
// Make newform 6034.2.a.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6034_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6034_2_a_j();".
// 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_6034_2_a_j();".
// 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 := [2, 1, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, -4, -2, 1], [3, -2, -3, 1]];
    Rf_basisdens := [1, 1, 1, 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_6034_a();"
function MakeCharacter_6034_a()
    N := 6034;
    order := 1;
    char_gens := [1725, 869];
    v := [1, 1];
    // 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;

function MakeCharacter_6034_a_Hecke(Kf)
    return MakeCharacter_6034_a();
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], [-1, 1, 0, 0], [0, 0, 0, 1], [1, 0, 0, 0], [-3, 0, 1, 0], [3, -1, 0, 1], [-4, -1, 0, 0], [-1, 1, -2, 0], [-2, 2, -1, 1], [5, 0, 1, 0], [-2, -1, 0, -2], [2, -1, -1, 0], [-3, -1, -3, 5], [0, 0, -4, -2], [5, 1, 0, -3], [-5, 2, 1, 0], [-6, 1, -1, 1], [3, -1, 1, -1], [-6, -1, 0, 2], [-3, 4, -4, 1], [-1, 3, -2, -1], [8, -2, 2, -2], [-6, -1, -1, 2], [3, -6, 3, -3], [2, 2, 3, -3], [-1, -2, 2, -5], [-9, -1, 1, 1], [0, -3, 3, -4], [2, -1, 2, -2], [0, -6, 4, -2], [4, 1, 0, 0], [5, 1, 2, -7], [2, -3, 2, -2], [-2, 3, 6, -3], [-2, 0, -5, -1], [3, 0, 6, 2], [7, 1, -1, -4], [3, -4, 3, 2], [1, 2, 4, -5], [1, 3, 6, -8], [-3, 2, 2, 1], [15, -1, -3, -1], [-8, -2, 1, -2], [-14, 2, -7, 4], [2, -6, 2, 1], [5, -4, -1, 5], [-2, -2, 7, 2], [9, -4, 2, 4], [6, -6, 1, 1], [-7, 7, 0, 2], [-6, -1, -1, 4], [-11, 1, -9, 5], [-5, 6, 8, -1], [10, -9, 2, 0], [0, 0, -1, 2], [16, -3, 7, -3], [-3, -5, -5, 1], [-5, 7, -5, 5], [7, 6, -6, -2], [-6, 3, -10, 2], [-16, 6, 3, 1], [-6, -10, 6, 0], [-17, -3, 0, -3], [-5, -2, -10, 5], [-8, 13, -3, -2], [6, 3, -9, -6], [0, 8, -6, 4], [1, 4, 5, 7], [4, 6, -1, 4], [0, -2, 7, -12], [13, 0, 3, -2], [-16, 6, 5, -6], [15, 5, 3, -9], [3, -4, -3, 3], [-2, 10, -5, -2], [0, 7, 4, -4], [-8, -7, 3, -3], [10, -3, 9, -3], [0, -9, 5, 8], [10, -6, 5, -2], [-23, 0, -8, -1], [1, 7, 0, 3], [1, 0, 0, 0], [1, -1, -7, 0], [5, -3, 10, 4], [-23, 2, 4, 0], [-5, 1, -3, 2], [-5, -1, 2, -1], [-22, 7, -9, 3], [12, -11, 3, 5], [9, 0, 11, -15], [-12, 13, -4, -2], [-14, 2, -8, 8], [-2, -2, 8, -2], [-5, -5, 7, -7], [-25, 12, -3, 2], [-6, 7, -7, 12], [8, 6, -4, -10], [-4, 1, 9, 2], [-11, 2, -4, 10], [1, -8, 7, -10], [1, 17, -9, -3], [-13, 6, -4, 7], [-22, 10, -4, 1], [-12, -1, -4, 2], [7, 0, -2, 7], [1, -16, 6, -1], [27, -5, 4, -4], [8, -6, 4, -4], [-6, -1, 11, -4], [9, 5, 0, -9], [-13, 0, 1, -7], [-27, 7, -3, 3], [-6, 12, -13, 2], [6, -4, -1, -11], [-26, -7, 2, 4], [-20, -1, -7, 10], [-3, 7, 1, 8], [0, -6, -7, -2], [-4, 7, -3, -5], [11, 3, 3, 4], [11, -3, -7, -1], [-2, -10, 1, 1], [-5, 10, -11, 5], [-33, 6, 5, -5], [19, -9, 6, -1], [-17, -1, 1, 8], [7, 5, 5, 5], [-29, 0, -13, 5], [10, -11, -9, -3], [-14, 4, -1, 10], [-15, 7, -10, -7], [1, -10, -3, 4], [19, -6, -8, 9], [21, 11, -4, -15], [5, -5, -8, -10], [19, -11, -6, 6], [0, -6, -14, 14], [18, 12, -6, -12], [-2, 7, 6, -18], [-18, 11, 1, 7], [-4, -7, 2, -2], [4, -13, 6, 1], [7, -8, 3, -12], [-2, -15, 9, -6], [14, 4, 12, 8], [14, -1, 11, 2], [2, -19, 8, -6], [-20, 5, 0, 2], [23, -10, 14, -9], [24, -7, 19, -11], [-7, -8, 0, 10], [10, 6, 8, 10], [27, -12, -11, 0], [-34, 16, 5, -5], [-4, -3, 0, -9], [2, -6, 10, 5], [-12, -10, -7, 10], [3, -8, 6, 0], [-4, 4, -8, -8], [-22, 0, 9, -2], [-14, 16, 0, -9], [-19, 8, -17, -3], [3, -6, 14, 7], [-19, 12, -12, 6], [8, -16, 2, 3], [17, 2, -2, -11], [38, -6, -2, -1], [-15, -6, 5, 6], [16, 0, -2, 10], [-17, 5, -3, 14], [17, -5, -3, -8], [0, 13, -3, 13], [6, -3, 2, 9], [13, -2, -4, 14], [-8, -18, 3, -6], [7, 9, -8, 3], [-6, 8, -5, 10], [-9, 3, 14, 7], [7, 12, -6, 1], [-15, 7, 10, -6], [7, -4, -2, -1], [15, 11, 4, -1], [7, -1, 18, 1], [23, -12, 7, 7], [29, 4, -6, 3], [6, -14, -7, 8], [10, -23, 3, 1], [29, 1, 18, -17], [-15, 9, -17, 7], [-13, -5, -12, 10], [12, -4, -1, -17], [3, 7, 3, -19], [-10, 6, -8, 11], [-42, 9, -2, 0], [-11, 16, 11, 1], [-20, 19, -5, 0], [3, -3, -16, 19], [-7, 11, 1, -13], [0, 0, -5, 16], [-7, -1, 8, -3], [-8, -7, -2, 15], [8, -8, 3, -3], [-16, -13, -2, 10], [-36, -1, -4, 1], [36, -9, 12, -10], [8, 1, 16, 2], [-2, -12, -8, 26], [-6, 0, -8, -10], [-6, 10, -11, -2], [11, -11, 12, 10], [-7, 23, -7, 2], [-13, -5, -14, -10], [-19, -10, 5, -3], [13, 11, -11, 7], [-22, -6, 3, -18], [-11, 8, 14, -1], [4, 14, -3, -6], [5, -1, 1, -9], [0, 7, 6, -4], [-6, 8, 16, -13], [2, 0, -3, -9], [-6, 1, 0, -14], [17, 0, 12, 1], [21, -7, -2, 3], [37, 3, 14, -19], [-12, 1, 7, -10], [24, -7, 19, -14], [-7, 18, -15, 7], [-47, 11, -5, 1], [2, -16, -1, 12], [11, -29, 16, 1], [-16, -3, -16, 4], [-20, 12, 9, 10], [48, -7, 25, -14], [23, 7, 18, -18], [36, -21, -3, 1], [-25, -8, 9, 10], [11, 6, -7, -11], [14, -12, 18, 1], [29, 7, 4, 3], [24, 11, -2, -19], [2, 10, 6, -9], [15, 9, -7, 12], [42, -7, 3, 6], [-16, 16, -13, 2], [19, -7, -6, 11], [-28, 4, 3, -9], [38, -7, -13, -2], [-23, 11, -14, 6], [16, -19, 1, 13], [-24, 12, -4, -4], [-30, 9, 3, 12], [20, 6, -11, 16], [30, 16, -5, 4], [-2, -20, -8, 18], [19, -6, 10, -12], [16, 3, 2, 10], [12, 1, -6, -16], [37, 1, 11, -18], [-24, 5, 9, 0], [-31, 11, -19, 7], [-40, 5, -9, -6], [16, -26, -3, -2], [1, -2, 17, 5], [38, -8, 27, -14], [-27, 10, 14, -1], [-22, -6, 0, -14], [39, -11, 0, 5], [19, 2, -15, -11], [-19, 19, -22, 11], [-12, 26, -13, 5], [-2, -2, 8, 16], [0, 3, 6, 20], [50, 1, 5, 3], [-30, 9, -2, 8], [-25, 17, -22, -3], [30, -1, 9, 8], [-25, -2, 13, -1], [19, 1, 24, 5], [-27, -4, 16, 6], [16, -2, -13, -12], [2, -15, -6, 20], [-21, 22, -9, 11], [-19, 25, 12, -1], [6, 17, 8, 3], [0, 13, -1, -13], [24, -3, -16, 2], [-1, -8, 17, 1], [1, -7, 1, 12], [5, -23, 4, 13], [7, -20, -1, 6], [3, -2, 4, -21], [14, -30, 5, -7], [6, 4, -20, 7], [17, 11, -4, -3], [-6, 25, -15, 4], [11, 11, -17, 23], [-22, 6, -2, 12], [-8, 0, -1, 25], [8, 12, 6, -6], [-22, -4, -1, -20], [-1, -11, 5, -19], [17, -37, 21, 5], [-2, -10, 5, -6], [-33, 34, -19, 3], [-24, -5, -2, -6], [23, 10, 18, -4], [21, 3, 11, -3], [25, 13, -14, 16], [3, -8, -15, 25], [7, -16, 4, -16], [-31, 21, 12, -7], [-28, -7, -14, -12], [-27, -9, 9, 18], [-13, 27, -11, -11], [-32, 5, -5, 9], [-13, 15, 6, -13], [-16, 11, -4, -12], [-47, -6, 5, -6], [-33, -17, 0, 1], [13, -12, -22, 5], [9, -11, -4, 0], [34, -15, -13, 4], [18, -5, 12, -28], [-24, -8, -22, 29], [-18, 5, -13, -13], [-23, 0, -16, 12], [-26, -16, 9, -14], [-23, 15, -2, -8], [-14, 7, 9, 6], [-12, 1, -19, 25], [-18, 8, 13, -30], [22, -26, -16, 5], [9, -1, -7, 11], [18, 0, 12, -8], [45, -23, 20, -10], [-3, -16, -5, 30], [9, -10, 14, -25], [-9, -1, 6, 1], [12, -14, 16, 14], [5, 0, 2, -11], [-16, 10, 20, 14], [-6, -5, 18, 16], [8, -2, -4, 2], [13, -14, 7, -22], [5, 10, 17, -33], [-44, 2, -10, 11], [-8, -2, -29, -11], [-15, 14, -2, 15], [-12, -3, -7, 22], [13, 22, -12, -5], [-17, -1, -10, -13], [9, -20, 21, -24], [-11, 12, -16, 7], [-4, 29, -20, 25], [17, -5, 14, -12], [-34, 8, 13, 1], [61, 0, 17, -7], [-9, 5, -7, -9], [21, 12, 13, -39], [30, -4, -15, 20], [-9, 21, -5, -19], [-16, -18, 7, 24], [-5, 7, 2, -13], [8, 16, -23, -19], [31, -17, -9, 9], [24, 4, 9, 11], [0, 0, 4, 1], [-17, 2, 1, 17], [5, -9, -14, 27], [-5, 16, -13, 10], [16, 17, 14, -21], [-19, -4, 23, 22], [48, -20, 3, 8], [6, 7, 7, -11], [-14, 3, -19, 21], [1, 13, -9, 19], [-23, 0, -10, -18], [8, 9, 10, 16], [-22, 17, 5, -24], [-11, -9, 21, -7], [44, -33, 27, -1], [-16, -13, -1, 0], [-42, -6, 13, 2], [27, -5, 2, -5], [14, -4, -3, -2], [-30, 19, 6, 14], [40, -16, 31, 1], [6, -4, 13, -30], [3, -13, 9, 15], [26, 5, 19, 6], [23, -28, -13, 7], [11, -8, 19, 12], [-12, -23, 0, -3], [11, 16, -10, 11], [-3, -21, -18, 29], [-68, 12, -16, 0], [-16, -4, -2, 28], [1, 22, -14, 1], [-61, 1, -5, -1], [22, 10, -9, -2], [3, -25, -10, 7], [-7, 2, 22, 9], [-34, 6, -18, 2], [5, -33, 18, 4], [-48, -3, -14, 14], [9, -3, -20, 7], [32, -18, 5, -9], [-17, 2, -25, 25], [50, 4, -2, -4], [34, -12, 4, 0], [-15, 23, -17, -3], [-14, 10, 15, -11], [-38, -18, 12, 20], [10, -21, -21, -3], [40, -17, 16, -20], [-74, 4, 3, 6], [-14, 12, 12, 8], [5, -14, 14, 5], [-26, -26, 17, 8], [-47, 34, -15, 3], [15, 3, -1, -25], [24, 5, 5, 1], [-55, 14, -22, 19], [-7, -14, -9, 9], [32, -13, 12, -14], [-34, -7, 6, 22], [19, -24, 25, -1], [23, -20, -24, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6034_2_a_j();".
// 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_6034_2_a_j(:prec:=4)
    chi := MakeCharacter_6034_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6034_2_a_j();".
// 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_6034_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, -11, -5, 2, 1]>,<5,R![-1, 6, -7, -1, 1]>,<11,R![47, 88, 55, 13, 1]>],Snew);
    return Vf;
end function;