
// Make newform 3200.2.a.bl in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3200_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_3200_2_a_bl();".
// 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_3200_2_a_bl();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [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_3200_a();"
function MakeCharacter_3200_a()
    N := 3200;
    order := 1;
    char_gens := [1151, 901, 2177];
    v := [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_3200_a_Hecke(Kf)
    return MakeCharacter_3200_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], [1, 1], [0, 0], [1, -1], [-2, 0], [0, -2], [0, 2], [0, 2], [7, 1], [-2, 0], [2, -2], [2, 4], [8, -2], [1, -3], [5, -1], [4, 2], [-4, 2], [-6, 0], [-1, 3], [2, 2], [0, 2], [4, -4], [-3, -3], [-6, 4], [12, -2], [-10, 0], [-1, 1], [15, -1], [6, -4], [-6, -4], [9, -5], [6, -4], [-2, 0], [16, -2], [-2, -4], [10, -6], [-14, -4], [5, -3], [1, 7], [2, -4], [-16, 2], [2, -4], [-2, -6], [16, 2], [4, 2], [8, 4], [-10, -4], [11, 1], [-1, -5], [-6, -4], [-20, -2], [-4, 4], [4, -10], [-2, 0], [-6, -4], [11, -3], [-6, 8], [-2, 10], [2, -4], [8, 2], [-15, -3], [6, -8], [7, 3], [-2, 6], [2, -12], [16, -2], [-2, -8], [-14, -4], [15, -1], [-6, 4], [-14, -4], [-24, -4], [25, 3], [-14, -4], [20, 2], [-17, -3], [-6, -8], [8, -2], [-2, 0], [4, -6], [0, 2], [-2, 12], [-2, 10], [-4, -2], [24, 4], [-11, -7], [12, 6], [14, 8], [6, -16], [19, 9], [-17, 11], [4, -4], [-15, -1], [-30, -4], [28, 6], [15, 9], [10, 12], [2, 8], [9, 13], [6, -8], [-5, 7], [-22, -4], [-11, -3], [-12, 10], [10, 12], [-2, 8], [7, 7], [14, 0], [-4, 8], [0, 10], [-27, -9], [-24, -2], [-12, 6], [16, -2], [-30, 2], [-4, -2], [-11, 5], [-15, -1], [36, -6], [8, -6], [18, -8], [-8, -14], [8, -10], [-31, -3], [-30, 0], [-18, -4], [-18, 0], [-4, -12], [5, 3], [-34, 8], [0, -22], [15, -7], [-6, -10], [-2, 8], [-14, -8], [-6, 4], [4, 10], [15, -13], [12, -6], [-30, -8], [-2, 0], [30, 4], [-5, -3], [-5, -5], [-38, 0], [4, 0], [12, 2], [2, 12], [36, -6], [-25, -3], [-34, 8], [-20, 2], [21, -3], [13, 11], [7, -1], [-22, 14], [12, 8], [20, -2], [8, 2], [18, -12], [-33, -5], [-10, 16], [17, 7], [6, -8], [8, -14], [-1, 17], [-22, -2], [48, 6], [34, -4], [-32, 6], [-4, 2], [-30, 4], [30, -2], [12, 22], [0, 24], [2, -4], [-30, -4], [-30, 4], [11, 5], [-22, 0], [17, 19], [30, -4], [-16, -2], [48, 2], [-21, -15], [-2, -8], [18, -12], [9, -7], [-20, 18], [-30, 6], [54, 0], [-15, 13], [-6, 0], [10, 8], [23, -13], [16, -6], [12, 18], [-32, 6], [-18, 16], [-41, -7], [30, 8], [-10, -6], [-48, 6], [18, -16], [-4, -14], [-10, 0], [-16, -24], [-31, 17], [-22, -12], [30, -8], [-12, -2], [-2, -4], [39, -7], [-25, -17], [-12, -24], [-16, 6], [-3, -1], [-34, -16], [-19, -13], [-12, -14], [2, 8], [44, 0], [-54, 4], [-1, -11], [-5, 15], [-46, 0], [-12, -2], [-48, 8], [13, -5], [6, 24], [-28, 2], [-12, 6], [46, 10], [32, 10], [53, 9], [-19, -17], [36, -2], [8, 6], [-48, 6], [54, 6], [5, 21], [14, 8], [-29, -11], [-14, 4], [26, -12], [32, -4], [-43, -1], [30, -4], [-12, -14], [-21, 17], [-8, -18], [18, 16], [13, -13], [40, 10], [30, -8], [-16, -6], [-2, 32], [31, 15], [-38, 4], [-2, -8], [27, -7], [-1, -21], [34, -4], [-34, 16], [-12, -2], [52, -2], [-14, 12], [-10, 8], [57, -3], [34, -12], [18, 12], [-1, -5], [-36, -2], [48, -8], [14, 8], [7, 25], [67, 3], [-42, 0], [16, 10], [-6, -8], [-33, -11], [-10, -18], [-7, 7], [-22, -12], [-45, -13], [-6, -26], [-64, 2], [40, -10], [-44, 16], [16, 6], [6, 28], [43, -17], [44, -10], [-78, 4], [46, 8], [38, 4], [-14, 6], [38, 8], [8, -2], [7, 11], [6, -8], [-30, -20], [20, 4], [57, 1], [-10, 0], [32, -6], [19, -5], [38, 16], [-32, 12], [-14, 20], [-9, -35], [34, 4], [38, 8], [-19, 29], [13, 11], [-30, -20], [16, -6], [-6, -10], [66, 4], [8, -2], [2, 16], [70, -8], [-30, 16], [-5, -23], [26, 4], [20, 6], [44, -2], [53, 9], [13, 15], [8, 6], [30, 12], [-10, -16], [12, 4], [33, 9], [-14, -4], [-21, 3], [-18, 20], [-2, 8], [-24, -14], [-3, 7], [22, 8], [-66, 8], [-34, -12], [-14, 10], [26, 12], [-28, -10], [-22, 0], [19, -5], [-70, -10], [28, 10], [2, 32], [26, -20], [10, -32], [15, 21], [74, -4], [-8, -22], [12, -4], [54, 8], [0, 26], [19, -3], [-16, -26], [-50, -16], [41, 3], [32, -10], [-41, 3], [-8, 34], [52, 10], [-9, -31], [-24, -18], [18, -8], [12, -6], [-53, 17], [14, 4], [-22, -22], [0, 6], [0, 10], [-46, 22], [54, 8], [-4, -10], [-38, -12], [18, -28], [-70, 4], [-35, 19], [-2, -32], [36, 6], [-41, 25], [-18, 2], [-6, 12], [9, 29], [33, -21], [-48, 6], [18, 12], [16, 22], [11, -9], [-86, 2], [30, 16], [44, -12], [26, 32], [-10, 16], [-22, 24], [-26, -4], [-36, -26], [-59, 7], [54, 8], [-38, -16], [-62, -14], [36, -14], [-42, 24], [-27, -27], [-20, 30], [-30, 12], [82, 4], [-3, -7], [6, -12], [24, 10], [-66, 16], [-64, -8], [-3, -5], [-20, 14], [71, -7], [-14, 24], [-24, -26], [25, 11], [-52, -14], [-18, -8], [-38, -12], [37, -19], [48, -14], [-6, -36], [0, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3200_2_a_bl();".
// 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_3200_2_a_bl(:prec:=2)
    chi := MakeCharacter_3200_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_3200_2_a_bl();".
// 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_3200_2_a_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -2, 1]>,<7,R![-4, -2, 1]>,<11,R![2, 1]>,<13,R![-20, 0, 1]>],Snew);
    return Vf;
end function;