
// Make newform 2112.2.a.bh in Magma, downloaded from the LMFDB on 23 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_bh();".
// 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_bh();".
// 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, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 2, 0], [-5, 0, 2]];
    Rf_basisdens := [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_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, 0], [1, 0, 0], [0, 1, 0], [-1, 0, -1], [1, 0, 0], [0, 1, 0], [3, 1, -1], [1, 1, -1], [2, -1, 0], [-3, -1, 1], [-2, -2, 2], [0, -2, 2], [3, -1, -1], [3, 1, 1], [6, 1, 0], [2, -1, -2], [0, 2, 0], [-2, 1, -2], [-2, 2, 2], [10, -1, 0], [4, -2, -2], [-5, 2, -1], [-8, 0, 0], [2, 0, 0], [2, -2, 0], [5, -1, 1], [-12, 2, 0], [2, -2, 2], [0, 1, 0], [8, -2, 2], [-5, 2, -1], [2, 2, -2], [-4, -4, 2], [1, -3, -1], [3, -5, -1], [-1, 2, 3], [6, -2, 0], [-2, -2, -2], [2, -4, 2], [-3, -1, 1], [-8, -2, 0], [6, 2, 0], [-2, 5, 0], [2, 2, -4], [13, -1, 1], [-2, 0, -2], [-7, -5, 3], [0, 0, 0], [6, -2, 6], [8, 4, -2], [-9, -1, 3], [6, 0, -2], [4, 0, 2], [12, 2, -4], [-14, 0, 0], [-18, -2, 2], [-18, -1, 2], [-11, -6, 1], [12, -1, 0], [-7, 3, 1], [15, -3, -3], [-1, 3, -5], [-3, -7, 3], [10, -1, 0], [-4, 6, -2], [4, -5, 0], [-22, -2, 2], [-4, -6, 6], [-16, 4, 0], [-6, 3, -2], [6, 0, 4], [0, 2, 4], [-16, 4, 0], [-2, -1, 2], [-18, -2, -2], [6, 3, -4], [-8, -3, 0], [8, -4, 6], [0, -4, -2], [-8, -4, 6], [-16, -2, 0], [-10, -8, 4], [-2, 2, -6], [24, -2, 2], [-19, 4, -3], [4, 2, 4], [10, 2, 4], [-4, 4, 2], [-21, 3, -1], [-16, 2, 4], [4, -4, 0], [-2, -10, 2], [26, 0, 2], [6, -2, -2], [12, -4, 0], [-2, 0, -2], [-2, 3, -6], [-12, -4, 2], [-21, -3, 1], [-4, 1, -4], [3, -1, 5], [15, 3, 3], [-4, 4, -8], [23, -3, 3], [19, 3, -3], [-4, -4, 6], [4, 2, -4], [17, 3, 1], [10, -1, 0], [6, -4, -4], [-13, -2, 7], [22, 1, -2], [4, 6, -10], [-8, 0, 0], [-6, -6, 6], [2, -6, -4], [-4, -8, 4], [14, -1, 4], [-22, 1, 2], [8, 0, 4], [6, -2, -8], [10, 10, -4], [33, -1, 5], [-12, -10, 4], [-4, 4, 4], [7, -1, 3], [-8, -4, -2], [-2, -5, 4], [26, 0, 2], [16, 1, 0], [-17, 1, -3], [-24, 2, -4], [-18, -2, -6], [-14, 0, -8], [3, 1, 7], [-2, 6, 0], [26, -5, -2], [-5, 1, -7], [8, 1, 0], [9, -3, 1], [21, 3, -1], [17, -1, 5], [8, 0, -8], [-12, 8, 4], [-4, -10, 6], [14, 13, -8], [6, -3, 6], [-27, -7, 5], [10, 2, 2], [22, -3, 8], [18, -1, -2], [16, 6, 2], [-30, -6, 6], [-12, 2, 8], [16, 12, -8], [-2, -5, -4], [1, -6, 5], [-20, 0, 2], [-10, -4, -4], [17, 3, -3], [-24, 0, -4], [13, -7, -3], [3, -2, 7], [28, -2, -4], [-24, 0, 6], [-10, 3, 4], [6, -2, -6], [14, -1, 2], [-28, -4, 2], [10, 5, 2], [-26, 2, 2], [0, 8, 6], [-20, 6, 0], [12, -10, -2], [-20, -8, 4], [-20, -6, -2], [-17, -5, 1], [18, 5, -6], [7, -4, 7], [14, 3, 2], [22, -6, -6], [6, -2, 10], [-26, 4, -4], [29, 1, -3], [-10, 7, 4], [-26, -3, -2], [36, -1, 0], [-16, 4, 0], [28, 0, 2], [18, -2, -2], [20, 8, -2], [-12, -4, 4], [8, -4, 4], [36, 1, -4], [-32, 8, 4], [-22, 4, 0], [6, 0, 4], [-6, 2, 4], [1, 5, 1], [-8, 8, 0], [-5, 7, -1], [-9, 0, 7], [30, -6, 8], [-8, 6, -6], [0, 6, 0], [28, 7, 0], [-16, -6, 4], [-14, -2, -2], [15, 17, -5], [2, -6, -10], [-6, 14, -4], [-14, -11, 10], [14, 6, -6], [-4, 0, 0], [8, 2, 4], [-18, 10, -4], [-19, -8, -3], [33, 11, -7], [26, 7, -8], [26, 13, -6], [-36, 3, -8], [-49, 2, 3], [28, -4, 2], [24, 12, 0], [-44, 8, 0], [-30, 15, 6], [24, 6, -6], [42, 5, -4], [21, 4, -3], [-14, 2, 2], [36, -2, -2], [17, -5, 11], [7, -4, -1], [39, 1, 3], [24, 4, 4], [24, 4, -8], [14, -12, 4], [-13, -9, -1], [0, -14, 8], [10, -3, -4], [0, 0, -12], [-13, 9, -15], [26, 10, -2], [-50, 2, 0], [-3, -9, 5], [8, 6, -4], [-14, 12, 2], [-12, -12, 8], [51, -1, 5], [20, -6, 0], [48, 7, -4], [31, -9, 3], [-2, 9, -8], [26, 6, 0], [7, 3, -13], [24, -8, 8], [-12, -2, 6], [-3, -13, 7], [-32, 3, -12], [30, 2, 0], [-7, -14, -3], [-2, 10, -6], [8, -7, 0], [-18, -13, 2], [-26, 2, 0], [6, -6, 2], [-10, -3, 6], [-12, 6, -2], [5, 9, -5], [41, -5, 5], [-28, 4, 10], [24, 16, -8], [28, -8, -2], [-19, 12, -3], [-16, 2, -14], [-38, 0, -6], [-4, 4, -8], [-14, 5, -6], [-18, 0, -12], [28, 0, -8], [10, 6, -2], [10, 8, 10], [48, -2, -4], [18, -9, -10], [-5, -5, 5], [-22, 11, 8], [-6, -2, 8], [5, 3, 9], [-2, -4, 6], [-11, -1, 13], [28, -7, 4], [-12, -10, 4], [37, -1, 5], [-4, 4, 0], [-10, 7, 2], [-25, -9, 11], [46, 0, 6], [-10, -13, 10], [-4, 0, -4], [45, -1, -1], [30, 6, -8], [-17, 11, 3], [7, 4, -9], [8, -16, 4], [2, -10, -10], [-16, -2, -6], [8, 0, 4], [-38, 4, -16], [18, 13, -12], [-42, -13, 2], [18, 8, -6], [0, -9, 12], [3, 7, 7], [-4, 20, 0], [30, 2, 2], [-48, 2, 2], [-8, 12, 12], [-28, 0, -4], [-18, 12, 4], [-27, 1, -3], [-27, 5, 3], [-8, 8, -6], [15, 3, -5], [-14, -4, 10], [41, -13, -7], [16, -8, -6], [2, 2, -2], [8, 0, -12], [10, -8, -14], [-3, 3, 1], [-20, 15, -8], [28, 5, -4], [-29, -4, 3], [-34, -10, 6], [-25, -11, 13], [-4, -4, -16], [-44, 0, 2], [-27, -1, -3], [10, 8, 0], [19, 2, -9], [24, -8, -10], [-10, 2, 8], [11, -5, -9], [-4, 12, -4], [6, -9, 10], [44, -4, -4], [34, -6, 4], [-30, -6, 2], [32, 10, -4], [10, -9, -10], [-15, 19, 3], [-14, -12, 0], [-34, 15, 2], [3, -10, 7], [-18, -7, 6], [45, 1, -3], [-38, -5, 8], [38, 6, -10], [53, 9, -3], [2, -15, 4], [-38, -15, 2], [-11, 15, 5], [26, 23, -16], [26, 22, -14], [18, -2, 14], [18, 0, 8], [-53, 3, -1], [11, -2, 7], [26, -10, 4], [40, 2, 0], [-50, 6, -2], [0, -4, 16], [-15, -17, 5], [25, 8, 9], [10, -8, 8], [-4, 12, 0], [-66, -4, 6], [-22, 0, 8], [9, -3, 9], [16, -16, 6], [48, 1, -8], [34, 6, -12], [-53, -10, 7], [15, 17, 3], [39, 1, 5], [-26, -19, 8], [-6, -2, 6], [8, 10, -6], [7, 1, 5], [-46, 9, 4], [6, 6, -20], [22, -11, 14], [-32, 10, 8], [-16, -4, 12], [-90, -3, 0], [12, 16, -6], [3, -8, -13], [-14, -6, -4], [-38, -18, 10], [5, -13, 1], [38, 1, -10], [0, -8, -2], [-5, -8, 11], [10, -2, 20], [-53, -13, 7], [-21, 8, -13], [32, 14, -14], [43, -9, -1], [-34, 10, -6], [12, -8, -16], [-6, -16, 8], [-17, 11, 11], [32, -20, -4], [19, 7, -3], [-40, -14, 2], [-12, 3, -16], [-2, 12, -2], [20, -10, 16], [-34, -2, -8], [54, 4, 6], [-22, 9, -14], [2, -7, -6], [-6, -5, 0], [-14, -18, 2], [20, 4, -10], [50, -1, -10], [24, 4, -12], [-57, -9, 3], [-2, 18, -2], [14, -10, 10]];
    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_bh();".
// 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_bh(:prec:=3)
    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_bh();".
// 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_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2112_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, -16, 0, 1]>,<7,R![-56, -16, 4, 1]>,<13,R![-8, -16, 0, 1]>,<17,R![64, -4, -8, 1]>,<19,R![32, -24, -2, 1]>,<23,R![32, -4, -6, 1]>],Snew);
    return Vf;
end function;