
// Make newform 1764.2.i.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1764_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1764_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_1764_2_i_d();".
// 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_1764_2_i_d();".
// 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 := [3, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [6, 0, 1, 5, -1, 1], [0, -3, 2, -5, 1, -1], [6, -21, 19, -16, 5, -2], [-9, 33, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 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_1764_i();"
function MakeCharacter_1764_i()
    N := 1764;
    order := 3;
    char_gens := [883, 785, 1081];
    v := [3, 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_1764_i_Hecke();"
function MakeCharacter_1764_i_Hecke(Kf)
    N := 1764;
    order := 3;
    char_gens := [883, 785, 1081];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 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, 0, 0], [0, 0, 1, 1, 0, -1], [0, -1, -1, -1, 0, 1], [0, 0, 0, 0, 0, 0], [2, -3, 0, 2, -1, -1], [-2, 4, 0, -2, 1, 0], [1, 2, 1, 2, 1, -1], [1, 1, 0, 1, -1, -3], [-4, 1, 2, 1, -4, -2], [-1, -3, -2, -3, -1, 2], [0, 0, -3, -3, 0, 0], [-2, 1, 0, -2, 0, 3], [2, -1, 0, 2, -1, -3], [0, -1, 3, -1, 0, -3], [-7, -1, -3, -1, -1, 0], [-2, 1, 0, 1, -2, 0], [-10, -1, -2, 0, -1, 0], [-5, 1, -6, -8, 1, 0], [2, 1, 3, 1, 1, 0], [6, -3, -8, -2, -3, 0], [0, 2, 3, 2, 0, -3], [-5, 1, -3, -5, 1, 0], [7, 2, 1, 2, 7, -1], [-4, 5, 0, -4, -1, 3], [-1, 1, 6, 1, -1, -6], [1, -1, 0, 1, -5, -1], [-4, -3, 0, -3, -4, 0], [0, 6, 0, 0, 9, -6], [-4, -6, -9, -6, -4, 9], [6, -4, 0, 6, -6, -8], [-3, 6, 9, -3, 6, 0], [1, -3, -4, -3, 1, 4], [-4, 1, 0, -4, -4, 7], [1, 1, 0, 1, -3, -3], [-3, -3, -3, -3, -3, 3], [-3, 0, 0, -3, 4, 6], [1, 3, 3, -3, 3, 0], [-1, -1, 0, -1, 4, 3], [2, -6, 0, 2, -1, 2], [-12, 6, 7, -5, 6, 0], [0, 0, 3, 0, 0, -3], [0, 6, 3, -9, 6, 0], [10, -2, 3, 7, -2, 0], [0, -5, -6, 4, -5, 0], [-4, -1, -8, -6, -1, 0], [4, 5, 12, 5, 4, -12], [-4, 8, 0, -4, -11, 0], [9, 4, 0, 4, 9, 0], [1, -2, 0, 1, -20, 0], [11, 4, 0, 4, 11, 0], [8, -15, 0, 8, -10, -1], [10, -8, 0, 10, -5, -12], [7, -8, 0, 7, 5, -6], [14, -7, -3, 11, -7, 0], [-2, 4, -6, 4, -2, 6], [-1, 11, 0, -1, -1, -9], [11, -5, -4, -5, 11, 4], [2, -1, 0, 2, -13, -3], [3, -3, 0, 3, 8, -3], [23, -2, -1, -2, 23, 1], [0, -2, -3, 1, -2, 0], [-1, 14, 0, -1, 8, -12], [0, 7, 9, -5, 7, 0], [-1, -10, -5, 15, -10, 0], [8, -9, 0, 18, -9, 0], [9, 0, 3, 3, 0, 0], [-4, 3, 9, 3, 3, 0], [9, -12, 0, 9, 6, -6], [-1, -7, -15, -1, -7, 0], [-24, 2, 0, 2, -24, 0], [-3, -6, 0, -3, 3, 12], [-9, 17, 0, -9, 18, 1], [4, -11, 0, 4, -10, 3], [12, -5, -9, -5, 12, 9], [20, -5, 12, 22, -5, 0], [6, 1, -5, 1, 6, 5], [-1, 7, 0, -1, -19, -5], [6, -9, 0, 6, 0, -3], [19, 7, 9, 7, 19, -9], [8, -2, -6, -2, -2, 0], [5, -10, 0, 5, 5, 0], [2, -4, 3, -4, 2, -3], [17, 11, 6, 11, 17, -6], [-25, 4, 6, -2, 4, 0], [4, 1, 6, 4, 1, 0], [6, 6, -2, -14, 6, 0], [-18, -3, -7, -1, -3, 0], [-3, -1, 9, 11, -1, 0], [-13, 4, 11, 4, -13, -11], [-9, 24, 0, -9, 13, -6], [-10, 3, 0, -10, 14, 17], [5, -5, 0, 5, -19, -5], [-11, 3, 6, 3, -11, -6], [-2, 1, 0, -2, -26, 3], [-16, -6, -6, -6, -16, 6], [7, 4, 9, 1, 4, 0], [16, 1, 12, 1, 16, -12], [-9, 6, -6, 6, -9, 6], [3, 0, 0, 3, 10, -6], [8, -10, 0, 8, 5, -6], [-2, -5, -12, -5, -2, 12], [-1, 9, -2, 9, -1, 2], [-27, 6, -13, -25, 6, 0], [-3, -12, -6, 18, -12, 0], [4, 7, 0, -14, 7, 0], [-6, -13, -18, -13, -6, 18], [-3, -5, -11, -5, -3, 11], [3, 9, 0, 3, -3, -15], [-4, 2, 0, -4, 2, 0], [23, 3, 3, 3, 23, -3], [3, 0, -6, 0, 3, 6], [9, -6, -12, -6, 9, 12], [-4, 3, 0, -4, -19, 5], [-13, 26, 0, -13, 13, 0], [-17, 2, 9, 5, 2, 0], [-5, 2, 0, -5, -29, 8], [-3, 18, 0, -3, 16, -12], [30, 6, 9, 6, 30, -9], [-17, -8, -15, -8, -17, 15], [14, -8, 5, -8, 14, -5], [-18, -5, 0, 10, -5, 0], [17, -5, -3, -5, 17, 3], [-18, -3, 6, 12, -3, 0], [-2, -20, -18, -20, -2, 18], [-11, 0, 3, 3, 0, 0], [-14, -2, 2, 6, -2, 0], [16, -10, -12, 8, -10, 0], [11, -4, 0, 11, -16, -18], [-10, -11, -3, -11, -10, 3], [-23, -2, -12, -2, -23, 12], [-7, 5, 6, 5, -7, -6], [-3, 15, 0, -3, -6, -9], [28, 4, 9, 4, 28, -9], [18, -15, -21, 9, -15, 0], [-2, 2, 16, 2, -2, -16], [3, -12, 0, 3, 11, 6], [21, 8, 8, 8, 21, -8], [-24, 12, 15, -9, 12, 0], [-1, 5, 0, -1, -10, -3], [17, 18, 19, 18, 17, -19], [-7, -15, -12, 18, -15, 0], [-15, 9, 13, -5, 9, 0], [16, 0, 0, 0, 0, 0], [-10, 5, 24, 14, 5, 0], [-8, 0, 3, 0, -8, -3], [-19, 6, -8, 6, -19, 8], [-2, 2, -9, 2, -2, 9], [-7, 14, 0, -7, -13, 0], [-25, 6, 6, 6, -25, -6], [3, -13, 0, 3, 24, 7], [-27, -14, -15, -14, -27, 15], [28, -2, 3, 7, -2, 0], [-14, -4, -15, -7, -4, 0], [-4, -13, -6, -13, -4, 6], [12, 0, 0, 12, -21, -24], [-1, 17, 12, 17, -1, -12], [6, -3, 0, 6, 25, -9], [-32, 7, -18, -32, 7, 0], [26, -5, 9, 19, -5, 0], [-24, 12, 13, -11, 12, 0], [29, 5, -4, -14, 5, 0], [-10, 8, 16, 0, 8, 0], [5, -7, 0, 5, -20, -3], [7, 1, 0, 7, 10, -15], [13, -8, 0, 16, -8, 0], [-5, -2, 0, -5, -17, 12], [30, 8, 15, 8, 30, -15], [-4, -4, 0, -4, 42, 12], [4, -13, -21, 5, -13, 0], [28, 22, 15, 22, 28, -15], [9, 14, 0, 9, 0, -32], [0, -27, 0, 0, -13, 27], [-3, 6, 6, 6, -3, -6], [0, 0, 0, 0, 33, 0], [7, -8, 12, 28, -8, 0], [-4, -15, 0, -4, -7, 23], [-4, -19, -27, -19, -4, 27], [0, -25, -25, -25, 0, 25], [-31, 1, -12, -14, 1, 0], [-32, 12, 3, -21, 12, 0], [-19, -7, -6, 8, -7, 0], [-4, -12, 1, -12, -4, -1], [2, 8, 0, 2, 30, -12], [9, -5, 0, 9, -27, -13], [19, -17, -22, 12, -17, 0], [-7, 11, 0, -7, -37, 3], [22, -5, -3, -5, 22, 3], [3, -3, 0, 3, 10, -3], [15, 15, 9, 15, 15, -9], [24, 0, -3, 0, 24, 3], [-21, 24, 0, -21, 2, 18], [6, 3, -9, 3, 6, 9], [9, -24, 0, 9, -26, 6], [-25, 2, -1, -5, 2, 0], [-18, -15, -9, -15, -18, 9], [-1, -1, 14, 16, -1, 0], [3, -5, -3, 7, -5, 0], [-19, 7, -15, -29, 7, 0], [27, -6, 9, -6, 27, -9], [10, -26, 0, 10, 7, 6], [-5, 10, 18, -2, 10, 0], [-9, -24, -30, -24, -9, 30], [-27, -5, -18, -5, -27, 18], [-10, 5, 0, -10, -19, 15], [-29, 4, 3, -5, 4, 0], [23, 5, 21, 5, 23, -21], [-10, 20, 0, -10, 12, 0], [-26, 7, 0, 7, -26, 0], [-16, 14, 12, 14, -16, -12], [-15, 8, -15, -31, 8, 0], [-12, 15, 0, -12, 11, 9], [13, -19, 0, 13, -41, -7], [14, 2, -6, 2, 14, 6], [31, -2, 1, 5, -2, 0], [10, -11, -3, 19, -11, 0], [34, -9, 12, 30, -9, 0], [27, 11, 12, -10, 11, 0], [-12, 5, 0, -12, 30, 19], [5, -15, 0, 5, 26, 5], [-7, 24, 0, -7, 38, -10], [23, 21, 6, 21, 23, -6], [12, -12, 0, 12, -10, -12], [24, -12, -12, -12, 24, 12], [16, -35, 0, 16, -16, 3], [2, -16, 0, 2, -16, 12], [13, -12, -9, -12, 13, 9], [-7, 8, 35, 19, 8, 0], [0, 1, 10, 1, 0, -10], [5, -4, -3, 5, -4, 0], [-12, 0, 9, 9, 0, 0], [20, -16, -45, -13, -16, 0], [-2, -6, 0, -6, -2, 0], [-4, 14, 0, -4, 30, -6], [-43, 5, 3, -7, 5, 0], [-36, -8, -18, -8, -36, 18], [10, -23, 0, 10, -32, 3], [-12, 14, 18, 14, -12, -18], [3, -24, 0, 3, -15, 18], [-6, 15, 0, -6, -30, -3], [41, -1, 1, 3, -1, 0], [-8, 17, 0, -8, 25, -1], [-11, -2, 0, -11, 26, 24], [7, 6, 3, -9, 6, 0], [13, -20, 0, 13, -14, -6], [-15, 26, 0, -15, 21, 4], [34, -17, -9, 25, -17, 0], [-20, -11, -24, -2, -11, 0], [7, -2, -19, -15, -2, 0], [-31, 3, 21, 15, 3, 0], [-33, 12, 0, -24, 12, 0], [12, -39, 0, 12, -17, 15], [10, -26, 0, 10, -14, 6], [11, -7, 0, 14, -7, 0], [-17, -6, 3, -6, -17, -3], [23, -23, 0, 23, -13, -23], [-3, -17, 0, -3, -24, 23], [16, 12, 3, 12, 16, -3], [15, -21, 0, 15, -27, -9], [-6, -12, -5, 19, -12, 0], [4, -20, 0, 4, 5, 12], [25, -32, 0, 25, -19, -18], [13, -4, 7, -4, 13, -7], [9, -2, 9, 13, -2, 0], [15, -2, 12, 16, -2, 0], [-8, 4, 0, -8, 4, 0], [-10, 6, 6, -6, 6, 0], [65, -7, 3, 17, -7, 0], [-12, -13, -25, -13, -12, 25], [2, 5, 0, 2, -39, -9], [-27, -3, 12, 18, -3, 0], [21, -18, -33, -18, 21, 33], [18, -22, -21, -22, 18, 21], [14, -22, 0, 14, 5, -6], [-3, -10, -15, -10, -3, 15], [-5, 4, 0, -5, -42, 6], [8, -16, 0, 8, -1, 0], [14, -5, -10, -5, 14, 10], [-13, 23, 0, -13, 47, 3], [-10, 26, 0, -10, 42, -6], [40, -2, -3, 1, -2, 0], [21, 0, 30, 30, 0, 0], [11, -4, -21, -13, -4, 0], [-30, -9, 0, -9, -30, 0], [-29, -13, -18, -13, -29, 18], [1, 10, 3, 10, 1, -3], [-1, 27, 0, -1, 11, -25], [4, -26, -9, -26, 4, 9], [13, -28, 0, 13, 22, 2], [-13, 14, 0, -13, 6, 12], [16, -11, -33, -11, -11, 0], [-9, -6, 0, -9, 9, 24], [-2, -20, -24, -20, -2, 24], [36, -12, -9, -12, 36, 9], [-1, -15, 0, -1, 17, 17], [-10, 31, 36, 31, -10, -36], [20, 8, -9, -25, 8, 0], [16, 3, -12, -18, 3, 0], [22, 10, 36, 10, 22, -36], [12, -3, 0, 12, 36, -21], [-23, 10, 6, 10, -23, -6], [23, 4, 0, 4, 23, 0], [-3, 6, 0, -3, 9, 0], [5, 33, 30, 33, 5, -30]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1764_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1764_2_i_d();".
// 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_1764_2_i_d(:prec:=6)
    chi := MakeCharacter_1764_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_1764_2_i_d();".
// 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_1764_2_i_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1764_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -4, 17, 2, 5, -1, 1]>],Snew);
    return Vf;
end function;