
// Make newform 4704.2.a.bx in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4704_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_4704_2_a_bx();".
// 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_4704_2_a_bx();".
// 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, -3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [1, -2, -2, 1], [-3, -4, 2, 0], [-2, 4, 2, -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_4704_a();"
function MakeCharacter_4704_a()
    N := 4704;
    order := 1;
    char_gens := [1471, 1765, 3137, 4609];
    v := [1, 1, 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_4704_a_Hecke(Kf)
    return MakeCharacter_4704_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 := [[0, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 1], [0, 0, 0, 0], [-1, 0, -1, 0], [2, 0, -1, -1], [1, 1, 1, 0], [-2, -1, -1, -1], [-3, 0, 1, 0], [0, 1, -1, 1], [2, 3, -1, -1], [0, -4, 0, 0], [5, -3, -1, -2], [2, -2, 0, 2], [4, -1, 1, 3], [0, 2, 0, 2], [0, 3, 1, 3], [4, 0, 1, -1], [2, 4, 2, 0], [-3, 4, 1, 0], [2, 1, 0, 2], [-4, 2, -2, -2], [0, 2, 2, 2], [7, 5, -1, 0], [10, -3, 0, 2], [1, -2, 0, 1], [6, 3, 3, -1], [-3, 0, 1, 4], [-6, 2, 0, -2], [-2, 8, 0, 0], [2, 0, -2, -4], [-4, -6, -2, 2], [4, 5, -1, -3], [-4, -8, 0, 0], [4, -10, 0, -2], [10, -6, 0, 2], [0, -2, 1, -5], [2, -8, 2, 0], [4, 3, -3, -1], [7, -10, 0, -1], [-7, 8, -3, -4], [14, 6, 3, -1], [5, 12, 1, 0], [0, 4, 2, -4], [4, 2, 0, -6], [4, 4, 0, 4], [12, 0, 0, -4], [-4, -8, 4, 0], [-8, -9, -3, -1], [10, 4, -3, 1], [0, -9, 1, -1], [-3, 0, 1, 0], [10, 5, -2, 4], [-4, 5, -5, -3], [5, 3, 1, -4], [-1, -8, -1, 4], [11, 8, 0, 3], [6, 3, -5, -1], [0, 4, -6, -4], [-8, 3, 5, 3], [-10, 3, 3, 3], [9, 4, -2, 3], [6, -11, 1, 5], [12, -3, 3, 1], [4, -1, 0, -4], [0, -6, 4, 2], [12, 0, -4, 0], [4, -2, -2, -2], [-5, 8, 3, 0], [0, -12, -1, -3], [-5, -1, 3, 0], [-1, -4, -1, 4], [4, 2, 2, -6], [2, -4, 4, -4], [8, -12, -4, -4], [8, -8, 0, -8], [12, 1, 3, 1], [-8, 2, -1, -3], [-16, -1, 1, -1], [-2, -5, 6, 0], [4, 13, 3, -3], [-4, 0, 2, 8], [-7, -8, 5, 0], [0, -3, -4, -4], [24, 8, 0, 0], [1, 4, 1, -8], [-18, 0, 0, 0], [10, 0, -4, 0], [3, 4, -6, -7], [10, -4, -2, 4], [-12, 9, -1, -7], [0, -5, 1, -1], [26, -6, 0, 2], [11, -8, -1, -4], [-2, -10, 0, -2], [20, 2, 2, -6], [3, -10, -2, 5], [7, -9, 3, 4], [-16, 10, 2, 10], [-8, 4, -2, -4], [0, 2, 10, 6], [8, 2, -4, 2], [-8, 3, 1, -5], [-4, 9, -5, -7], [12, 2, -2, -2], [4, 13, 0, 4], [-24, 3, 1, 3], [-15, 7, -1, 2], [-11, 4, 1, 8], [-4, 15, 0, -4], [8, 12, -4, 4], [2, -12, -2, -8], [-16, -11, -5, 5], [-12, 14, -6, -10], [18, -4, -2, -4], [-24, -1, 1, 7], [14, -13, 3, 3], [-4, -5, -3, -1], [8, 5, -5, -3], [-13, -4, -1, -4], [20, -6, 5, 3], [6, 4, 6, 4], [-17, -2, -4, -5], [-15, 4, 1, 8], [4, -6, -2, -6], [0, -17, 1, -1], [-14, 18, 0, 6], [-8, 10, -2, -6], [22, -17, -1, -5], [-6, 16, -3, -7], [6, 12, 2, 4], [-3, 12, -3, -4], [0, 20, 0, 0], [-6, -14, 0, -2], [-9, 17, -5, -4], [-16, 9, 0, 8], [-17, 6, 4, -5], [4, 14, 2, 6], [-11, 10, 4, 9], [2, -4, 4, -12], [-4, -6, 6, 10], [12, 22, 4, -2], [16, 10, 2, -2], [-35, -12, -3, 0], [-2, -2, 1, 1], [0, -1, -3, -5], [-16, 0, 7, 5], [15, -3, -1, 0], [-10, -3, -7, -3], [-27, 12, 1, 0], [6, 20, 2, 0], [-11, 3, -1, 6], [22, 6, -8, -2], [20, -7, 7, 5], [10, -18, -4, -2], [5, -16, 1, 0], [-14, -6, 4, 6], [3, 11, -3, -2], [-32, 5, 2, -2], [11, 10, 2, 1], [-15, -8, 1, 0], [-2, 8, -4, 0], [-6, 16, -2, -12], [-8, 4, 0, -4], [-28, 1, 3, -7], [20, 4, 0, -4], [10, -10, 8, 10], [4, -6, -3, 11], [12, -4, 2, 4], [-15, -24, 2, -1], [-3, -12, -3, -8], [0, -10, 3, 1], [21, -8, 5, 4], [2, -14, 4, 10], [-12, 4, 0, 4], [-5, -15, 1, -6], [12, 10, 10, 2], [20, -5, 1, -5], [10, -5, -1, 7], [-28, -10, -3, -5], [-4, -2, -10, -10], [28, 12, 4, -4], [-20, 6, -6, -6], [-9, 17, 5, 2], [9, 4, -7, 4], [-1, 24, -2, 1], [14, 8, 4, 8], [-12, -12, 4, -12], [2, 6, -8, -10], [-12, -6, 2, 2], [-10, -7, -2, 8], [7, 0, 3, 4], [28, -16, 0, -4], [-15, -16, 6, -5], [-15, -8, -11, -4], [-15, 3, 7, 10], [18, 0, -10, -8], [-36, -2, 2, 2], [11, -1, -7, 2], [4, 27, -3, 7], [-4, -6, 4, 10], [-32, 6, -6, -10], [10, 6, -7, 5], [10, -1, 8, 2], [-20, 2, 6, 10], [-17, -8, 2, 5], [-20, 8, 4, 0], [31, 0, 7, 8], [6, -4, 12, 4], [10, -13, -9, -13], [-12, 16, 4, 8], [7, -6, -6, 5], [-2, -6, 12, 10], [-4, 2, -10, -6], [-16, -9, 5, 3], [12, 3, -6, -6], [-2, 14, -4, 10], [23, -11, 5, 10], [-15, -16, 9, 4], [-20, 17, 3, 1], [-26, 10, 4, -2], [34, -3, 5, 1], [-2, 16, 12, 8], [36, 12, -4, 8], [8, 0, -4, -8], [-32, -8, 4, -4], [15, 19, -3, 2], [31, 0, -5, 8], [-2, 7, -9, -13], [9, 12, 5, 12], [10, -10, 4, 10], [22, 1, 5, 1], [34, -10, 0, 10], [22, 12, -12, -4], [-26, 17, 5, 1], [0, -2, 10, 14], [6, 7, 12, 2], [4, -10, 0, -2], [19, -4, 3, -8], [-20, -8, 12, 0], [-5, -24, -5, -8], [16, -8, 4, -8], [-28, -20, 0, 4], [8, 0, 2, -8], [-23, -1, -7, -8], [16, 5, -1, -7], [-2, 5, 1, -7], [8, 13, -1, -3], [-12, -14, 6, -2], [-21, -24, 3, 4], [-10, -12, 0, 4], [3, 5, -7, -6], [1, 4, -3, -16], [22, 15, -8, -2], [-43, -10, 8, -3], [-11, 8, 1, -4], [4, 6, -6, 2], [6, -11, 1, -11], [29, 0, -2, -9], [-12, -21, 2, -6], [28, 10, 2, -6], [7, 16, -1, -8], [2, 0, -5, 11], [-20, -12, 3, -3], [37, -1, 3, -6], [16, 0, 4, 8], [20, 14, 0, -10], [13, -27, 5, 16], [-32, -10, 2, -2], [12, 7, -3, 7], [20, -10, 5, -5], [-36, -12, 8, 4], [-26, -21, -4, -6], [-12, -4, 4, 0], [10, -11, 6, -12], [-10, -1, -1, -5], [-15, 32, 1, -8], [-20, 12, -2, -12], [-8, -4, -6, -12], [32, -17, 5, 7], [-28, 24, 4, 0], [-6, 8, -6, 8], [52, -2, -2, 6], [28, -2, 11, 13], [8, -2, 6, -2], [1, 28, 1, -4], [-2, -16, 4, -16], [16, 2, 4, -6], [14, 7, -1, -13], [7, 9, 7, -8], [-4, 18, -4, -14], [4, -12, 4, 4], [-20, -15, 11, -7], [8, 11, 9, 3], [14, 36, 4, 4], [5, -6, 4, 9], [-32, -18, -6, 6], [-3, -8, -4, -7], [0, 39, -3, -1], [4, 4, 4, -4], [8, -19, -6, -10], [36, -17, 5, 7], [-34, 16, -2, 8], [15, 8, 7, -8], [12, 6, 2, -22], [-4, 6, 6, -18], [31, -4, 3, -4], [42, 8, 7, 7], [11, -20, 3, -4], [-4, -36, 6, 4], [-20, 0, -12, -8], [-16, 7, 1, -17], [4, -1, -3, -1], [-10, -6, -8, -2], [-35, 28, 1, 8], [18, 9, -8, -6], [24, -2, -2, -18], [-3, -4, -7, 8], [-14, 13, 0, 18], [-16, -1, 1, 15], [-2, -5, 3, 11], [-50, -16, 2, 0], [27, 8, -2, 5], [-2, -10, 12, -6], [14, -8, -4, -16], [-6, -5, -10, -20], [-46, 0, -6, 8], [10, -3, -3, -3], [-3, -12, 5, 12], [-4, 33, -5, -7], [-14, -16, 4, 0], [-16, 13, 11, 5], [-26, -5, -5, -9], [20, 23, -7, -1], [-10, -20, -2, 8], [-28, -25, 9, -1], [-22, 8, 2, 12], [-9, 5, 3, -20], [42, -5, -8, 2], [6, 3, 11, 15], [-10, -12, 4, 4], [-4, 7, 13, 7], [13, -28, 0, 5], [-40, 24, -8, -16], [-20, -29, 1, 3], [-31, -8, -7, 8], [14, -6, 1, -7], [2, 18, -4, 6], [-8, -12, -4, -12], [11, -2, 0, -5], [50, -21, -1, -5], [-52, 16, 2, 8], [32, -10, -4, 14], [4, -20, 8, 4], [-18, -16, -16, 0], [27, -15, 1, 10], [-40, -24, 8, -8], [24, 2, 2, -14], [14, 4, 4, 12], [13, -24, -7, -8], [2, 4, -4, 20], [1, 5, 19, 14], [5, 20, 5, -4], [23, 16, 3, 12], [0, -4, 0, -4], [26, -36, 2, 4], [13, 32, 0, -11], [-36, -6, 2, -6], [-8, -36, -6, 4], [3, 20, 3, 8], [-22, -33, 3, -1], [-13, 12, -9, -16], [12, -10, -8, -18], [38, -21, 3, 7], [44, 0, -2, -8], [4, 6, 10, -2], [-45, -16, -9, 0], [-44, 3, 8, 12], [33, -5, 3, 6], [62, 1, -4, -6], [67, 2, 0, -5], [22, 32, 4, 0], [-8, -8, 0, 8], [-30, 24, -4, 0], [0, 6, 6, 22], [-52, -6, 2, 2], [-26, -6, 0, -2], [42, -8, 1, 5], [8, 14, -14, 6], [-24, -1, 5, -13], [2, -2, -8, 6], [11, 16, 12, -1], [-9, -24, 11, 4], [10, -13, -1, 3], [13, 12, -3, 4], [-40, -10, -6, 14], [-12, 12, 0, 12], [-5, -23, -15, -14], [36, 6, 6, 14], [24, 27, 5, 3], [-8, 22, 9, 19], [2, -12, 8, 4], [-6, -12, -2, -12], [37, -15, -5, -14], [-23, 16, 8, 9], [0, -4, -4, 12], [36, -8, -6, 0], [6, -4, 4, 4], [20, -20, 12, 4], [4, 9, -9, -23], [-52, 27, -2, -2], [-12, 21, 7, 21], [-21, -4, -1, -4], [22, -22, 12, 10], [-14, -2, -8, 6], [-17, -16, -4, -5], [-7, -12, -3, 0], [-10, -35, -7, 1], [13, 35, -9, -2], [-8, -35, -9, 9], [-4, -1, -3, 23], [24, 12, 11, 1], [-13, -4, 7, -24], [-52, -6, -2, 2], [-16, 19, 14, 10], [13, -24, 2, 3], [-13, -4, -1, 12], [4, -7, 3, -7], [32, -18, -2, -2], [28, -7, -1, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4704_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4704_2_a_bx();".
// 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_4704_2_a_bx(:prec:=4)
    chi := MakeCharacter_4704_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_4704_2_a_bx();".
// 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_4704_2_a_bx( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4704_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![28, 24, -8, -4, 1]>,<11,R![16, -48, -20, 4, 1]>,<13,R![68, 80, -4, -8, 1]>,<19,R![64, -64, -8, 8, 1]>,<31,R![-64, 128, -40, -8, 1]>],Snew);
    return Vf;
end function;