
// Make newform 2736.2.bm.o in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_bm();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_bm_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_2_bm_o();".
// 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_2736_2_bm_o();".
// 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 := [27, -9, -6, 3, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-45, 21, 9, 1, 5, 1], [9, -33, 18, 1, -4, 1], [9, 12, 0, -2, -1, -2], [18, -12, 9, 2, -8, 2], [27, 12, -3, 4, -1, -2]];
    Rf_basisdens := [1, 27, 27, 27, 27, 9];
    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_2736_bm();"
function MakeCharacter_2736_bm()
    N := 2736;
    order := 6;
    char_gens := [1711, 2053, 1217, 1009];
    v := [3, 6, 6, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_bm_Hecke();"
function MakeCharacter_2736_bm_Hecke(Kf)
    N := 2736;
    order := 6;
    char_gens := [1711, 2053, 1217, 1009];
    char_values := [[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0], [0, 0, 0, 0, 0, 0], [1, 0, -1, 1, 1, 0], [1, 1, 0, 2, 0, -1], [0, -1, 0, 0, 0, 1], [0, 0, -2, 0, 1, -1], [1, 1, -1, 1, 1, -2], [1, 1, 0, 1, 1, -2], [2, 0, 2, 1, -1, 2], [-2, 0, 2, -1, -1, 1], [0, 0, -1, 0, 0, 0], [-1, 2, -2, -2, 4, -2], [2, 0, -2, -2, -2, 0], [4, 0, -1, -4, -1, 0], [-4, 0, 0, -2, 0, 4], [-8, 0, 0, -4, 0, -1], [1, 0, -1, 1, 1, 0], [0, 2, 0, -2, -3, -1], [0, 2, 0, 7, 2, -1], [-5, -1, -1, -5, 1, 2], [3, -2, 2, 3, -2, 4], [-4, 0, 3, -4, -3, 0], [-1, -3, 1, -2, -2, 3], [0, 0, -4, 0, 2, 1], [-3, -1, 1, 3, 1, 0], [0, 2, 0, -3, 3, -1], [-5, 3, 0, 0, 0, 3], [-5, -1, -1, 0, 0, -1], [-1, 3, -1, 1, -1, 0], [2, 5, 0, 4, 0, -5], [0, -4, 0, -4, 0, 2], [7, -3, -1, -7, -1, 0], [0, 6, 0, 9, 3, -3], [-4, 0, -6, -2, 3, -4], [-5, -2, -1, -5, 1, 4], [0, 2, 4, 0, 0, 2], [-11, 2, 4, -11, -4, -4], [-3, -3, 2, -6, -4, 3], [0, 4, 0, 3, 3, -2], [-1, 7, 1, 1, 1, 0], [5, 2, 1, 0, 0, 2], [6, 0, -2, 3, 1, 3], [4, 3, 2, 8, -4, -3], [-2, -3, 1, 2, 1, 0], [-10, -3, -8, 0, 0, -3], [12, 0, 2, 6, -1, 4], [10, -2, -3, 10, 3, 4], [4, 0, -5, 4, 5, 0], [-18, -1, 0, 0, 0, -1], [-2, -1, -5, 0, 0, -1], [3, 1, -9, 3, 9, -2], [2, -1, 0, 4, 0, 1], [-14, 0, 0, -7, 0, 4], [8, 0, 4, 4, -2, -4], [-12, 0, -4, -6, 2, -5], [9, -1, -3, -9, -3, 0], [10, -5, 0, -10, 0, 0], [4, 2, 2, -4, 2, 0], [-1, 1, -3, 0, 0, 1], [-6, 0, 6, -3, -3, -2], [0, 4, 6, 0, 6, 0], [4, 0, 2, 8, -4, 0], [-15, 1, 5, -15, -5, -2], [3, 8, 1, 6, -2, -8], [0, -2, 0, -7, 1, 1], [6, 0, 2, 3, -1, 4], [8, -2, -7, 0, 0, -2], [10, 3, -1, -10, -1, 0], [-6, -5, 0, 6, 0, 0], [14, -1, -3, 0, 0, -1], [-9, 0, -3, 0, 0, 0], [-2, 8, 6, 2, 6, 0], [-12, 0, -6, -6, 3, 0], [-13, 1, -1, -26, 2, -1], [-4, -6, -7, 0, 0, -6], [-14, -1, 2, -14, -2, 2], [0, -2, 0, 16, 4, 1], [-16, -3, 5, -16, -5, 6], [0, -11, -2, 0, -2, 0], [6, 0, -2, 3, 1, -9], [-2, -5, -2, -4, 4, 5], [9, -1, -3, -9, -3, 0], [0, -8, 0, -6, 0, 4], [16, 0, 14, 8, -7, 9], [13, 5, -2, 13, 2, -10], [22, 0, 2, 11, -1, -5], [5, -11, 7, 10, -14, 11], [-22, -3, -1, 0, 0, -3], [4, -5, 14, 4, -14, 10], [13, 9, -2, 26, 4, -9], [8, -4, 4, 16, -8, 4], [22, 0, -10, 11, 5, 5], [10, -2, -2, 0, 0, -2], [9, -5, 1, -9, 1, 0], [13, -3, -2, -13, -2, 0], [-18, 0, -6, -9, 3, -9], [-28, 0, 0, -14, 0, 2], [9, 2, -3, 18, 6, -2], [0, -6, 0, 7, -4, 3], [0, -12, 0, -9, 2, 6], [0, 0, 0, 30, 5, 0], [0, -10, 0, -15, -9, 5], [3, -9, -9, 0, 0, -9], [-2, -16, 2, -4, -4, 16], [-1, 11, 2, -2, -4, -11], [17, 1, -1, 0, 0, 1], [7, 2, -3, -7, -3, 0], [0, 2, 0, 0, -6, -1], [0, 2, 0, 16, 4, -1], [-10, 5, 3, -20, -6, -5], [14, 4, 9, 0, 0, 4], [-19, -9, 3, -19, -3, 18], [0, -4, 0, 13, 7, 2], [-3, -9, 6, -6, -12, 9], [-22, 0, 4, -11, -2, -1], [-12, -10, -6, 12, -6, 0], [9, -1, -4, -9, -4, 0], [5, 11, -1, 10, 2, -11], [-27, 1, 9, 0, 0, 1], [0, 2, 0, 4, 4, -1], [4, 0, 12, 2, -6, 2], [-4, 5, -7, -8, 14, -5], [-2, 8, 0, -4, 0, -8], [-28, -3, -2, 0, 0, -3], [4, -6, -2, 8, 4, 6], [-3, 2, -9, -3, 9, -4], [0, -14, 0, 10, 3, 7], [13, 2, -3, -13, -3, 0], [-14, -2, -1, 14, -1, 0], [11, -11, -7, 0, 0, -11], [5, -13, -2, -5, -2, 0], [30, -1, 6, 30, -6, 2], [0, -6, 0, 1, 6, 3], [14, -3, -7, 14, 7, 6], [30, 3, 0, 0, 0, 3], [0, 8, 0, -7, -8, -4], [16, 0, -4, 8, 2, 0], [-9, 9, 8, 0, 0, 9], [1, 1, 1, 2, -2, -1], [1, 4, 11, 0, 0, 4], [0, 0, 0, -14, 6, 0], [0, -2, 0, -15, -3, 1], [-16, 0, 4, -8, -2, 12], [0, 14, 0, -5, 7, -7], [-12, -5, -3, -24, 6, 5], [-18, 3, -12, -18, 12, -6], [-1, -2, 8, -1, -8, 4], [-4, 4, -2, 4, -2, 0], [26, 0, 8, 13, -4, -1], [2, -6, 4, 0, 0, -6], [9, 0, -10, -9, -10, 0], [-39, 0, 3, 0, 0, 0], [8, 0, 6, 4, -3, -2], [0, 12, 0, 7, 7, -6], [-2, 4, -6, -2, 6, -8], [-28, -4, 4, 0, 0, -4], [19, -1, 2, 38, -4, 1], [25, -4, -1, 25, 1, 8], [0, -6, 0, 10, -7, 3], [-8, 0, -8, -4, 4, 4], [13, -9, -7, -13, -7, 0], [-1, 8, 7, 1, 7, 0], [18, 2, -3, -18, -3, 0], [56, 0, -2, 56, 2, 0], [-2, 8, -4, -4, 8, -8], [-2, 9, 2, -2, -2, -18], [6, -10, 13, 6, -13, 20], [0, 0, 0, -11, 6, 0], [0, 4, 4, 0, 4, 0], [0, 12, 0, 2, 2, -6], [-13, 8, 13, 0, 0, 8], [0, -13, -7, 0, -7, 0], [7, 11, -1, -7, -1, 0], [-12, 7, -1, 0, 0, 7], [0, -14, 0, -17, 2, 7], [0, -16, 0, 0, -6, 8], [16, 0, -10, 8, 5, -6], [26, -2, 4, 26, -4, 4], [-17, -5, -14, 0, 0, -5], [-13, 8, -4, -13, 4, -16], [-10, 0, 10, -5, -5, 11], [19, -3, -7, 0, 0, -3], [6, 0, -4, 3, 2, 12], [6, -7, -6, -6, -6, 0], [17, -3, 1, 34, -2, 3], [-40, 0, -2, 0, 0, 0], [26, 0, -6, 13, 3, -3], [-17, -9, 0, -17, 0, 18], [-5, 9, -1, -10, 2, -9], [2, -7, 2, 4, -4, 7], [6, 0, 0, 3, 0, 10], [28, 0, -12, 14, 6, -4], [0, -4, 1, 0, 0, -4], [2, 20, 4, -2, 4, 0], [6, 0, 6, 3, -3, 12], [20, 0, -4, 10, 2, 8], [0, 10, 0, 5, 9, -5], [7, -4, 4, 7, -4, 8], [3, -3, -15, 0, 0, -3], [0, 2, -2, 0, 4, -2], [2, 0, 10, 1, -5, -6], [0, 6, 0, -17, -7, -3], [2, 13, -1, -2, -1, 0], [8, 1, -9, -8, -9, 0], [-5, 0, -1, 5, -1, 0], [0, 24, 0, 0, 12, -12], [-12, 0, 0, -6, 0, -6], [0, -24, 0, -24, 0, 12], [-6, 0, 6, -6, -6, 0], [-29, 1, -6, 0, 0, 1], [-8, -7, 13, -8, -13, 14], [0, 2, 0, -13, 5, -1], [4, 0, 9, 8, -18, 0], [0, 14, 0, -30, 18, -7], [21, 2, 9, 0, 0, 2], [14, 0, 0, 7, 0, 6], [-6, 2, -4, 6, -4, 0], [6, 23, -6, 12, 12, -23], [-19, -7, -17, 0, 0, -7], [26, -3, -8, 26, 8, 6], [0, 0, 2, 0, -1, -23], [-5, 1, 6, 0, 0, 1], [4, 5, 4, -4, 4, 0], [5, 9, 8, -5, 8, 0], [22, -2, -10, 22, 10, 4], [0, 14, 0, 24, 11, -7], [-6, -8, -4, -12, 8, 8], [3, -10, 9, 3, -9, 20], [-14, 6, -18, -14, 18, -12], [22, -2, -2, 44, 4, 2], [0, 24, 0, 29, 6, -12], [0, -14, 0, 21, -10, 7], [-29, 5, 7, 0, 0, 5], [-15, 7, 3, -30, -6, -7], [4, -16, 2, 8, -4, 16], [2, 7, 10, -2, 10, 0], [2, -9, -13, 0, 0, -9], [36, -4, 0, 0, 0, -4], [12, 2, 0, -12, 0, 0], [0, 2, 0, 48, 6, -1], [-4, -11, -2, -4, 2, 22], [14, -2, -8, 28, 16, 2], [24, 0, -18, 12, 9, 6], [14, 0, -16, 7, 8, 0], [5, 4, 5, -5, 5, 0], [-19, -3, 7, -38, -14, 3], [0, 0, 12, 0, -6, -12], [0, 20, 0, 43, 7, -10], [-45, 3, 8, -45, -8, -6], [-16, 0, 24, -8, -12, -5], [9, -4, -10, 0, 0, -4], [-3, 2, -21, -3, 21, -4], [19, -3, -7, 38, 14, 3], [42, 0, -6, 21, 3, -5], [-11, -4, 5, -11, -5, 8], [-18, 0, 10, -9, -5, -13], [0, 14, 0, 2, 7, -7], [37, 7, -12, 0, 0, 7], [12, 6, -12, -12, -12, 0], [0, 2, 0, 12, -7, -1], [0, 10, 0, 5, 10, -5], [3, 6, -15, 3, 15, -12], [22, 3, 7, 22, -7, -6], [-1, -12, -2, 1, -2, 0], [0, 22, 0, 12, -6, -11], [-17, -5, 10, 0, 0, -5], [-29, -5, 1, -58, -2, 5], [-6, -6, 6, 0, 0, -6], [0, 26, 0, 29, 4, -13], [0, 0, 0, 6, 6, 0], [-10, 3, -11, -20, 22, -3], [-49, 1, 3, 0, 0, 1], [-12, -2, -2, -12, 2, 4], [14, -12, -4, 28, 8, 12], [-24, 0, -6, -12, 3, -7], [16, -12, 3, -16, 3, 0], [-28, 2, 2, -56, -4, -2], [-2, 23, 11, 2, 11, 0], [-2, 0, 16, -1, -8, 12], [-20, 0, -16, -10, 8, 2], [-43, -6, -11, 0, 0, -6], [26, -4, -4, 52, 8, 4], [44, 0, 0, 22, 0, 7], [13, 6, -2, 26, 4, -6], [-1, 11, -3, 1, -3, 0], [6, 0, -6, 3, 3, -25], [7, -6, 10, 0, 0, -6], [2, 0, -2, 1, 1, -16], [28, -6, -1, -28, -1, 0], [-15, -7, -3, -30, 6, 7], [-6, 8, 12, 0, 0, 8], [22, 1, 0, 44, 0, -1], [-30, 0, -22, -15, 11, -17], [-32, -8, -22, 0, 0, -8], [7, 4, 6, -7, 6, 0], [4, 13, 2, 0, 0, 13], [0, -12, 0, 8, 16, 6], [12, -9, 18, 12, -18, 18], [22, 1, -10, 22, 10, -2], [-9, -15, 4, -18, -8, 15], [-1, -7, -3, -1, 3, 14], [-10, -14, 4, 10, 4, 0], [-14, 0, 0, -7, 0, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_bm_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_2_bm_o();".
// 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_2736_2_bm_o(:prec:=6)
    chi := MakeCharacter_2736_bm();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2736_2_bm_o();".
// 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_2736_2_bm_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_bm();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![144, -96, 88, -8, 12, -2, 1]>,<7,R![243, 0, 163, 0, 25, 0, 1]>,<11,R![48, 0, 64, 0, 16, 0, 1]>,<23,R![13872, -4080, -416, 240, 28, -12, 1]>,<31,R![3, -9, 1, 1]>],Snew);
    return Vf;
end function;