
// Make newform 2112.2.a.bf in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2112_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_2112_2_a_bf();".
// 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_2112_2_a_bf();".
// 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_2112_a();"
function MakeCharacter_2112_a()
    N := 2112;
    order := 1;
    char_gens := [2047, 133, 1409, 1729];
    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_2112_a_Hecke(Kf)
    return MakeCharacter_2112_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, 0], [-1, -1], [-1, 1], [-1, 0], [-3, 1], [0, 2], [2, -2], [-1, -3], [-4, 2], [0, 0], [-6, 0], [2, -4], [-4, 4], [1, 3], [-9, -1], [-10, -2], [-5, 3], [8, 0], [-1, -3], [-6, 0], [5, 3], [-4, -4], [2, 0], [-4, 2], [-8, -2], [2, -2], [-8, 4], [5, -7], [-2, -4], [-3, -5], [-16, 0], [-12, -2], [6, 2], [-2, 0], [9, -1], [12, -2], [20, 0], [-2, 10], [12, 2], [-6, 2], [4, -2], [5, -1], [-12, -2], [8, 6], [-10, -6], [12, 4], [0, 0], [-4, 0], [4, 6], [-2, 0], [22, 2], [-4, -10], [2, 10], [-14, 0], [16, 0], [11, 3], [15, 1], [-1, -1], [8, 10], [12, -4], [-10, -8], [-26, 2], [-9, -3], [2, 8], [1, -3], [-8, 4], [-18, 4], [0, 8], [-23, -3], [-2, -12], [-2, -6], [28, -4], [-7, 5], [12, -8], [-1, 13], [-21, 3], [8, 10], [0, -6], [28, -2], [-6, -6], [2, 0], [-28, -4], [-10, 12], [25, -1], [34, 2], [16, 2], [0, -14], [10, -4], [-10, 2], [16, 4], [-32, 0], [-6, -2], [12, -8], [8, -4], [6, -6], [7, 7], [12, -2], [8, 8], [-15, -3], [10, 6], [14, -8], [16, 4], [24, 2], [-26, -6], [4, -6], [-14, -6], [8, 2], [-9, 13], [10, 8], [-23, -9], [-37, -5], [-6, 8], [8, 16], [16, 0], [0, 18], [36, 4], [27, -7], [-19, 9], [-40, 4], [-12, -2], [12, -2], [12, 2], [-2, 6], [0, 0], [10, 12], [-12, -10], [-9, 13], [-14, 6], [-11, 9], [-8, -8], [6, -14], [-16, 8], [30, -4], [-32, -6], [20, -2], [-21, -5], [4, 12], [-17, -9], [-2, -8], [-24, 8], [-28, 10], [8, -8], [-16, -16], [26, 8], [21, -1], [-1, -25], [-34, -8], [12, -8], [-19, -1], [-27, 1], [-34, 4], [-4, -20], [2, -10], [-12, -4], [-9, 5], [-21, -3], [-16, 10], [10, -8], [8, 6], [8, 20], [14, -8], [-39, -9], [22, -10], [-28, -2], [-1, 21], [-8, 8], [33, -3], [32, 10], [1, 13], [-12, 12], [-12, -2], [-2, 18], [-46, 0], [-12, -12], [14, 12], [-38, -2], [-39, 5], [-53, -3], [13, -7], [12, -4], [-16, 0], [6, -4], [-26, 16], [3, 1], [5, -7], [-17, 7], [-12, 4], [4, 22], [4, 4], [-32, 6], [12, -4], [4, 8], [-29, -13], [-40, -4], [-42, 12], [-10, 12], [-40, -6], [22, -8], [0, 8], [26, -12], [-25, -7], [36, -2], [22, 12], [-38, -6], [-3, 17], [6, 2], [36, -12], [-20, 6], [-12, 4], [8, -6], [-15, -3], [32, -8], [20, 16], [22, 10], [44, 10], [-11, -5], [16, 2], [39, 13], [-55, -3], [-53, 3], [-39, 15], [20, -2], [20, -4], [28, -16], [-11, 25], [6, 12], [9, -13], [-31, -1], [20, 0], [-30, 0], [36, -4], [-5, -11], [12, -10], [4, 8], [20, 4], [42, -8], [14, -24], [46, -10], [-15, -5], [-16, -12], [-36, 4], [-32, 8], [-8, -14], [24, 18], [34, -2], [-10, 18], [40, -4], [18, -18], [-46, -2], [-13, -13], [-10, 0], [33, -13], [-4, -6], [26, 4], [-28, 12], [26, -16], [8, -24], [-11, 17], [-48, 10], [19, -11], [-52, -12], [5, -15], [-27, -15], [20, -2], [-8, -8], [-43, -15], [-6, -16], [-26, 18], [-4, 18], [20, 6], [-8, -24], [-40, -10], [-63, 7], [18, 8], [26, 6], [-32, 12], [11, 3], [30, 12], [28, 16], [36, -4], [-6, 22], [26, -18], [13, 1], [14, -14], [-21, -7], [-28, -22], [32, 6], [2, 14], [72, 2], [11, 27], [-10, -26], [-32, 2], [-16, 28], [-39, 13], [-18, 0], [6, 2], [-1, 23], [16, 0], [28, 4], [4, 6], [30, -8], [19, 21], [-24, -12], [40, -8], [6, 28], [-16, 4], [50, 16], [-39, -13], [-11, 9], [-6, -18], [65, 5], [18, -12], [40, 12], [-52, -4], [42, 16], [-28, -24], [-44, 20], [42, 8], [70, -8], [2, -26], [-28, 10], [50, 12], [6, 10], [0, 10], [4, 2], [-52, 4], [24, -20], [58, 14], [24, -18], [15, -9], [-33, 7], [-1, -39], [-20, -24], [-12, -4], [-24, 4], [8, -14], [-16, 26], [2, -40], [-3, -5], [4, -10], [4, 38], [34, 12], [24, 16], [-29, -5], [12, 12], [48, -6], [-52, -16], [-26, -6], [7, -1], [-20, 12], [-82, -4], [-5, 3], [1, -1], [-13, 19], [-10, -16], [51, 9], [4, 40], [-26, 8], [-3, 15], [-25, -1], [-72, -6], [47, 5], [24, -20], [-32, 24], [2, -24], [-26, 0], [17, 23], [16, 26], [-10, -10], [44, 8], [20, 20], [20, 26], [33, 15], [-2, 20], [-40, 12], [-62, -2], [-6, 8], [-42, -8], [8, -30], [7, 31], [-28, -18], [25, 15], [28, -18], [56, -8], [5, -17], [4, 4], [50, 0], [40, -24], [69, 7], [0, 38], [-43, -23], [-2, 22], [28, 0], [-19, -9], [-36, -10], [51, 13], [32, -14], [16, 4], [-64, -2], [-21, -13], [36, -18], [27, -27], [-60, -10], [50, -4], [-13, 5], [-50, -4], [18, -28], [-32, -16], [-20, 0], [-6, 24], [-38, -28], [-4, 0], [2, -26], [10, 40], [-31, -3], [-30, 6], [-2, -38], [0, 10], [-30, 6], [-11, 17], [-1, 23], [-61, 17], [-8, -28], [20, -30], [-17, -17], [44, 0], [30, 0], [8, 20], [-40, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2112_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2112_2_a_bf();".
// 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_2112_2_a_bf(:prec:=2)
    chi := MakeCharacter_2112_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_2112_2_a_bf();".
// 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_2112_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2112_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 2, 1]>,<7,R![-4, 2, 1]>,<13,R![4, 6, 1]>,<17,R![-20, 0, 1]>,<19,R![-16, -4, 1]>,<23,R![-44, 2, 1]>],Snew);
    return Vf;
end function;