
// Make newform 1014.2.e.n in Magma, downloaded from the LMFDB on 17 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_n();".
// 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_n();".
// 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], [-1, 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 := [[0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, -2, -1, 0, 0], [2, 1, 0, 0, 2, -2], [0, 1, -1, 3, 3, 3], [0, 0, 0, 0, 0, 0], [2, 2, 0, 0, 0, -2], [-4, 4, 0, 0, 4, 4], [0, -4, 4, -2, -2, -2], [0, -3, 3, -1, -1, -5], [-5, 0, 5, -5, 0, 0], [0, 6, -6, 2, 2, 0], [0, -2, 2, -6, -6, -2], [-6, 4, 0, 0, 2, 6], [-4, 0, 4, -4, 0, 0], [4, 0, -2, 5, 0, 0], [-4, 5, 0, 0, 2, 4], [-8, 2, 0, 0, 0, 8], [0, -2, 2, 2, 2, 2], [-8, 4, 0, 0, 2, 8], [8, 0, -4, 7, 0, 0], [10, 0, 3, 2, 0, 0], [0, 0, -3, 10, 0, 0], [0, 8, -8, 6, 6, 0], [-8, 4, 0, 0, -3, 8], [0, -3, 3, -4, -4, 2], [-2, 0, 0, -5, 0, 0], [0, 7, -7, 6, 6, 4], [16, 0, -8, 4, 0, 0], [4, 0, 0, 0, -8, -4], [0, -4, 4, 3, 3, 2], [-2, 0, 2, 5, 0, 0], [-4, 0, 0, 0, 0, 4], [2, 4, 0, 0, 10, -2], [-3, -7, 0, 0, -1, 3], [-7, 0, -9, -3, 0, 0], [-16, 0, 4, -2, 0, 0], [10, -10, 0, 0, 0, -10], [0, -8, 8, -6, -6, 0], [5, 5, 0, 0, 3, -5], [0, -6, 6, 3, 3, -10], [8, 0, 0, -2, 0, 0], [-2, 6, 0, 0, -4, 2], [0, -8, 8, -12, -12, -2], [0, 9, -9, -8, -8, -6], [9, -9, 0, 0, -1, -9], [0, 2, -2, 8, 8, -4], [0, -8, 8, 4, 4, 12], [2, -6, 0, 0, 3, -2], [-6, 0, 14, 0, 0, 0], [-8, 0, 6, -2, 0, 0], [14, 0, -12, 2, 0, 0], [8, 4, 0, 0, 5, -8], [8, -3, 0, 0, 2, -8], [0, -2, 2, 0, 0, 10], [0, 14, -14, 14, 14, 20], [-12, -6, 0, 0, 1, 12], [0, 4, -4, -3, -3, 2], [10, -20, 0, 0, -6, -10], [-12, 0, -8, 6, 0, 0], [0, 8, -8, 12, 12, 6], [15, -21, 0, 0, -11, -15], [2, 0, -10, -8, 0, 0], [2, 0, -4, 2, 0, 0], [12, 0, -12, 7, 0, 0], [22, 0, -3, 4, 0, 0], [4, -6, 0, 0, -2, -4], [9, 0, 11, 5, 0, 0], [0, -1, 0, 0, 2, 0], [0, -16, 16, -2, -2, -12], [0, -18, 18, 0, 0, -6], [14, 0, 0, 14, 0, 0], [0, -9, 9, 2, 2, -2], [-12, -2, 0, 0, 20, 12], [0, 8, -8, 0, 0, 12], [-2, -4, 0, 0, -2, 2], [3, 0, 15, 11, 0, 0], [-28, 2, 0, 0, 8, 28], [0, 2, -2, -8, -8, 6], [0, 1, 0, 0, 0, 0], [0, -23, 23, -9, -9, -17], [-4, 0, 6, -16, 0, 0], [0, 4, -4, 2, 2, 24], [-25, 9, 0, 0, 9, 25], [0, -21, 21, -22, -22, -14], [1, 0, -15, 21, 0, 0], [0, -12, 0, 0, 8, 0], [0, 1, -1, -3, -3, -15], [10, -15, 0, 0, -4, -10], [-4, 0, 12, 4, 0, 0], [-18, 0, 18, -17, 0, 0], [0, 10, -10, -6, -6, -6], [2, -12, 0, 0, -27, -2], [0, -7, 7, -13, -13, -17], [0, 0, 0, 0, 0, 0], [-4, 6, 0, 0, 26, 4], [0, 4, -4, -12, -12, 2], [10, 0, 16, 4, 0, 0], [0, -2, 2, 10, 10, 4], [14, 0, -6, -2, 0, 0], [4, 0, -16, -4, 0, 0], [0, 8, -8, 0, 0, -6], [22, -18, 0, 0, -25, -22], [0, 4, -4, 4, 4, 0], [10, 0, 0, -8, 0, 0], [-16, 0, 11, -28, 0, 0], [0, 23, -23, 25, 25, 5], [-14, 0, 24, -4, 0, 0], [6, 0, 14, 20, 0, 0], [0, -3, 3, 5, 5, 5], [-6, 1, 0, 0, 2, 6], [0, -18, 18, -10, -10, -26], [4, 14, 0, 0, 12, -4], [-16, 0, 8, -10, 0, 0], [-14, 27, 0, 0, 18, 14], [-26, -12, 0, 0, 14, 26], [6, -6, 0, 0, -20, -6], [0, 10, -10, -18, -18, -8], [0, 23, -23, 16, 16, 2], [-4, -3, 0, 0, -10, 4], [0, -18, 18, -6, -6, 8], [0, 21, -21, 20, 20, 16], [-14, 0, 15, -8, 0, 0], [-14, 22, 0, 0, 21, 14], [0, 4, -4, -16, -16, -14], [18, 0, 16, -4, 0, 0], [-20, 2, 0, 0, -2, 20], [-22, 0, 0, 0, -16, 22], [14, 0, 19, 6, 0, 0], [-2, 0, -10, 4, 0, 0], [0, -18, 18, -34, -34, -6], [0, 6, -6, 18, 18, -10], [0, 3, -3, -13, -13, -17], [0, -16, 16, -8, -8, 8], [-4, 8, 0, 0, -16, 4], [0, 4, -4, -8, -8, 6], [6, 7, 0, 0, 4, -6], [10, -14, 0, 0, -10, -10], [5, 17, 0, 0, 23, -5], [0, 6, -6, -6, -6, -30], [16, 0, 20, 12, 0, 0], [0, 10, -10, 15, 15, -8], [10, 5, 0, 0, 6, -10], [-9, 0, -1, 7, 0, 0], [0, -8, 8, 18, 18, 10], [0, -2, 0, 0, -18, 0], [38, 0, -6, 2, 0, 0], [0, 0, -26, 4, 0, 0], [-24, 0, 30, -8, 0, 0], [6, 0, -10, -14, 0, 0], [-14, 2, 0, 0, 2, 14], [0, -14, 14, -8, -8, 18], [-6, 0, 6, 4, 0, 0], [0, 14, -14, 22, 22, -2], [0, 8, -8, 24, 24, 6], [-6, 0, -18, -16, 0, 0], [6, -16, 0, 0, -17, -6], [36, 6, 0, 0, -14, -36], [15, 0, -15, 15, 0, 0], [-1, 0, 3, 19, 0, 0], [0, -10, 10, -1, -1, -30], [-2, 0, 0, 0, 26, 2], [-13, 0, -27, -1, 0, 0], [13, 13, 0, 0, -17, -13], [0, -26, 26, -4, -4, -2], [-18, 0, -14, 28, 0, 0], [0, -8, 8, -29, -29, -18], [16, -28, 0, 0, -26, -16], [11, 0, -15, -9, 0, 0], [-30, 0, 0, -16, 0, 0], [20, 0, 8, 0, 0, 0], [26, 2, 0, 0, -24, -26], [-36, 14, 0, 0, 18, 36], [34, -32, 0, 0, -12, -34], [4, 0, -28, 12, 0, 0], [2, -16, 0, 0, -18, -2], [0, 4, -4, 6, 6, 38], [-26, 0, -13, -8, 0, 0], [0, 32, -32, -4, -4, 0], [0, -36, 36, -18, -18, -16], [30, 0, -12, 15, 0, 0], [9, 0, -3, 17, 0, 0], [38, 0, -22, 20, 0, 0], [-6, 0, 16, -18, 0, 0], [0, -6, 6, -6, -6, -56], [22, -3, 0, 0, -20, -22], [34, 0, -30, 36, 0, 0], [22, 0, 4, 6, 0, 0], [0, -9, 9, -8, -8, -8], [-10, -4, 0, 0, 18, 10], [23, -15, 0, 0, 5, -23], [20, 9, 0, 0, -2, -20], [-18, 0, 6, -24, 0, 0], [0, -26, 26, -15, -15, -24], [-34, 2, 0, 0, -7, 34], [0, 20, -20, 4, 4, 40], [11, 0, -15, 3, 0, 0], [-28, 40, 0, 0, 18, 28], [-20, 0, 22, -24, 0, 0], [-14, 0, -22, 16, 0, 0], [-22, 24, 0, 0, 36, 22], [-37, -7, 0, 0, 9, 37], [0, 0, 0, 28, 28, 20], [16, 0, -28, 35, 0, 0], [0, 29, -29, 18, 18, 20], [0, -31, 31, -45, -45, -9], [-10, 0, 20, -29, 0, 0], [-12, -13, 0, 0, 14, 12], [0, 30, -30, 32, 32, 20], [-28, 14, 0, 0, 22, 28], [0, 8, -8, 13, 13, -36], [-13, 0, 15, -21, 0, 0], [0, 8, -8, -4, -4, -24], [-44, 12, 0, 0, 0, 44], [-14, 14, 0, 0, -12, 14], [-12, 0, 9, -48, 0, 0], [45, 0, 3, -7, 0, 0], [12, 16, 0, 0, -8, -12], [0, -16, 16, -30, -30, -26], [28, 0, -10, 23, 0, 0], [0, 16, -16, -8, -8, 6], [-34, 0, 20, 1, 0, 0], [-32, 0, -18, -2, 0, 0], [-13, -19, 0, 0, -3, 13], [0, 24, -24, -4, -4, -4], [-8, 0, 2, 0, 0, 0], [0, 16, -16, 4, 4, 34], [4, -28, 0, 0, -2, -4], [26, 12, 0, 0, 11, -26], [18, 0, 2, 5, 0, 0], [0, -14, 14, -34, -34, -26], [0, -16, 16, -52, -52, -6], [0, 15, -15, 10, 10, 62], [-16, 0, 32, -22, 0, 0], [28, 0, 16, 2, 0, 0], [10, 0, 4, -18, 0, 0], [4, 13, 0, 0, 4, -4], [0, 19, -19, 40, 40, 22], [-4, -20, 0, 0, 4, 4], [0, 30, -30, -18, -18, -10], [0, -17, 17, -35, -35, 25], [0, 4, -4, -6, -6, 10], [-28, 8, 0, 0, -4, 28], [0, 16, -16, 44, 44, 4], [32, -8, 0, 0, 8, -32], [-18, 0, -10, -18, 0, 0], [15, 9, 0, 0, 1, -15], [0, -9, 9, -31, -31, 25], [-36, 2, 0, 0, -10, 36], [0, 18, -18, 10, 10, 10], [0, 2, -2, 14, 14, -12], [0, -30, 30, -36, -36, 6], [-14, 0, 20, -26, 0, 0], [0, 9, -9, 16, 16, 16], [-46, 0, 61, -28, 0, 0], [4, 29, 0, 0, 34, -4], [-2, -24, 0, 0, -34, 2], [0, 38, -38, 14, 14, 38], [6, 0, -3, -28, 0, 0], [-5, 29, 0, 0, 41, 5], [-27, 0, 7, 21, 0, 0], [0, -18, 18, 13, 13, -10], [-14, 0, -14, -16, 0, 0], [0, 12, -12, 28, 28, -2], [0, 6, 0, 0, -24, 0], [36, 16, 0, 0, 4, -36], [0, 2, 0, 0, -1, 0], [22, 0, 20, 12, 0, 0], [32, -24, 0, 0, -6, -32], [12, 22, 0, 0, 3, -12], [-32, 0, 28, -20, 0, 0], [-6, 0, 38, -18, 0, 0], [0, 28, -28, 33, 33, 4], [6, -20, 0, 0, -11, -6], [-32, -5, 0, 0, -8, 32], [0, 5, -5, -7, -7, 5], [18, -46, 0, 0, -33, -18], [18, 0, -4, 40, 0, 0], [-16, 1, 0, 0, 40, 16], [32, -45, 0, 0, -10, -32], [0, -22, 22, -24, -24, 4], [0, 8, -8, -19, -19, 10], [0, 0, -16, -32, 0, 0], [0, 0, 0, -12, -12, -50], [26, 0, 8, 32, 0, 0], [-8, 0, 40, -44, 0, 0], [12, 0, -12, 37, 0, 0], [40, 0, -30, 35, 0, 0], [1, -5, 0, 0, -21, -1], [14, -6, 0, 0, 6, -14], [0, -33, 33, -40, -40, -34], [-3, -27, 0, 0, -37, 3], [0, -14, 14, -16, -16, 40], [-30, -26, 0, 0, 19, 30], [-48, 48, 0, 0, 30, 48], [-26, 0, 7, 28, 0, 0], [14, 0, -20, 5, 0, 0], [0, -19, 19, 12, 12, 42], [0, -15, 15, 27, 27, 15], [0, -38, 38, -12, -12, -34], [7, -19, 0, 0, -27, -7], [28, 0, -14, -17, 0, 0], [0, 19, -19, 10, 10, 74]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1014_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-29, -16, 1, 1]>,<7,R![1, 20, 409, -182, 61, -9, 1]>],Snew);
    return Vf;
end function;