
// Make newform 1568.2.i.n in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1568_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1568_i_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_1568_2_i_n();".
// 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_1568_2_i_n();".
// 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, 1, 2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-1, -2, 2, -1], [-1, 6, -2, 1], [2, 0, 0, 1]];
    Rf_basisdens := [1, 2, 2, 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_1568_i();"
function MakeCharacter_1568_i()
    N := 1568;
    order := 3;
    char_gens := [1471, 197, 1473];
    v := [3, 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_1568_i_Hecke();"
function MakeCharacter_1568_i_Hecke(Kf)
    N := 1568;
    order := 3;
    char_gens := [1471, 197, 1473];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 1, 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 := 2;
    raw_aps := [[0, 0, 0, 0], [-1, -1, -1, 0], [0, -1, 1, 1], [0, 0, 0, 0], [2, 2, -2, 0], [-3, 0, 0, 1], [0, 0, -2, 0], [0, -1, -1, -1], [0, 4, 0, 0], [0, 0, 0, -2], [2, 2, 2, 0], [0, 0, 2, 2], [4, 0, 0, 2], [2, 0, 0, 2], [0, -6, 2, 2], [10, 10, 0, 0], [-7, -7, 1, 0], [0, -9, 1, 1], [-4, -4, 0, 0], [4, 0, 0, -4], [6, 6, 4, 0], [0, 4, -4, -4], [7, 0, 0, 1], [0, 6, 0, 0], [-8, 0, 0, -2], [5, 5, 3, 0], [0, 10, 2, 2], [0, 0, -4, -4], [-4, -4, 2, 0], [-8, 0, 0, -2], [0, 0, 0, 4], [0, -5, 3, 3], [-6, -6, 4, 0], [1, 0, 0, -1], [0, -6, -4, -4], [0, 0, -4, 0], [3, 3, 1, 0], [0, 10, 6, 6], [-10, 0, 0, -6], [0, 1, 3, 3], [16, 16, 4, 0], [1, 0, 0, 5], [0, -4, -4, -4], [4, 4, 2, 0], [2, 0, 0, 4], [2, 2, -6, 0], [-12, 0, 0, 0], [-4, 0, 0, -4], [-15, -15, 1, 0], [0, 1, 11, 11], [0, 2, 8, 8], [4, 0, 0, -8], [8, 8, -2, 0], [-11, 0, 0, 3], [0, 14, 0, 0], [-20, -20, -4, 0], [7, 7, 5, 0], [0, 24, 0, 0], [10, 10, 8, 0], [26, 0, 0, 0], [5, 5, 5, 0], [15, 0, 0, -1], [-13, 0, 0, 5], [8, 8, 0, 0], [0, 4, -2, -2], [0, 18, 4, 4], [0, 18, -2, -2], [-12, 0, 0, 10], [-2, -2, 2, 0], [-11, 0, 0, -7], [2, 2, 8, 0], [0, -8, 4, 4], [4, 4, 12, 0], [0, -6, -4, -4], [-18, 0, 0, 6], [0, 2, -6, -6], [0, 0, -2, 0], [0, 13, -1, -1], [0, -28, -2, -2], [-8, -8, 6, 0], [-11, 0, 0, -5], [-2, 0, 0, 0], [28, 28, 0, 0], [4, 0, 0, -6], [0, 32, 0, 0], [0, 4, 8, 8], [10, 0, 0, -8], [0, -12, -2, -2], [-15, 0, 0, -3], [0, 0, 0, -16], [0, 3, -13, -13], [-22, -22, -6, 0], [-20, -20, 0, 0], [20, 0, 0, 8], [0, 4, -8, -8], [-4, 0, 0, 4], [0, 21, -9, -9], [-20, -20, -6, 0], [0, 11, 3, 3], [0, -14, -4, -4], [-18, 0, 0, 6], [-18, -18, -4, 0], [-29, -29, -5, 0], [0, 4, -10, -10], [10, 10, -2, 0], [-6, -6, 0, 0], [-3, 0, 0, 3], [0, -6, 12, 12], [0, 0, 8, 0], [6, 0, 0, -16], [0, 8, 8, 8], [-16, -16, 6, 0], [12, 0, 0, 10], [-19, -19, 13, 0], [-36, 0, 0, -4], [-28, -28, -6, 0], [37, 0, 0, 3], [2, 2, 10, 0], [0, 4, 6, 6], [6, 0, 0, 14], [-7, -7, 7, 0], [6, 0, 0, -4], [0, -31, -5, -5], [12, 12, 16, 0], [0, 3, -5, -5], [-28, 0, 0, -6], [0, 24, 10, 10], [0, -2, 22, 22], [10, 0, 0, -10], [0, 7, 1, 1], [-10, -10, -6, 0], [16, 0, 0, 4], [0, 36, 0, 0], [8, 0, 0, 14], [0, 8, 2, 2], [-4, 0, 0, 10], [29, 29, -5, 0], [-19, -19, -3, 0], [41, 0, 0, 5], [4, 4, -6, 0], [15, 0, 0, 17], [0, 22, 8, 8], [-16, -16, -16, 0], [24, 0, 0, -4], [9, 9, -1, 0], [26, 0, 0, -2], [-7, 0, 0, -3], [-32, -32, -2, 0], [0, 45, 5, 5], [0, -12, 4, 4], [0, 0, -6, -6], [2, 0, 0, -12], [32, 0, 0, -4], [0, 2, -14, -14], [8, 8, 4, 0], [-8, 0, 0, -12], [0, 0, 16, 16], [0, 16, -14, -14], [18, 0, 0, -4], [-11, -11, 11, 0], [0, -18, -6, -6], [34, 0, 0, 8], [12, 0, 0, 8], [0, -25, -1, -1], [-14, -14, -12, 0], [-10, -10, -2, 0], [24, 24, 0, 0], [-11, -11, -13, 0], [28, 0, 0, 2], [0, -7, 19, 19], [-20, -20, 8, 0], [-7, 0, 0, -11], [0, 4, -12, -12], [-34, -34, -8, 0], [18, 18, -6, 0], [34, 0, 0, -12], [-24, 0, 0, -4], [32, 32, -10, 0], [36, 0, 0, 4], [0, -29, -3, -3], [0, 36, 0, 0], [-3, 0, 0, -5], [36, 0, 0, 10], [0, -18, -16, -16], [-8, -8, 4, 0], [11, 11, 1, 0], [10, 10, -8, 0], [-7, -7, 9, 0], [0, 38, -4, -4], [46, 46, -2, 0], [0, -34, -8, -8], [4, 0, 0, -16], [0, 24, 12, 12], [0, -3, -1, -1], [-16, -16, -4, 0], [-58, -58, -4, 0], [12, 12, -6, 0], [0, 32, -14, -14], [-30, 0, 0, -12], [0, -48, 0, 0], [-6, -6, 24, 0], [-14, 0, 0, 14], [0, 19, -11, -11], [46, 46, 4, 0], [11, 0, 0, -3], [45, 45, 3, 0], [0, 2, 18, 18], [0, -14, 6, 6], [0, 0, 0, -2], [15, 15, -25, 0], [0, -20, 6, 6], [7, 0, 0, 7], [16, 0, 0, 12], [0, 3, 19, 19], [-40, -40, -8, 0], [0, 24, 2, 2], [-16, -16, 16, 0], [-20, -20, 18, 0], [0, 16, 8, 8], [-44, 0, 0, 10], [0, -12, -2, -2], [-40, 0, 0, 8], [0, 10, 8, 8], [0, -28, -8, -8], [-9, 0, 0, -23], [28, 0, 0, 10], [0, -26, 16, 16], [4, 4, 16, 0], [0, 12, -20, -20], [0, -18, 2, 2], [4, 4, 18, 0], [35, 35, 3, 0], [-8, 0, 0, 24], [14, 14, -8, 0], [-9, 0, 0, -23], [-16, -16, 24, 0], [0, -24, -6, -6], [0, -6, 28, 28], [36, 0, 0, -8], [-12, 0, 0, -12], [34, 34, -2, 0], [0, -53, 3, 3], [20, 20, 4, 0], [0, -42, 8, 8], [-6, 0, 0, 12], [0, 14, 6, 6], [16, 0, 0, 0], [-3, -3, -11, 0], [22, 22, 2, 0], [-24, 0, 0, 0], [-6, 0, 0, -20], [0, 30, 8, 8], [-68, -68, 4, 0], [6, 0, 0, 0], [17, 17, -9, 0], [0, 32, -4, -4], [-20, -20, -14, 0], [3, 3, 3, 0], [-17, 0, 0, -1], [0, 48, -14, -14], [12, 12, -8, 0], [34, 0, 0, -6], [17, 17, -1, 0], [-25, 0, 0, 15], [16, 16, -26, 0], [0, 1, -23, -23], [-22, 0, 0, -4], [14, 0, 0, 8], [24, 0, 0, -4], [12, 12, 10, 0], [0, 1, 3, 3], [-38, -38, -10, 0], [-2, -2, -12, 0], [0, -24, 4, 4], [22, 0, 0, -24], [0, 4, 12, 12], [0, -38, -2, -2], [38, 38, -12, 0], [0, 16, -14, -14], [0, 39, 15, 15], [-36, -36, 20, 0], [-24, -24, 8, 0], [-14, 0, 0, 30], [39, 0, 0, 7], [0, -41, 7, 7], [0, 72, -4, -4], [38, 38, -8, 0], [26, 0, 0, -20], [-50, -50, -2, 0], [30, 0, 0, 24], [-14, -14, -32, 0], [17, 17, 9, 0], [0, -36, 6, 6], [55, 0, 0, 9], [46, 0, 0, 8], [59, 59, -7, 0], [0, -40, 0, 0], [-11, 0, 0, -15], [0, -11, -19, -19], [1, 0, 0, 15], [0, 56, -6, -6], [0, 20, -18, -18], [8, 8, 8, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1568_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1568_2_i_n();".
// 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_1568_2_i_n(:prec:=4)
    chi := MakeCharacter_1568_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1568_2_i_n();".
// 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_1568_2_i_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1568_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, -8, 8, 2, 1]>,<5,R![16, 8, 8, -2, 1]>,<11,R![256, 64, 32, -4, 1]>],Snew);
    return Vf;
end function;