
// Make newform 43.2.e.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_43_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_43_e_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_43_2_e_a();".
// 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_43_2_e_a();".
// 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, 1, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_43_e();"
function MakeCharacter_43_e()
    N := 43;
    order := 7;
    char_gens := [3];
    v := [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_43_e_Hecke();"
function MakeCharacter_43_e_Hecke(Kf)
    N := 43;
    order := 7;
    char_gens := [3];
    char_values := [[0, -1, 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 := [[0, -1, 1, -1, 1, -1], [0, 0, 0, 1, 0, 1], [0, -1, 1, -1, 0, 0], [-1, 0, -2, 1, -1, 2], [-3, 1, -3, 0, -2, 2], [-1, 4, -2, 4, -1, 0], [0, -2, 2, 0, 0, 0], [4, -5, 4, -4, 5, -4], [2, -1, 2, 0, 4, -4], [0, 0, 0, 6, 0, 0], [0, 3, 4, -4, 4, 3], [-4, 0, -1, 4, -4, 1], [0, 2, -2, -6, -2, 2], [0, -7, 2, -2, 0, -2], [9, -8, 6, -6, 8, -9], [-3, 6, -6, 3, 0, 1], [1, 2, -2, -1, 0, 3], [4, -4, 0, -2, 4, -2], [-6, 4, -2, 2, -4, 6], [10, -4, 8, -8, 4, -10], [-1, -4, -4, -4, -1, 0], [-5, 0, -3, 2, -2, 3], [1, -1, 0, 6, -3, 6], [-2, 2, 0, 9, -3, 9], [2, 8, 2, 0, -5, 5], [-7, 4, -11, 11, -4, 7], [2, 2, -2, -2, 0, -1], [-1, 1, 0, -1, 2, -1], [4, -8, 4, 0, 6, -6], [12, -3, 3, -12, 0, 1], [0, 0, -9, 9, -9, 0], [0, -6, 7, 4, 7, -6], [7, -11, 5, -11, 7, 0], [6, -3, 3, -6, 0, -1], [0, 4, 0, 0, 1, -1], [-13, 13, 0, 8, -6, 8], [-2, 11, -6, 6, -11, 2], [-4, 11, -4, 0, -1, 1], [-5, -1, 6, -1, -5, 0], [-2, 0, 1, 4, -4, -1], [-2, 0, 8, 4, -4, -8], [2, -16, 2, 0, 9, -9], [6, -2, 12, -12, 2, -6], [-2, 2, 0, -1, 5, -1], [-4, 4, 0, 1, 5, 1], [3, -10, 10, -3, 0, 5], [9, 2, 1, -1, -2, -9], [11, -7, -1, -7, 11, 0], [0, -3, -7, 3, -7, -3], [0, 17, -6, 18, -6, 17], [-12, 12, 0, 12, -16, 12], [-9, 1, -10, 10, -1, 9], [-2, 0, 0, 2, 0, 1], [0, 0, 6, -14, 14, -6], [-4, 0, -3, 0, 0, 3], [-3, 10, -13, 10, -3, 0], [-6, -7, -6, 0, -17, 17], [1, 1, -12, 1, 1, 0], [1, 0, 11, -11, 0, -1], [6, 12, 6, 0, -1, 1], [0, 0, 0, 4, 22, 4], [-10, 6, -8, 6, -10, 0], [1, 0, -13, 5, -5, 13], [-1, -6, 6, 1, 0, 2], [0, -12, -2, -2, -2, -12], [1, 7, -7, -1, 0, -14], [10, -11, 9, -11, 10, 0], [14, 0, 15, -8, 8, -15], [13, -13, 0, -13, 4, -13], [-7, -5, 18, -5, -7, 0], [-2, 17, -2, 0, 2, -2], [12, -18, 4, -4, 18, -12], [-4, -10, -4, 0, 11, -11], [0, 5, 1, -19, 1, 5], [-20, 16, -15, 16, -20, 0], [-7, 19, -7, 7, -19, 7], [0, 22, -15, 11, -15, 22], [-7, 5, -5, 7, 0, 3], [0, -10, 2, -2, 2, -10], [8, -8, 0, -7, -5, -7], [9, 15, 9, 0, 2, -2], [0, -17, 0, 0, 4, -4], [30, 0, 6, 0, 0, -6], [1, -1, 0, 8, 9, 8], [4, -10, 4, 0, 3, -3], [6, -18, 4, -18, 6, 0], [4, -16, 9, -9, 16, -4], [0, 6, -6, 0, 0, -29], [0, 10, -23, -5, -23, 10], [-3, 12, -12, 3, 0, 8], [-25, 0, -21, 23, -23, 21], [-4, 0, -7, -16, 16, 7], [0, 4, 3, 4, 3, 4], [-9, 9, 0, -18, -10, -18], [-9, -11, 11, 9, 0, 5], [-16, 10, 9, 10, -16, 0], [-22, 0, -7, 7, -7, 7], [-11, -8, -15, -8, -11, 0], [18, 0, 4, -21, 21, -4], [3, -3, 0, -14, 2, -14], [0, -4, -16, 12, -16, -4], [0, 23, 9, -4, 9, 23], [6, -11, 5, -5, 11, -6], [-13, -5, 5, 13, 0, -2], [0, 9, -24, 11, -24, 9], [2, -2, 0, 21, 4, 21], [14, -14, 1, -1, 14, -14], [11, -10, 11, 0, 18, -18], [-19, 19, 0, 14, 2, 14], [-5, 0, -1, -5, 5, 1], [0, 21, -4, 21, 0, 0], [15, 7, 15, 0, 4, -4], [7, -19, 16, -16, 19, -7], [11, -5, 5, -11, 0, -11], [0, -7, 5, -12, 5, -7], [-15, -7, -14, 14, 7, 15], [0, -18, 4, -4, 4, -18], [0, -19, 0, 8, 0, -19], [-4, -24, 2, -24, -4, 0], [0, -2, -13, 18, -13, -2], [15, -9, 9, -15, 0, -25], [-31, 29, -24, 24, -29, 31], [0, 6, 0, 0, 13, -13], [-26, 26, -31, 26, -26, 0], [-14, 14, 0, 12, -9, 12], [14, -9, 29, -9, 14, 0], [1, -1, 0, -12, 22, -12], [0, -14, 7, -22, 7, -14], [12, -20, 25, -25, 20, -12], [0, 2, 1, 8, 1, 2], [-16, 15, -11, 15, -16, 0], [0, 3, -9, 35, -9, 3], [3, -12, 3, 0, 9, -9], [13, 0, 0, -13, 0, 5], [12, -13, -2, -13, 12, 0], [-3, -14, 20, -14, -3, 0], [16, 0, 17, 20, -20, -17], [11, 7, 3, 7, 11, 0], [11, 24, 11, 0, -6, 6], [27, -7, 14, -7, 27, 0], [-4, 0, 32, -19, 19, -32], [-2, 13, -6, 6, -13, 2], [13, 0, -13, 15, -15, 13], [19, -5, 5, -19, 0, 21], [0, 22, 2, -7, 2, 22], [8, -8, 0, 2, 11, 2], [-34, 0, -21, 29, -29, 21], [-11, 11, 0, -1, -4, -1], [0, 0, -27, -4, 4, 27], [-32, 32, 0, -7, -12, -7], [2, -2, 2, -2, 0, 9], [26, -26, 0, -2, 9, -2], [23, -16, 23, 0, -5, 5], [3, 7, -7, -3, 0, 21], [27, -19, 30, -30, 19, -27], [11, -9, 8, -9, 11, 0], [-23, 37, -37, 23, 0, 34], [30, -21, 21, -30, 0, -32], [23, -10, 23, 0, 2, -2], [-31, 11, -9, 11, -31, 0], [-23, 0, -16, 24, -24, 16], [19, 0, 10, -37, 37, -10], [1, -1, 0, -12, -3, -12], [10, -10, 0, -6, 18, -6], [0, 18, -5, 4, -5, 18], [-25, 0, -20, -3, 3, 20], [0, -26, 46, -5, 46, -26], [-1, 18, 21, 18, -1, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_43_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_43_2_e_a();".
// 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_43_2_e_a(:prec:=6)
    chi := MakeCharacter_43_e();
    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_43_2_e_a();".
// 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_43_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_43_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 3, 9, 13, 11, 5, 1]>],Snew);
    return Vf;
end function;