
// Make newform 441.2.f.f in Magma, downloaded from the LMFDB on 15 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_f_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_441_2_f_f();".
// 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_441_2_f_f();".
// 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, -9, 36, -3, 90, -36, 40, -8, 9, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [72, 342, 936, -87, 402, 18, 95, -3, 9, 2], [-63, -270, -204, -123, -30, -68, -8, -12, 1, -2], [639, 1395, -27, 1893, -417, 786, -106, 159, -24, 17], [-180, 684, -351, 1680, -780, 663, -176, 156, -39, 16], [504, 369, 1620, 1020, -57, 624, -4, 141, -24, 20], [-306, 477, -45, 732, -219, 330, -43, 69, -12, 8], [-234, -1476, -1458, -5214, 1794, -2604, 403, -591, 123, -71], [720, -2898, -432, -7260, 2850, -3264, 632, -732, 165, -82]];
    Rf_basisdens := [1, 1, 189, 63, 567, 567, 567, 189, 567, 567];
    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_441_f();"
function MakeCharacter_441_f()
    N := 441;
    order := 3;
    char_gens := [344, 199];
    v := [1, 3];
    // 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_441_f_Hecke();"
function MakeCharacter_441_f_Hecke(Kf)
    N := 441;
    order := 3;
    char_gens := [344, 199];
    char_values := [[-1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 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]];
    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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1, -1, -1, 0], [2, 0, 0, 0, 0, 0, -2, 1, -1, 0], [3, -1, 0, 0, -1, 1, 0, 1, 0, 0], [0, -1, 1, -1, 0, 1, 0, 0, 1, 1], [1, 0, 2, 0, -2, 1, -1, 1, -3, 2], [0, 1, 2, 1, 0, 0, 1, -2, -2, 0], [0, 0, -1, 0, 1, -1, 0, -1, 2, -1], [0, -2, 0, -2, 0, 2, 0, 0, 0, 2], [-2, 0, -1, 0, 1, -1, 2, -2, 3, -1], [0, -1, -4, -1, -3, 0, 0, 1, 1, 0], [0, 0, -1, 1, 1, 0, -5, 2, 2, 0], [3, 1, 2, 0, 2, -1, 0, -2, 2, 0], [-7, 0, -1, 0, 1, 0, 7, -1, 2, 1], [0, 2, 2, -1, 1, 0, 1, -1, -1, 0], [2, 0, -2, 0, 2, 5, -2, 1, 1, -1], [0, 0, 2, 3, 1, 0, 0, -1, 2, -3], [4, 0, 0, 3, 1, 0, 0, -1, 0, -3], [0, 3, 1, 0, -1, 0, -2, -2, -2, 0], [0, -2, 0, 4, -2, 0, 1, -2, -2, 0], [8, -3, -1, 2, -1, 3, 0, 1, -1, -2], [0, -1, -6, -1, -2, 0, 4, 4, 4, 0], [0, -5, -3, -3, -1, 0, 0, 2, 2, 0], [-4, 0, 1, 0, -1, -3, 4, -2, 1, 3], [-2, 6, 4, 1, 4, -6, 0, -4, 4, -1], [3, 4, 0, 1, 0, -4, 0, 0, 0, -1], [3, 0, -1, 0, 1, -3, -3, 0, 1, 3], [7, -4, -2, -1, -3, 4, 0, 3, -2, 1], [-12, 0, 2, 0, -2, -5, 12, 0, -2, 0], [0, 1, -1, -1, 2, 0, -1, 3, 3, 0], [-5, 0, -4, 0, 4, -2, 5, 0, 4, 1], [-5, 0, -2, 0, 2, 1, 5, 3, -1, -4], [0, -3, 4, 3, -1, 0, -2, -5, -5, 0], [0, 0, 1, 0, -1, 5, 0, 0, -1, -4], [1, -1, 2, 2, 4, 1, 0, -4, 2, -2], [0, 0, 3, 0, -3, 6, 0, 2, -5, 0], [0, 5, -2, -3, -1, 0, -9, 1, 1, 0], [-7, 0, -1, -1, -1, 0, 0, 1, -1, 1], [5, 1, -1, -2, -8, -1, 0, 8, -1, 2], [0, 2, 8, -2, 7, 0, -8, -1, -1, 0], [1, 0, 6, 0, -6, 0, -1, -3, -3, 0], [-5, 4, -3, 1, -3, -4, 0, 3, -3, -1], [-12, 11, 6, -1, 6, -11, 0, -6, 6, 1], [1, 0, 3, 0, -3, 6, -1, -1, -2, 3], [0, 2, 3, -1, 3, 0, -8, 0, 0, 0], [0, 5, -7, -2, 0, 0, -5, 7, 7, 0], [11, 0, 2, 0, -2, 3, -11, -1, -1, -6], [12, -4, -4, 0, -9, 4, 0, 9, -4, 0], [7, 0, 2, 0, -2, 5, -7, 3, -5, -4], [0, -3, 2, -3, -2, 0, 5, -4, -4, 0], [5, 2, 0, 2, 3, -2, 0, -3, 0, -2], [1, 0, 5, 0, -5, 3, -1, 0, -5, -1], [0, -6, -4, 4, 2, 0, 4, 6, 6, 0], [-3, 5, 1, -6, 1, -5, 0, -1, 1, 6], [-4, 4, -3, -2, 3, -4, 0, -3, -3, 2], [0, 4, 3, -5, 3, 0, -1, 0, 0, 0], [0, -6, 0, 3, 2, 0, -3, 2, 2, 0], [1, 0, 4, 0, -4, 1, -1, 1, -5, 7], [-2, 0, -2, 0, 2, -4, 2, 0, 2, -1], [-8, 6, -1, 0, 3, -6, 0, -3, -1, 0], [-10, 0, 1, 0, -1, 0, 10, 7, -8, 7], [0, -7, -10, -1, -3, 0, 4, 7, 7, 0], [0, 0, -5, 5, -1, 0, -4, 4, 4, 0], [0, -5, -5, 1, -9, 0, 5, -4, -4, 0], [-1, 0, 5, 0, -5, 2, 1, 2, -7, -1], [14, 0, 1, 0, -1, 5, -14, 7, -8, 1], [0, -9, 8, 6, 2, 0, 2, -6, -6, 0], [0, 3, 10, 5, 7, 0, -1, -3, -3, 0], [8, -11, -3, 2, -7, 11, 0, 7, -3, -2], [0, -3, 0, -3, 5, 0, -10, 5, 5, 0], [5, 0, -2, 0, 2, -1, -5, -4, 6, -7], [-2, 11, 4, -1, 9, -11, 0, -9, 4, 1], [-10, 0, 1, 0, -1, 8, 10, -4, 3, 1], [0, 10, 1, -7, 0, 0, -4, -1, -1, 0], [9, 2, 2, -1, -5, -2, 0, 5, 2, 1], [-8, 0, 3, 0, -3, -2, 8, 2, -5, -1], [2, 0, 0, 0, 0, 6, -2, -6, 6, 3], [-25, 0, -3, 0, 3, -8, 25, -7, 10, 4], [0, 0, 2, 6, 4, 0, -5, 2, 2, 0], [-9, 10, 7, -3, 3, -10, 0, -3, 7, 3], [-8, 12, 5, 3, 9, -12, 0, -9, 5, -3], [0, -9, -1, 0, -3, 0, -7, -2, -2, 0], [0, -3, -7, -1, -2, 0, 17, 5, 5, 0], [-15, 9, -3, -6, 3, -9, 0, -3, -3, 6], [0, -2, 5, 4, 7, 0, -7, 2, 2, 0], [0, -3, -4, 3, -7, 0, -12, -3, -3, 0], [5, 0, -8, 0, 8, -4, -5, -5, 13, -10], [-6, -11, 0, 0, 1, 11, 0, -1, 0, 0], [0, -2, 7, 6, -2, 0, -18, -9, -9, 0], [3, 1, -1, 4, -3, -1, 0, 3, -1, -4], [-19, 0, 4, 0, -4, -14, 19, -7, 3, -2], [-1, 0, 3, 0, -3, -4, 1, 3, -6, 2], [6, 3, 3, -6, -3, -3, 0, 3, 3, 6], [-2, 0, -11, 0, 11, -9, 2, 5, 6, 5], [-3, -4, 6, -3, -4, 4, 0, 4, 6, 3], [-7, 13, 13, 7, 20, -13, 0, -20, 13, -7], [7, -1, -5, 8, -2, 1, 0, 2, -5, -8], [0, 0, 0, 3, -4, 0, -2, -4, -4, 0], [-5, -10, -1, 4, 5, 10, 0, -5, -1, -4], [-9, 0, -11, 0, 11, -16, 9, -5, 16, 0], [0, -4, -10, -5, -5, 0, -5, 5, 5, 0], [14, 0, -2, 0, 2, -4, -14, -1, 3, -1], [-7, -5, -9, -5, -12, 5, 0, 12, -9, 5], [0, 8, 13, 5, 9, 0, -10, -4, -4, 0], [-3, 7, -4, 0, -10, -7, 0, 10, -4, 0], [-8, 0, -6, 0, 6, -6, 8, -12, 18, -1], [0, -1, 1, -7, 3, 0, -8, 2, 2, 0], [19, 0, 7, 0, -7, 7, -19, 6, -13, -2], [1, -7, -9, 8, 0, 7, 0, 0, -9, -8], [-8, 0, -10, 0, 10, 8, 8, 0, 10, -4], [0, -7, 7, 5, 5, 0, 19, -2, -2, 0], [-3, -5, -11, 7, -3, 5, 0, 3, -11, -7], [0, -11, 2, -1, -1, 0, 26, -3, -3, 0], [-9, 0, -1, 0, 1, -10, 9, -12, 13, -7], [10, 6, -9, 1, -1, -6, 0, 1, -9, -1], [-2, 0, -7, 0, 7, 7, 2, -5, 12, -4], [0, -19, -3, 2, -1, 0, -4, 2, 2, 0], [18, 0, -9, 0, 9, -7, -18, -4, 13, -7], [0, 4, 9, -11, 9, 0, -13, 0, 0, 0], [0, -12, -17, -11, -7, 0, 1, 10, 10, 0], [-3, 1, 5, -3, 8, -1, 0, -8, 5, 3], [0, 3, 1, 3, -4, 0, 2, -5, -5, 0], [9, -6, 4, 6, 14, 6, 0, -14, 4, -6], [0, 20, 9, -4, 12, 0, -9, 3, 3, 0], [0, 4, 0, -6, -11, -4, 0, 11, 0, 6], [0, 10, 5, -2, -4, 0, 6, -9, -9, 0], [-6, 0, 0, 0, 0, -4, 6, 3, -3, -4], [-3, 7, -7, 1, -8, -7, 0, 8, -7, -1], [-2, 0, 15, 0, -15, 2, 2, -5, -10, 5], [24, 0, -7, 0, 7, 7, -24, -3, 10, 1], [9, -14, -9, 1, -14, 14, 0, 14, -9, -1], [8, 0, -3, 0, 3, 17, -8, -2, 5, -11], [5, 0, 2, 0, -2, -3, -5, -1, -1, 6], [-17, 9, 8, 13, 12, -9, 0, -12, 8, -13], [7, 0, -1, 0, 1, -2, -7, 5, -4, -8], [-3, 0, 8, 0, -8, -3, 3, -2, -6, -3], [-12, 13, -2, 6, -3, -13, 0, 3, -2, -6], [4, -9, 0, 15, 0, 9, 0, 0, 0, -15], [0, 4, 11, 0, -2, 0, -15, -13, -13, 0], [6, 0, -6, 0, 6, -2, -6, 0, 6, 1], [21, 3, 8, 3, 7, -3, 0, -7, 8, -3], [-1, 10, -5, -5, 3, -10, 0, -3, -5, 5], [0, -4, 2, -10, 8, 0, -10, 6, 6, 0], [0, 9, -6, 3, -3, 0, -7, 3, 3, 0], [0, -6, -17, 2, -11, 0, 35, 6, 6, 0], [-22, 0, -5, 0, 5, -9, 22, -2, 7, -3], [-9, 2, -1, -3, -6, -2, 0, 6, -1, 3], [-16, 0, 3, 0, -3, -7, 16, 9, -12, 5], [0, 9, -5, 9, -1, -9, 0, 1, -5, -9], [8, -6, 1, 9, 11, 6, 0, -11, 1, -9], [7, 0, 4, 0, -4, -12, -7, 2, -6, -4], [0, 0, -4, -12, 4, 0, -8, 8, 8, 0], [0, 17, 15, 2, 11, 0, -19, -4, -4, 0], [-1, -15, -4, -3, -8, 15, 0, 8, -4, 3], [0, 8, 8, 6, -5, 0, 6, -13, -13, 0], [23, -14, 7, 10, 13, 14, 0, -13, 7, -10], [13, 0, 9, 0, -9, -5, -13, -9, 0, 11], [0, -2, -2, 9, -10, 0, 9, -8, -8, 0], [2, -7, -16, -4, -11, 7, 0, 11, -16, 4], [-5, 0, 3, 0, -3, -3, 5, -4, 1, 6], [3, 10, 9, 0, 10, -10, 0, -10, 9, 0], [28, 0, -3, 0, 3, 12, -28, 15, -12, 2], [0, -1, -13, -11, -6, 0, -11, 7, 7, 0], [17, -6, -3, 0, 0, 6, 0, 0, -3, 0], [0, -7, -2, -7, -7, 0, 34, -5, -5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_2_f_f();".
// 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_441_2_f_f(:prec:=10)
    chi := MakeCharacter_441_f();
    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_441_2_f_f();".
// 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_441_2_f_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, -9, 36, -3, 90, -36, 40, -8, 9, -2, 1]>,<5,R![9, 54, 378, -294, 402, -51, 79, 16, 21, 4, 1]>],Snew);
    return Vf;
end function;