4, 0], [-33, 7, 26, -14], [-1, -2, 3, 4], [-11, -5, 16, 10], [-4, 14, 14, 14], [9, 5, -14, -10], [7, 2, 9, 0], [-4, 8, -23, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_456_bf_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_456_2_bf_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_456_2_bf_b(:prec:=4)
    chi := MakeCharacter_456_bf();
    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_456_2_bf_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_456_2_bf_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_456_bf();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 6, 1, -3, 1]>],Snew);
    return Vf;
end function;