
// Make newform 462.2.i.f in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_462_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_462_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_462_2_i_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_462_2_i_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 := [4, -18, 27, -19, 12, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [3, -1, 1, 0, 0, 0], [-18, 43, -28, 22, -5, 2], [22, -56, 37, -31, 7, -3], [50, -121, 82, -64, 15, -6], [24, -62, 44, -33, 8, -3]];
    Rf_basisdens := [1, 1, 2, 1, 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_462_i();"
function MakeCharacter_462_i()
    N := 462;
    order := 3;
    char_gens := [155, 199, 211];
    v := [3, 2, 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_462_i_Hecke();"
function MakeCharacter_462_i_Hecke(Kf)
    N := 462;
    order := 3;
    char_gens := [155, 199, 211];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0], [1, 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+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 := [[-1, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, -1, 0, 0, 1, 0], [0, -1, 0, -1, 1, -1], [0, 0, 1, 0, 0, 0], [0, 2, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [1, 1, 0, -2, 0, -1], [-3, 0, -4, -2, 1, -1], [4, 0, 0, -1, 0, 1], [-2, -2, 0, -2, 4, -4], [-3, 1, -4, -2, 0, -1], [5, 0, 0, -1, 0, 1], [2, -2, 0, 1, 0, -1], [5, 0, 4, -2, 1, -1], [0, 0, 0, 0, -2, 0], [1, 1, 4, 1, 2, 2], [2, 3, 4, 4, -5, 2], [-1, -1, 1, -1, 2, -2], [2, 0, 0, -1, 0, 1], [2, 2, 2, 2, -2, 4], [-4, -3, -4, 0, 3, 0], [-3, -4, 0, 1, 0, -1], [12, -2, 12, 0, 2, 0], [1, 0, 0, 2, 0, -2], [-1, -1, -6, -1, -4, -2], [4, 2, 2, -4, 0, -2], [-8, -5, -9, -2, 6, -1], [2, 2, 4, 2, -5, 4], [-6, 0, 0, 2, 0, -2], [2, -3, 0, 1, 0, -1], [6, 6, 8, 4, -8, 2], [0, 0, 0, 0, -2, 0], [12, 0, 0, -3, 0, 3], [-9, -1, -8, 2, 0, 1], [-1, -1, 0, -1, 5, -2], [-3, -3, -6, -3, 0, -6], [4, 5, 5, 2, -6, 1], [6, 6, 0, -2, 0, 2], [14, 2, 16, 4, -4, 2], [-1, -1, -10, -1, 4, -2], [12, -2, 0, -2, 0, 2], [-8, -4, -8, 0, 4, 0], [2, 2, 0, 2, 2, 4], [4, -2, 0, 3, 0, -3], [2, 2, -10, 2, 0, 4], [8, 2, 0, -4, 0, 4], [-2, 2, 0, 2, 0, -2], [1, 1, 5, 1, -2, 2], [2, 0, 0, -4, 2, -2], [-7, 2, -9, -4, 0, -2], [0, 2, 0, -4, 0, 4], [2, 2, 2, 2, 4, 4], [-8, -6, 0, 1, 0, -1], [16, -2, 16, 0, 2, 0], [-4, -4, -14, -4, -2, -8], [-2, -2, -12, -2, 7, -4], [-8, -4, -8, 0, 4, 0], [0, 0, 8, 0, -6, 0], [-11, 0, 0, 2, 0, -2], [-4, -4, 0, -4, 10, -8], [-10, 0, 0, 3, 0, -3], [-2, -4, 0, 2, 0, -2], [-1, -1, -4, -1, -3, -2], [-25, -1, -26, -2, 2, -1], [4, 5, 8, 8, -9, 4], [0, 5, 5, 10, -10, 5], [0, 10, 0, -2, 0, 2], [-1, -1, -15, -1, 6, -2], [12, 5, 0, 0, 0, 0], [-2, -2, -18, -2, 6, -4], [10, 8, 10, 0, -8, 0], [0, 0, 20, 0, -6, 0], [-2, 5, -4, -4, -3, -2], [-1, 4, 0, 1, 0, -1], [-5, 1, 0, 10, -6, 5], [0, 0, -4, 0, -1, 0], [-13, -1, -10, 6, -2, 3], [2, -4, 0, -4, 6, -2], [6, 6, -12, 6, -10, 12], [0, -6, 0, -1, 0, 1], [-20, -2, 0, 3, 0, -3], [4, 4, 8, 4, 6, 8], [11, -4, 0, -2, 0, 2], [1, -4, 4, 6, 1, 3], [19, -5, 18, -2, 6, -1], [4, -2, 0, -4, 0, 4], [10, -4, 14, 8, 0, 4], [-28, -4, 0, 5, 0, -5], [-22, -2, 0, 2, 0, -2], [-21, 1, -24, -6, 2, -3], [0, 0, -2, 0, -6, 0], [-2, -2, -30, -2, -2, -4], [-3, -12, 0, 5, 0, -5], [-2, -10, 0, 4, 8, 2], [4, 6, 0, -2, 0, 2], [4, 2, 0, -8, 2, -4], [0, 0, -2, 0, 8, 0], [-20, 6, -16, 8, -10, 4], [-16, -1, -12, 8, -3, 4], [2, -2, 0, 3, 0, -3], [-3, -3, -26, -3, 4, -6], [0, 0, 0, 0, 0, 0], [-15, -7, -18, -6, 10, -3], [1, 1, 10, 1, -2, 2], [-3, -3, 9, -3, 10, -6], [-30, -2, 0, -2, 0, 2], [11, -5, 6, -10, 10, -5], [-4, -4, -8, -4, 5, -8], [24, 0, 0, -2, 0, 2], [-12, -1, -8, 8, -3, 4], [-4, -4, -4, -4, 9, -8], [4, 2, 0, -6, 0, 6], [3, 3, 19, 3, 6, 6], [2, 2, 0, 6, 0, -6], [-2, -2, -12, -2, 0, -4], [-16, 0, 0, 6, 0, -6], [4, 4, 16, 4, 1, 8], [0, -3, -4, -8, 7, -4], [35, 4, 0, 1, 0, -1], [1, 1, -10, 1, 4, 2], [-2, -4, 0, 6, 0, -6], [25, -3, 28, 6, 0, 3], [-7, -7, -12, -7, 14, -14], [0, 7, 3, 6, -10, 3], [2, 2, 0, 3, 0, -3], [15, 1, 16, 2, -2, 1], [9, 4, 8, -2, -3, -1], [2, -8, 0, 4, 0, -4], [-6, 1, -8, -4, 1, -2], [6, 6, -10, 6, -12, 12], [-32, 2, 0, -2, 0, 2], [10, 8, 8, -4, -6, -2], [4, 4, 0, 3, 0, -3], [-20, -1, -21, -2, 2, -1], [28, 6, 0, -10, 0, 10], [-2, -2, -28, -2, 11, -4], [9, 9, 16, 9, -8, 18], [0, -7, 0, -2, 0, 2], [1, 1, -15, 1, -2, 2], [-42, 0, 0, 2, 0, -2], [-10, 6, -8, 4, -8, 2], [-6, -6, -12, -6, 6, -12], [21, 0, 0, -5, 0, 5], [1, 1, 48, 1, 0, 2], [0, 9, 0, -8, 0, 8], [12, 3, 0, 0, 0, 0], [4, 4, 9, 4, -4, 8], [2, 3, 1, -2, -2, -1], [28, 1, 32, 8, -5, 4], [0, 7, 4, 8, -11, 4], [-2, 0, 0, 2, 0, -2], [17, 0, 0, -9, 0, 9], [-12, 0, -6, 12, -6, 6], [-3, -3, -9, -3, -6, -6], [14, -15, 0, 1, 0, -1], [19, -6, 24, 10, 1, 5], [-10, 8, -4, 12, -14, 6], [-14, 0, 0, 6, 0, -6], [3, 3, -30, 3, -6, 6], [11, -9, 6, -10, 14, -5], [15, -12, 0, 5, 0, -5], [30, -5, 0, 1, 0, -1], [12, 2, 18, 12, -8, 6], [8, 8, -4, 8, -12, 16], [-3, -3, -36, -3, -1, -6], [0, 0, 8, 0, -10, 0], [0, 0, 40, 0, -2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_462_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_462_2_i_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_462_2_i_f(:prec:=6)
    chi := MakeCharacter_462_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(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_462_2_i_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_462_2_i_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_462_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, 36, 81, 8, 9, 0, 1]>,<13,R![32, -36, 0, 1]>],Snew);
    return Vf;
end function;