
// Make newform 6384.2.a.cb in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6384_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_6384_2_a_cb();".
// 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_6384_2_a_cb();".
// 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, 0, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -3, 0, 1], [0, -5, 0, 1], [-5, 0, 2, 0]];
    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_6384_a();"
function MakeCharacter_6384_a()
    N := 6384;
    order := 1;
    char_gens := [799, 4789, 2129, 913, 1009];
    v := [1, 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_6384_a_Hecke(Kf)
    return MakeCharacter_6384_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], [0, 1, 0, 0], [1, 0, 0, 0], [0, 1, -1, 0], [0, 0, 0, 0], [-1, 0, -1, -1], [1, 0, 0, 0], [-1, 0, 2, -1], [1, -1, 0, 1], [2, 1, -1, 0], [1, -1, -1, -1], [-2, 0, 2, 0], [3, 0, 2, 1], [2, 2, -3, 0], [3, -1, 0, -1], [0, 2, 2, 0], [2, 2, -2, 0], [2, 1, -1, -2], [3, 1, -2, 1], [-2, 0, 2, 0], [5, -2, -2, 1], [5, -1, 2, -1], [0, 2, 0, 2], [-2, 1, 1, 0], [0, -1, -4, 0], [7, -3, 1, -1], [3, -1, 2, 1], [3, -1, 5, 1], [-3, 1, -2, 1], [9, 0, 2, 1], [3, -1, 0, 1], [0, 0, 4, -2], [8, -4, -2, 0], [5, 1, 3, -1], [7, 0, 2, -1], [-4, 0, 0, -2], [5, -4, -2, -1], [6, 0, 2, -2], [-2, -2, 0, 0], [1, 1, 2, 3], [-3, -4, -4, -1], [-3, -4, -2, 1], [0, 2, -6, -2], [-3, -1, 3, -1], [6, 2, 4, 2], [6, 5, -3, -2], [12, -1, 1, -2], [0, 4, -6, 0], [-2, -6, 8, 0], [0, 0, -6, 2], [-9, 0, 0, -5], [-4, 3, -1, 2], [-3, -7, -4, -1], [-14, -2, 6, 0], [-7, -6, -2, 1], [6, -2, 2, 4], [8, 4, -4, 0], [7, 0, -4, -1], [1, 3, -6, -3], [-2, 0, -2, -6], [0, 0, 6, 2], [9, 2, 0, -1], [0, 6, 3, 2], [0, 2, -6, -2], [7, 3, -8, -1], [0, -9, -3, 0], [-12, 0, -6, -2], [-8, -3, 5, 0], [6, 0, 0, 4], [-7, -8, 5, 1], [-7, -2, 4, 5], [12, -2, -8, 4], [-1, -3, 5, 1], [14, -1, 1, 2], [8, -6, 8, 0], [23, -3, -1, 1], [-10, 6, -2, 0], [5, -3, 6, -3], [-7, -3, -3, -3], [1, -7, 2, -1], [1, -5, 5, -1], [-11, -5, -2, -1], [4, -3, 9, 2], [4, 5, 3, -2], [-4, -5, 3, -4], [-5, 5, -2, 3], [7, -4, 0, -1], [-2, 1, 2, -2], [24, -4, 4, 0], [-5, -3, 6, 1], [-26, 0, -3, 0], [-1, 8, 0, -1], [-2, -5, 3, -2], [-16, 2, -2, 4], [4, 2, -5, 6], [-2, -2, -2, -4], [-2, 0, -4, -4], [9, 2, 0, -1], [14, 0, 0, 4], [-4, -1, 3, -4], [19, 1, 7, 1], [-10, 4, -2, 2], [7, -1, 2, 3], [-8, -4, 0, 4], [-14, 0, 8, 0], [-5, 1, -12, -3], [-5, 0, 7, -1], [-5, -3, -6, 1], [0, 7, -5, 2], [15, -1, -5, -1], [13, -6, 2, -3], [0, 6, -2, 2], [-4, 0, 2, 4], [1, -2, 4, 3], [7, -1, -10, -1], [0, 2, 4, 4], [-24, -2, 1, -2], [-7, -3, 1, 3], [-9, -1, 6, 5], [6, 6, -6, 2], [-12, 4, 8, -6], [-16, 4, 4, 2], [-11, -5, 0, -1], [16, -4, -4, 4], [-7, 11, 3, -1], [17, 6, -2, -3], [-6, 4, 3, 0], [-16, 6, 10, -4], [-14, -2, 8, 0], [-3, 6, -4, -1], [-1, 3, -6, 9], [3, 2, 0, 3], [6, 2, -10, 4], [-15, 6, -7, 5], [8, 6, -2, -2], [-12, 8, 6, -2], [-15, 0, 10, -5], [-4, -2, 0, 2], [2, -2, -2, 4], [-18, 4, 8, -6], [31, -5, 1, 1], [-1, -6, 16, 1], [9, 1, -2, 3], [-5, 10, -2, 1], [8, -10, -2, -4], [2, 2, -2, 0], [-2, -6, 4, 4], [-16, -10, 2, 0], [-3, 9, -6, 3], [7, -1, -5, -7], [-7, -8, -7, 1], [16, -2, -2, -4], [12, 12, -2, 0], [-2, 5, -11, 2], [5, -13, 2, -1], [-16, 0, 8, -8], [-29, 4, -1, 3], [8, 0, -12, -2], [-20, -2, -2, 6], [8, -3, -9, 4], [-1, 1, 6, -1], [19, 2, -12, 1], [-16, -2, 2, 0], [-15, -7, 6, 1], [-12, -12, 12, 4], [5, 4, -8, -3], [4, 8, 0, -2], [0, 0, -14, -2], [-22, -3, -10, -6], [3, -3, -8, -7], [-8, 12, 8, -4], [-33, 2, 6, -5], [-9, 4, -4, 3], [-13, -1, -1, 3], [19, -2, -9, 7], [11, -6, -12, 1], [11, 5, 1, -7], [22, 3, -3, -4], [16, 6, 6, 2], [-10, -4, 0, 6], [2, 8, 0, 2], [-19, -9, 1, 7], [-16, -10, -4, -2], [-7, -8, 6, 1], [-10, -3, 2, -2], [17, 5, 3, 3], [9, -2, 4, 11], [-2, -14, 16, 0], [-42, -2, -5, 0], [-27, -7, 5, -3], [-4, 1, -5, 0], [-4, 5, -5, -4], [-10, 0, -8, -4], [-12, -5, 5, -4], [-30, 2, -2, 4], [15, -8, 4, 3], [-2, 10, 6, 4], [-7, 0, -7, 1], [2, 0, -7, -4], [-5, 15, 0, -1], [11, 7, -5, 3], [9, 2, -6, -1], [-3, -3, 9, 5], [-11, -5, 0, 11], [24, -11, 4, 0], [-8, 4, 6, -4], [25, -13, 0, -5], [12, -6, 12, 6], [21, -8, 10, -1], [19, -8, 4, 3], [-4, -3, 12, -4], [-1, -6, -8, -7], [18, -6, 4, 2], [-24, -6, -6, -4], [-1, -13, -1, 7], [5, -2, 12, -5], [-32, 6, -6, -2], [-11, 11, -18, -1], [-13, 17, -5, 5], [-3, 11, -3, -1], [-7, 9, -11, 9], [-13, 1, -2, -9], [16, 0, -12, 8], [20, 4, 4, 0], [24, 4, 12, 6], [-48, -6, 2, 2], [-33, -9, -2, 1], [-10, -1, -7, 8], [10, 5, 9, 6], [22, 4, -8, -4], [3, -8, -6, 9], [15, -12, 16, -1], [-41, -1, 10, -5], [-14, -2, 2, -2], [-36, -12, 3, 2], [-30, 0, -14, -4], [11, 1, -7, 5], [-24, 5, 17, -4], [12, -6, 8, -4], [35, 1, -12, 1], [-12, -4, -8, -8], [-24, 8, -2, 4], [-25, 7, -13, -3], [12, 2, 6, 14], [-14, 0, -1, 0], [-6, 4, 14, 2], [-14, 2, -12, 2], [-2, -9, 3, -2], [1, 14, 4, -3], [4, -12, -12, 6], [-25, 14, -1, 3], [1, -4, 0, 9], [13, -7, 5, 5], [-32, -9, -8, -4], [14, -11, 11, 2], [26, -6, 18, 4], [-13, 12, -2, 5], [-22, 4, 12, 0], [6, -12, 8, -8], [41, -6, -6, 1], [11, 13, -4, -7], [-14, -14, 12, 0], [-2, 2, -2, -2], [-15, -6, 9, 5], [-19, 2, -12, -1], [-5, -3, -8, -9], [-29, -2, -1, -9], [-4, 17, 5, 0], [11, -7, -9, -3], [-21, 0, 0, 1], [36, -1, 7, 0], [4, -16, -14, 2], [-26, 4, 0, -2], [32, 11, 3, 2], [10, 16, -2, 2], [14, 9, -23, -2], [29, 7, -9, -1], [-20, -8, 16, -2], [7, 1, 8, -7], [-8, 6, 4, 8], [23, -6, -12, 9], [-12, 2, 3, 6], [-7, -2, -2, 3], [-40, -6, 10, 0], [17, 16, -20, -3], [-30, 0, 8, 4], [-1, 15, 6, 7], [-2, 6, 8, 2], [28, -8, 4, -6], [-21, -11, 15, 9], [31, 9, 10, 5], [-7, 15, 2, 5], [10, -12, 16, -6], [-9, -9, 11, 5], [-10, -21, 10, 2], [26, -1, 25, 4], [-6, -1, -18, 2], [-18, -20, -2, 2], [0, 18, 4, 4], [-48, 2, 6, 2], [21, 15, 12, 5], [-14, 12, -4, 2], [29, 7, 0, -1], [9, 4, -14, 3], [-4, 14, 4, 2], [37, -5, -8, -1], [10, 10, -14, -6], [3, 12, 12, 7], [-9, -14, 18, 7], [-2, -10, -13, 0], [23, -13, 5, -3], [-13, 11, 6, 3], [6, -5, 17, -2], [15, 6, -8, 3], [-20, -13, -5, 6], [21, -9, 2, -1], [-11, -5, 6, -1], [56, -4, -2, 2], [18, -14, 14, -8], [2, 18, 14, -2], [41, 0, 4, 5], [8, -18, 14, -2], [29, 2, 0, -3], [-16, 0, -6, -10], [8, -9, 3, -10], [-38, 1, -11, -2], [20, -2, -10, 8], [5, -21, 18, -1], [19, 5, -19, 1], [-8, -8, -8, 10], [9, 3, -18, -7], [18, 12, 6, 2], [23, 3, 18, 9], [-8, 2, 6, 12], [35, 1, -6, 9], [27, 7, -13, -3], [-68, 4, -2, 2], [-8, -18, 16, 6], [18, -22, -10, 2], [-37, -11, 1, -3], [22, 4, 0, 8], [16, 18, -4, 2], [23, -10, 2, -13], [-11, -7, -2, -11], [-11, -7, -10, -1], [6, 0, -14, -8], [24, 3, -17, 0], [8, -2, -2, 4], [16, -13, 0, -8], [-38, -14, -10, -2], [0, 0, 22, -2], [-29, -13, -9, 1], [9, 5, -11, 9], [9, 13, -11, 7], [-12, -4, 12, 6], [-26, -14, 5, 12], [-27, 1, -20, -1], [12, 4, 2, -6], [27, 5, 14, 1], [32, 8, -16, -6], [11, 16, 11, 7], [23, 9, 26, 5], [21, 11, 8, 7], [22, 6, 4, 6], [10, 2, -6, 8], [-10, -5, 2, -14], [-35, -20, 10, 5], [0, -6, -6, -2], [26, -5, 13, -6], [-41, -6, 4, 13], [9, 8, 8, -11], [55, -7, -2, -5], [-4, 0, -8, 12], [2, 22, 2, -4], [10, 14, -22, -10], [17, -12, 2, -3], [-8, 14, -14, 6], [21, -10, -11, 9], [38, -21, 7, -8], [-20, -20, 16, 6], [22, -10, -10, 0], [1, 14, -16, -5], [34, -4, -4, 0], [53, 6, -16, -1], [-22, -8, -10, 2], [33, -10, 20, 3], [-20, -12, 14, -2], [-3, 8, 14, -1], [-2, 4, 6, -10], [8, 4, 2, 2], [-10, -10, 0, -4], [-14, 17, -13, 2], [-16, 0, 4, -4], [7, -9, -2, 9], [-30, 14, 0, -4], [29, -1, 3, -11], [-14, -14, 0, 4], [-8, 23, 13, 0], [-3, -15, 7, 11], [-6, -10, -14, -6], [-10, -6, -18, 0], [-9, 4, 24, -5], [12, 10, 2, 6], [12, 6, -10, 6], [24, -20, -10, 4], [40, 0, 12, -2], [61, 3, -2, 5], [56, -10, 6, -4], [-21, -3, 10, -7], [-2, 7, 3, -12], [39, -11, 3, 1], [-27, -19, 6, -1], [-43, -8, 14, -1], [2, -8, 8, -4], [54, -7, -14, 2], [13, -19, 2, 3], [20, 5, -1, -6], [39, -10, 3, -9], [10, 18, -18, 2], [33, -1, 6, 1], [9, -22, -6, 7], [-5, 22, -2, 7], [6, 4, 8, 6], [45, -3, -15, 1], [-46, 2, 4, -4], [45, -3, 6, -9], [62, -12, 6, -4], [4, 22, 8, -8], [24, -16, -9, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6384_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6384_2_a_cb();".
// 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_6384_2_a_cb(:prec:=4)
    chi := MakeCharacter_6384_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_6384_2_a_cb();".
// 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_6384_2_a_cb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6384_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![32, 0, -14, 0, 1]>,<11,R![32, 0, -20, 0, 1]>,<13,R![0, 1]>,<17,R![16, -152, -38, 4, 1]>,<23,R![-608, -416, -68, 4, 1]>],Snew);
    return Vf;
end function;