
// Make newform 1014.2.e.k in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1014_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1014_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_1014_2_e_k();".
// 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_1014_2_e_k();".
// 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, -2, 5, 0, 3, -1, 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, -2, 5, -9, 3, -1], [18, -6, 15, -14, 9, -3], [-2, -34, -6, -10, -1, -4], [10, -25, -9, -15, 5, -6]];
    Rf_basisdens := [1, 1, 13, 13, 13, 13];
    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_1014_e();"
function MakeCharacter_1014_e()
    N := 1014;
    order := 3;
    char_gens := [677, 847];
    v := [3, 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_1014_e_Hecke();"
function MakeCharacter_1014_e_Hecke(Kf)
    N := 1014;
    order := 3;
    char_gens := [677, 847];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1]];
    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, 0, 0, 0, 1], [-1, 0, 0, 0, 0, 1], [0, 0, -2, 1, 0, 0], [0, -1, 1, -2, -2, -2], [1, -3, 0, 0, -1, -1], [0, 0, 0, 0, 0, 0], [0, 2, -2, 4, 4, -2], [0, 0, 0, 4, 4, 0], [-6, 0, 0, 0, 2, 6], [-7, 5, 0, 0, 3, 7], [-5, 0, 1, -5, 0, 0], [4, -2, 0, 0, 2, -4], [6, -2, 0, 0, -2, -6], [0, -4, 4, 2, 2, 2], [0, 0, 4, 0, 0, 0], [4, 0, 6, 3, 0, 0], [0, 1, -1, 2, 2, 4], [0, 2, -2, 0, 0, 4], [-2, 2, 0, 0, 10, 2], [0, 0, 0, -6, -6, -4], [0, 0, 4, -1, 0, 0], [-2, 0, -5, -6, 0, 0], [0, 0, -9, -2, 0, 0], [4, 0, 0, 0, -2, -4], [0, 4, -4, 3, 3, 0], [-6, 9, 0, 0, 4, 6], [-2, 0, -4, 7, 0, 0], [4, -5, 0, 0, -2, -4], [-8, 0, 4, 0, 0, 0], [0, 0, 0, -4, -4, -8], [2, 0, 0, 0, 9, -2], [-6, 0, 6, -9, 0, 0], [0, 8, -8, 12, 12, 4], [0, -4, 4, -2, -2, -2], [0, 9, -9, 3, 3, 15], [-7, 0, 11, -11, 0, 0], [0, 0, -4, -2, 0, 0], [0, 6, -6, 4, 4, -2], [-12, 16, 0, 0, 6, 12], [0, -3, 3, 7, 7, 7], [10, -14, 0, 0, -17, -10], [-4, 0, 4, 6, 0, 0], [0, -10, 10, -4, -4, -10], [-2, 8, 0, 0, 4, 2], [-6, 5, 0, 0, 0, 6], [0, 5, -5, -7, -7, -7], [12, -10, 0, 0, -4, -12], [-4, 8, 0, 0, 4, 4], [0, -6, 6, -1, -1, -18], [-18, 0, -2, -4, 0, 0], [-4, 0, -10, 18, 0, 0], [14, 0, 4, -2, 0, 0], [0, -4, 4, 3, 3, -16], [0, 9, -9, -6, -6, 8], [-6, 10, 0, 0, 4, 6], [0, 10, 0, 0, 10, 0], [0, -10, 10, 9, 9, 12], [-6, 0, 0, 0, 7, 6], [0, 4, -4, -6, -6, 6], [12, 0, -4, -2, 0, 0], [18, 4, 0, 0, 0, -18], [0, 3, -3, 1, 1, 13], [18, 0, -18, 8, 0, 0], [6, 0, 0, 6, 0, 0], [-4, 0, -4, 15, 0, 0], [-10, 0, 7, -4, 0, 0], [0, -14, 14, -10, -10, -4], [1, 0, -13, 5, 0, 0], [0, 3, -3, 2, 2, -8], [24, -16, 0, 0, -14, -24], [-2, 2, 0, 0, 20, 2], [6, 0, 0, 2, 0, 0], [-14, 9, 0, 0, 14, 14], [0, 6, -6, -16, -16, -4], [-12, 12, 0, 0, 4, 12], [0, 0, 0, 6, 6, -18], [9, 0, 1, -15, 0, 0], [0, -10, 10, -12, -12, -4], [18, 2, 0, 0, -16, -18], [0, -9, 9, -24, -24, -8], [-3, 13, 0, 0, 3, 3], [8, 0, 6, -4, 0, 0], [-12, -12, 0, 0, -2, 12], [0, -1, 1, 15, 15, 7], [6, -11, 0, 0, -18, -6], [19, 0, -13, 7, 0, 0], [0, 8, -8, 12, 12, 0], [9, -17, 0, 0, -5, -9], [0, -19, 19, -12, -12, -22], [4, 0, 12, 4, 0, 0], [18, 0, -2, 13, 0, 0], [-2, -6, 0, 0, 14, 2], [0, -16, 16, -1, -1, -6], [-11, 13, 0, 0, 7, 11], [-12, 0, 8, -16, 0, 0], [0, -2, 2, -14, -14, 4], [14, 12, 0, 0, 12, -14], [-14, 0, 24, -16, 0, 0], [-20, 10, 0, 0, 18, 20], [2, 0, -6, -2, 0, 0], [4, 0, 8, 0, 0, 0], [6, 8, 0, 0, 8, -6], [0, 6, -6, 11, 11, 18], [-24, 32, 0, 0, 24, 24], [10, 0, -16, -8, 0, 0], [16, 0, -5, -4, 0, 0], [-9, 19, 0, 0, 29, 9], [-2, 0, -16, 16, 0, 0], [18, 0, -10, 8, 0, 0], [-19, 11, 0, 0, 11, 19], [0, -1, 1, 14, 14, -18], [-6, 26, 0, 0, -2, 6], [0, 14, -14, -4, -4, -8], [4, 0, -20, -6, 0, 0], [0, 5, -5, 6, 6, -10], [0, 0, 0, 18, 18, -6], [0, 14, -14, 8, 8, -2], [-4, 6, 0, 0, 22, 4], [2, -13, 0, 0, -16, -2], [0, 25, -25, 6, 6, 20], [-8, -6, 0, 0, 18, 8], [16, -13, 0, 0, -20, -16], [18, 0, 13, 0, 0, 0], [0, 14, -14, 9, 9, -2], [2, -12, 0, 0, -12, -2], [-26, 0, 16, -12, 0, 0], [0, -14, 14, -26, -26, 0], [0, 32, -32, 8, 8, 26], [2, 0, 19, -10, 0, 0], [2, 0, 10, -16, 0, 0], [-10, 2, 0, 0, 6, 10], [-30, 6, 0, 0, -6, 30], [-25, -15, 0, 0, -3, 25], [-8, 8, 0, 0, -4, 8], [0, 0, 0, 8, 8, -8], [14, -12, 0, 0, -16, -14], [0, 19, -19, 12, 12, 46], [0, 22, -22, 22, 22, 6], [0, 1, -1, -13, -13, -25], [38, -14, 0, 0, -10, -38], [36, 0, -8, 16, 0, 0], [16, -26, 0, 0, -17, -16], [0, -13, 13, 10, 10, 22], [-3, 0, -3, -11, 0, 0], [-2, 4, 0, 0, 18, 2], [0, -30, 30, -22, -22, -8], [6, 0, -10, 22, 0, 0], [-16, 0, 18, -28, 0, 0], [-28, 0, 2, -16, 0, 0], [10, 0, -22, 34, 0, 0], [0, -2, 2, -6, -6, 34], [-22, 6, 0, 0, 4, 22], [-26, 0, -14, 0, 0, 0], [30, 6, 0, 0, -2, -30], [6, 16, 0, 0, 0, -6], [42, 0, -14, 8, 0, 0], [0, -4, 4, 5, 5, -26], [0, -14, 14, 14, 14, 8], [-33, 0, 25, -25, 0, 0], [-19, 0, 13, -7, 0, 0], [14, -18, 0, 0, -5, -14], [0, -12, 12, -6, -6, 6], [-5, 0, -7, 7, 0, 0], [0, -23, 23, 3, 3, -1], [-18, 2, 0, 0, 16, 18], [-6, 0, 34, 4, 0, 0], [-2, -12, 0, 0, 17, 2], [0, 8, -8, 18, 18, 0], [11, 0, -7, -1, 0, 0], [-10, 0, -8, 0, 0, 0], [-44, 0, -8, -8, 0, 0], [0, 18, -18, 8, 8, -10], [0, 30, -30, 2, 2, -4], [0, -32, 32, -12, -12, -22], [-12, 0, 36, -12, 0, 0], [0, -8, 8, 26, 26, 6], [2, 16, 0, 0, 14, -2], [-42, 0, -7, -8, 0, 0], [32, 0, 0, 0, -12, -32], [28, -24, 0, 0, -22, -28], [-2, 0, -24, 3, 0, 0], [43, 0, 15, 11, 0, 0], [-10, 0, -22, 12, 0, 0], [18, 0, 12, 6, 0, 0], [-8, 18, 0, 0, -2, 8], [0, 1, -1, -4, -4, -26], [-10, 0, 6, -4, 0, 0], [-2, 0, -24, 14, 0, 0], [8, 25, 0, 0, -16, -8], [0, -12, 12, 22, 22, -10], [0, 39, -39, 27, 27, 23], [0, -19, 19, -18, -18, -12], [26, 0, -14, -8, 0, 0], [8, 2, 0, 0, 33, -8], [0, 2, -2, -27, -27, 10], [-4, -24, 0, 0, -4, 4], [11, 0, 9, -29, 0, 0], [0, 0, 0, -14, -14, -24], [36, 0, -30, 8, 0, 0], [6, 0, -14, 0, 0, 0], [0, 32, -32, 4, 4, 14], [0, 35, -35, 15, 15, 19], [44, 0, 0, 0, 4, -44], [0, 0, 12, -21, 0, 0], [4, 1, 0, 0, -6, -4], [-11, -23, 0, 0, 23, 11], [-34, 0, 8, 7, 0, 0], [0, 15, -15, -18, -18, -4], [-20, 22, 0, 0, 32, 20], [0, 26, -26, 38, 38, 8], [-28, 8, 0, 0, 11, 28], [9, 0, -39, 9, 0, 0], [24, -8, 0, 0, -28, -24], [0, -4, 4, -24, -24, -20], [0, -38, 38, -32, -32, -18], [12, 0, 9, -8, 0, 0], [-19, 0, 23, -7, 0, 0], [0, -24, 24, -8, -8, -24], [-14, 12, 0, 0, -14, 14], [36, 0, -10, 1, 0, 0], [-6, 28, 0, 0, -12, 6], [6, 0, -8, -11, 0, 0], [24, 0, -38, 18, 0, 0], [0, 7, -7, 19, 19, -13], [-4, 8, 0, 0, 4, 4], [-8, 0, 14, -16, 0, 0], [-42, 40, 0, 0, 40, 42], [0, 28, -28, 26, 26, 20], [0, 40, -40, 25, 25, 18], [22, 0, -2, -1, 0, 0], [34, -2, 0, 0, 18, -34], [-34, -8, 0, 0, 0, 34], [2, 1, 0, 0, 22, -2], [28, 0, -28, 6, 0, 0], [8, 0, 4, 42, 0, 0], [22, 0, -8, -10, 0, 0], [0, -29, 29, 4, 4, -12], [30, 7, 0, 0, -8, -30], [0, 36, -36, 20, 20, 12], [-6, -10, 0, 0, -42, 6], [11, -13, 0, 0, -7, -11], [6, 24, 0, 0, 10, -6], [0, 48, -48, 36, 36, 52], [-48, -4, 0, 0, 20, 48], [0, 4, -4, 36, 36, 4], [-10, 0, -10, -30, 0, 0], [0, -13, 13, -33, -33, 15], [-21, -13, 0, 0, 29, 21], [0, 38, -38, 22, 22, 36], [26, -6, 0, 0, -22, -26], [-8, 22, 0, 0, 38, 8], [-18, -18, 0, 0, -16, 18], [-14, 0, 36, -42, 0, 0], [-40, 39, 0, 0, 32, 40], [34, 0, -33, 36, 0, 0], [0, 1, -1, -30, -30, 20], [0, 24, -24, 14, 14, 10], [-34, 14, 0, 0, 34, 34], [22, 0, -1, -12, 0, 0], [0, -13, 13, -9, -9, -53], [13, 0, -13, -3, 0, 0], [-14, -18, 0, 0, -23, 14], [14, 0, 6, 8, 0, 0], [22, -16, 0, 0, -8, -22], [0, 30, -30, 40, 40, -4], [0, 32, -32, 12, 12, 32], [0, -2, 2, 31, 31, 24], [6, 0, 12, -12, 0, 0], [0, 4, -4, 38, 38, 12], [0, -14, 14, -29, -29, 4], [44, 0, -4, 12, 0, 0], [-6, 0, 26, -14, 0, 0], [-52, -4, 0, 0, 15, 52], [0, -40, 40, -9, -9, -10], [0, -11, 11, -8, -8, -32], [21, 23, 0, 0, 15, -21], [0, -6, 6, -5, -5, 54], [42, 0, 12, -24, 0, 0], [0, -33, 33, -16, -16, -24], [0, 39, -39, 6, 6, 16], [52, -62, 0, 0, -36, -52], [-22, -12, 0, 0, 7, 22], [32, 0, -20, 20, 0, 0], [-22, -20, 0, 0, -12, 22], [-54, 0, -4, -16, 0, 0], [16, 0, -4, 28, 0, 0], [-44, 0, -4, -11, 0, 0], [8, 0, 10, 21, 0, 0], [0, -7, 7, -11, -11, -31], [0, -6, 6, -10, -10, 18], [-10, 3, 0, 0, -16, 10], [0, -23, 23, 23, 23, 31], [32, -34, 0, 0, -8, -32], [0, -10, 10, -1, -1, 22], [0, -24, 24, -2, -2, 36], [-18, 0, -27, 12, 0, 0], [14, 0, 8, 25, 0, 0], [10, -39, 0, 0, -12, -10], [-3, 29, 0, 0, 39, 3], [30, -46, 0, 0, -44, -30], [0, 35, -35, 3, 3, 23], [4, 0, -30, 1, 0, 0], [14, 5, 0, 0, -34, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1014_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1014_2_e_k();".
// 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_1014_2_e_k(:prec:=6)
    chi := MakeCharacter_1014_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(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_1014_2_e_k();".
// 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_1014_2_e_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1014_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -4, 3, 1]>,<7,R![1, -4, 13, -14, 13, 3, 1]>],Snew);
    return Vf;
end function;