
// Make newform 560.2.bw.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_560_bw();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_560_bw_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_560_2_bw_e();".
// 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_560_2_bw_e();".
// 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 := [25, -5, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-5, -4, 4, 1], [5, 4, 0, -1]];
    Rf_basisdens := [1, 1, 20, 4];
    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_560_bw();"
function MakeCharacter_560_bw()
    N := 560;
    order := 6;
    char_gens := [351, 421, 337, 241];
    v := [6, 6, 3, 2];
    // 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_560_bw_Hecke();"
function MakeCharacter_560_bw_Hecke(Kf)
    N := 560;
    order := 6;
    char_gens := [351, 421, 337, 241];
    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, 0, 1, 0], [1, -1, -1, 0], [0, 0, 2, 1], [-1, -1, 2, 2], [2, -2, -2, 2], [1, 1, 2, 0], [-1, 2, 1, -1], [4, 0, -3, -2], [-1, 1, 1, 1], [-1, 1, 0, -2], [9, 0, -5, -1], [-7, -1, -1, -1], [-3, 1, 5, -1], [5, 0, -3, -1], [1, -3, -2, 0], [-1, 1, 0, -2], [2, -4, 5, 2], [1, 4, 5, 0], [-2, -2, -2, -2], [-3, 1, -2, 0], [-1, 2, -3, -1], [1, -3, 1, 3], [0, 0, -7, 0], [-4, 0, 8, 0], [5, 2, -7, -4], [-8, 0, 5, 2], [-6, 0, 1, -4], [-3, -2, 5, 4], [-6, 2, 10, -2], [2, 4, -8, -4], [1, -2, -15, 1], [-7, 7, 0, 0], [0, -4, -4, -4], [-2, 4, 1, -2], [-17, 1, 16, -2], [1, 3, 4, 0], [1, 0, 1, 3], [-5, 3, 7, -3], [1, 0, -5, -9], [3, 1, -4, -2], [-21, 1, 1, 1], [-3, 6, 13, -3], [-7, 7, 0, 0], [12, -4, -20, 4], [-13, -1, 14, 2], [-6, 6, 6, 6], [2, -4, 0, 4], [9, -3, 6, 0], [1, -2, -1, 1], [19, 0, -9, 1], [-6, -2, -2, -2], [3, -3, 0, 6], [14, -2, -2, -2], [21, 0, -9, 3], [5, 6, 11, 0], [-23, 2, 21, -4], [3, -6, -3, 3], [9, 1, 10, 0], [6, 0, 0, 0], [1, -7, -6, 0], [-4, 0, 8, 0], [-3, -7, 13, 7], [3, -3, 0, 6], [-11, 0, 11, 11], [-17, 0, 11, 5], [3, -6, 5, 3], [-6, 2, 10, -2], [-7, 0, -7, 0], [7, 1, 1, 1], [-7, -3, -10, 0], [-5, 10, -15, -5], [5, 2, 7, 0], [9, 0, -5, -1], [-2, 6, 6, 6], [-8, 0, 1, -6], [11, 5, -16, -10], [-15, 0, 7, -1], [0, 0, -3, 0], [-3, -4, 7, 8], [0, 4, 4, 4], [-1, 1, 1, 1], [3, -5, 2, 10], [14, -2, -26, 2], [-3, 6, -11, -3], [14, 0, -7, 0], [13, 3, 3, 3], [-11, 0, 3, -5], [-14, 0, 0, 0], [3, -1, -5, 1], [-2, 0, -3, -8], [3, -3, 0, 6], [-7, -7, -14, 0], [22, -2, -2, -2], [5, -10, -23, 5], [-1, 11, -9, -11], [2, -4, -23, 2], [3, -3, 0, 6], [-11, 0, 5, -1], [2, -4, 9, 2], [-11, 13, 9, -13], [5, -1, 4, 0], [-7, 8, 1, 0], [5, -10, -13, 5], [3, 1, -4, -2], [-3, -1, -4, 0], [2, -8, 4, 8], [31, 0, -17, -3], [-1, -1, 2, 2], [14, 0, 0, 0], [-12, 0, 1, -10], [5, -15, -10, 0], [-4, -8, 16, 8], [-21, 7, 14, -14], [-20, 4, 4, 4], [-15, 4, 11, -8], [10, -16, -4, 16], [-11, 6, -5, 0], [-5, 0, -5, -15], [-34, 2, 2, 2], [9, -2, -7, 4], [4, 8, -16, -8], [-41, 0, 23, 5], [5, -8, -3, 0], [5, -10, 9, 5], [-1, -3, -3, -3], [2, -4, -19, 2], [5, -10, 9, 5], [7, -5, -9, 5], [41, 0, -21, -1], [-9, 7, 2, -14], [-9, 3, 15, -3], [1, -2, -19, 1], [10, 6, -26, -6], [-7, 14, 7, -7], [14, 0, 0, 0], [21, -3, 18, 0], [1, 0, 1, 0], [-20, -4, 44, 4], [17, -8, -9, 16], [-14, -2, -2, -2], [5, -10, -1, 5], [13, -18, -5, 0], [21, -7, -35, 7], [-21, -7, 28, 14], [28, -4, -4, -4], [10, -10, -10, 10], [21, -1, 20, 0], [5, -10, -19, 5], [24, 0, -11, 2], [39, 0, -17, 5], [-7, 11, 11, 11], [18, 8, -44, -8], [0, 0, 9, 18], [9, -20, -11, 0], [12, -4, -4, -4], [7, -14, 17, 7], [-4, 8, -31, -4], [0, 8, -8, -8], [-1, 1, 0, -2], [18, 0, -3, 12], [8, -12, -4, 12], [3, -1, -5, 1], [-7, 14, -7, -7], [-15, 3, -12, 0], [25, -2, 23, 0], [-5, 9, -4, -18], [1, 1, 2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_560_bw_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_560_2_bw_e();".
// 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_560_2_bw_e(:prec:=4)
    chi := MakeCharacter_560_bw();
    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_560_2_bw_e();".
// 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_560_2_bw_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_560_bw();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, -3, 1]>],Snew);
    return Vf;
end function;