
// Make newform 2736.2.s.x in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_s_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_s_x();".
// 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_s_x();".
// 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, 0, 0, -1, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 2, 2, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 2, 2], [0, 2, -2, 0, 0, 0], [0, 0, 0, 0, 2, -2]];
    Rf_basisdens := [1, 1, 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_2736_s();"
function MakeCharacter_2736_s()
    N := 2736;
    order := 3;
    char_gens := [1711, 2053, 1217, 1009];
    v := [3, 3, 3, 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_s_Hecke();"
function MakeCharacter_2736_s_Hecke(Kf)
    N := 2736;
    order := 3;
    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], [-1, 0, 1, 0, 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], [0, 0, 0, 0, -1, 1], [-1, 0, 0, 0, 0, 1], [-2, 2, 0, -1, 0, 0], [-1, 0, 1, 0, 0, 0], [0, -1, 0, -1, 1, -1], [-1, -1, 2, 1, 1, 0], [0, 0, 0, 0, -1, 0], [4, 1, -4, -2, -1, 0], [-5, 2, 0, -1, 0, 0], [-1, 2, 0, -1, 0, 1], [0, 0, 4, 0, -2, 2], [0, 0, 3, 0, -1, 1], [6, 0, -6, 0, 0, 0], [0, 1, 0, -2, -2, 0], [0, 0, 2, 0, -3, 3], [1, 0, -1, 0, -2, 0], [-1, -1, 1, 2, 0, 0], [0, -1, -2, -1, 3, -3], [0, 0, 3, 0, -2, 2], [0, 1, 9, 1, 2, -2], [-4, 2, 0, -1, 0, -3], [6, -1, -6, 2, 0, 0], [0, -1, -2, -1, 3, -3], [2, 1, -2, -2, 1, 0], [3, 0, 0, 0, 0, -1], [2, 2, 0, -1, 0, 1], [0, -1, -6, -1, 1, -1], [2, 6, 0, -3, 0, -2], [8, 0, -8, 0, 2, 0], [0, 1, 2, 1, 1, -1], [-2, -1, 2, 2, 5, 0], [1, 2, -1, -4, -1, 0], [0, 0, 10, 0, -1, 1], [4, 4, 0, -2, 0, 0], [0, -1, -7, -1, 3, -3], [9, -2, 0, 1, 0, 0], [-2, 2, 2, -4, -1, 0], [0, -1, -4, -1, -1, 1], [8, 0, 0, 0, 0, 1], [-14, 1, 14, -2, -1, 0], [-8, 6, 0, -3, 0, 2], [0, 1, -11, 1, -3, 3], [-10, -2, 0, 1, 0, 4], [-11, -1, 11, 2, -6, 0], [0, -2, -3, -2, -1, 1], [0, -5, 9, -5, 2, -2], [0, 2, 0, -1, 0, -6], [9, 4, 0, -2, 0, -2], [0, 1, 6, 1, -1, 1], [-4, -6, 0, 3, 0, 0], [-1, -2, 1, 4, 4, 0], [-6, 0, 6, 0, -2, 0], [8, -1, -8, 2, 6, 0], [0, 1, 18, 1, 1, -1], [0, -1, 6, -1, 4, -4], [0, 0, 16, 0, -4, 4], [14, 2, 0, -1, 0, -1], [0, -2, 0, 4, -1, 0], [0, 0, 4, 0, -4, 4], [-14, 0, 0, 0, 0, -4], [0, -3, 4, -3, 1, -1], [-2, -4, 0, 2, 0, 5], [-14, 1, 14, -2, 5, 0], [12, 2, -12, -4, -3, 0], [5, 0, 0, 0, 0, -1], [0, -3, 3, -3, 3, -3], [0, -3, -16, -3, -4, 4], [7, 0, 0, 0, 0, 8], [20, -4, 0, 2, 0, -1], [0, 0, -20, 0, 4, -4], [7, -5, -7, 10, 0, 0], [10, -6, 0, 3, 0, -5], [1, 0, 0, 0, 0, -3], [0, -3, -14, -3, -2, 2], [4, -1, -4, 2, 0, 0], [0, 4, 5, 4, 4, -4], [0, 3, 6, 3, 6, -6], [-10, -3, 10, 6, 1, 0], [-6, -10, 0, 5, 0, 0], [0, -5, -2, -5, -3, 3], [2, -2, -2, 4, 2, 0], [-15, -3, 15, 6, 5, 0], [0, 2, -5, 2, -5, 5], [4, -1, -4, 2, -3, 0], [0, 2, 0, -1, 0, -1], [13, 6, 0, -3, 0, -5], [0, -3, 2, -3, -2, 2], [1, -8, 0, 4, 0, -3], [8, 0, 0, 0, 0, -2], [-2, -1, 2, 2, -3, 0], [-16, 0, 0, 0, 0, 6], [0, 1, 6, 1, -3, 3], [0, 1, 33, 1, 0, 0], [4, 1, -4, -2, -1, 0], [10, 2, -10, -4, 2, 0], [-26, 0, 0, 0, 0, 3], [-15, -3, 15, 6, -2, 0], [11, -5, -11, 10, 1, 0], [23, -4, -23, 8, -3, 0], [6, -3, -6, 6, -5, 0], [-10, -2, 0, 1, 0, -1], [16, -8, 0, 4, 0, -2], [-13, -6, 0, 3, 0, -8], [18, 2, 0, -1, 0, 5], [0, -2, -10, -2, 3, -3], [-2, -1, 2, 2, -6, 0], [-2, 1, 2, -2, 8, 0], [3, -2, 0, 1, 0, -5], [-5, -10, 0, 5, 0, 0], [0, -5, -10, -5, -5, 5], [14, -4, -14, 8, -5, 0], [-5, 6, 0, -3, 0, -2], [1, 2, -1, -4, 9, 0], [0, 0, 2, 0, -2, 2], [0, -1, -15, -1, -6, 6], [-20, 2, 0, -1, 0, -5], [-2, 10, 0, -5, 0, 5], [-26, -3, 26, 6, 8, 0], [-26, 4, 26, -8, 0, 0], [-9, 6, 0, -3, 0, -5], [-10, -12, 0, 6, 0, 6], [12, -6, 0, 3, 0, 4], [-4, 8, 0, -4, 0, 8], [0, -4, -12, -4, 9, -9], [11, 2, -11, -4, 8, 0], [0, 4, 4, 4, -7, 7], [0, -3, 9, -3, -6, 6], [22, 6, 0, -3, 0, 13], [0, 3, 15, 3, 4, -4], [0, 7, 2, 7, 6, -6], [-19, 0, 19, 0, 1, 0], [0, 2, 15, 2, -4, 4], [-10, -10, 0, 5, 0, 8], [-27, 2, 27, -4, -2, 0], [22, -4, -22, 8, 2, 0], [19, -2, 0, 1, 0, -2], [26, 6, 0, -3, 0, 3], [16, 4, 0, -2, 0, 9], [-20, 2, 20, -4, 0, 0], [-20, 1, 20, -2, -1, 0], [-20, 4, 20, -8, -2, 0], [0, -3, 0, 6, -5, 0], [-11, 8, 0, -4, 0, 6], [0, 5, -4, 5, 2, -2], [0, -3, 25, -3, 3, -3], [0, -4, 10, -4, 4, -4], [19, 3, -19, -6, 8, 0], [2, -4, 0, 2, 0, -10], [0, -1, -3, -1, -3, 3], [-28, 4, 0, -2, 0, -1], [11, -2, -11, 4, 3, 0], [2, 0, -2, 0, -7, 0], [0, -4, -4, -4, 4, -4], [20, -8, 0, 4, 0, 2], [-7, 4, 0, -2, 0, -9], [0, -6, 12, -6, 7, -7], [-15, -3, 15, 6, -9, 0], [0, 2, 0, -4, -6, 0], [0, -5, 26, -5, -1, 1], [0, 4, -22, 4, 5, -5], [0, -1, 3, -1, -14, 14], [0, -2, -22, -2, 14, -14], [2, 8, 0, -4, 0, 10], [0, 1, 6, 1, 6, -6], [0, 3, 31, 3, -12, 12], [5, 3, -5, -6, -15, 0], [0, 2, -42, 2, 2, -2], [-20, 4, 20, -8, -6, 0], [-14, -12, 0, 6, 0, 5], [0, 4, -35, 4, -2, 2], [0, -1, 30, -1, -7, 7], [13, 10, 0, -5, 0, -3], [-11, 6, 11, -12, 7, 0], [-10, -6, 10, 12, -6, 0], [7, -10, -7, 20, 1, 0], [0, 4, 18, 4, 12, -12], [-1, -8, 0, 4, 0, -3], [0, 9, -11, 9, -3, 3], [20, 3, -20, -6, -3, 0], [36, 6, 0, -3, 0, -1], [23, 1, -23, -2, -11, 0], [0, 1, -24, 1, -6, 6], [-10, -14, 0, 7, 0, -11], [18, -12, 0, 6, 0, -6], [-34, -5, 34, 10, 3, 0], [0, 1, -17, 1, -2, 2], [-10, -2, 0, 1, 0, -1], [10, 6, 0, -3, 0, 2], [-1, 0, 1, 0, 12, 0], [0, 0, 0, 0, -6, 0], [5, -4, 0, 2, 0, -9], [0, 2, -30, 2, -4, 4], [4, 0, -4, 0, 11, 0], [-44, 4, 44, -8, -2, 0], [10, -5, -10, 10, 1, 0], [0, 5, -21, 5, 1, -1], [8, 6, 0, -3, 0, 5], [14, 8, 0, -4, 0, -2], [-22, 4, 22, -8, 9, 0], [24, 3, -24, -6, 5, 0], [0, 6, 19, 6, 12, -12], [0, 3, 35, 3, 9, -9], [0, 2, 12, 2, -9, 9], [-16, 8, 16, -16, -2, 0], [8, 0, -8, 0, 6, 0], [8, -6, -8, 12, -8, 0], [0, -2, 12, -2, -2, 2], [-25, -2, 0, 1, 0, 6], [0, 7, 1, 7, 1, -1], [-10, 1, 10, -2, 1, 0], [9, 6, 0, -3, 0, -10], [34, 3, -34, -6, 6, 0], [0, 4, 0, -2, 0, -7], [-27, -4, 27, 8, -2, 0], [0, -2, -20, -2, -4, 4], [36, -2, 0, 1, 0, 2], [48, 2, 0, -1, 0, 3], [0, -3, -34, -3, 2, -2], [47, -2, -47, 4, -4, 0], [37, -8, 0, 4, 0, -1], [0, -5, -28, -5, -4, 4], [0, -6, -7, -6, 15, -15], [0, -2, -4, -2, -4, 4], [-35, 2, 35, -4, 6, 0], [18, -4, 0, 2, 0, -6], [0, -4, -14, -4, -11, 11], [0, -4, -4, -4, -10, 10], [0, -4, 0, 2, 0, -8], [-17, 3, 17, -6, -1, 0], [-9, -3, 9, 6, 10, 0], [16, 10, 0, -5, 0, -9], [0, 2, 0, -1, 0, -1], [-8, 0, 0, 0, 0, 14], [0, -9, 26, -9, -2, 2], [13, -6, 0, 3, 0, -11], [16, 4, 0, -2, 0, 10], [0, 2, 20, 2, 4, -4], [-20, -1, 20, 2, -14, 0], [0, 3, 20, 3, -10, 10], [-36, -8, 0, 4, 0, -2], [-19, -5, 19, 10, 8, 0], [-15, 5, 15, -10, 5, 0], [0, -6, -6, -6, 9, -9], [-44, -6, 0, 3, 0, 1], [0, 8, 0, -16, -8, 0], [42, 2, -42, -4, -3, 0], [0, -1, -3, -1, 20, -20], [14, -7, -14, 14, 8, 0], [5, 10, 0, -5, 0, -11], [0, -2, -24, -2, -3, 3], [36, -2, 0, 1, 0, 11], [14, 5, -14, -10, 15, 0], [0, 6, -14, 6, 1, -1], [-8, 5, 8, -10, 11, 0], [19, -8, -19, 16, -6, 0], [3, 6, 0, -3, 0, 14], [0, 2, 16, 2, 10, -10], [-27, -1, 27, 2, -7, 0], [-19, -2, 19, 4, -1, 0], [0, 2, 18, 2, -3, 3], [0, 0, 23, 0, 6, -6], [0, 1, 17, 1, -1, 1], [14, 3, -14, -6, -4, 0], [53, -2, 0, 1, 0, 2], [-20, 6, 0, -3, 0, 3], [44, 12, 0, -6, 0, -4], [-7, 1, 7, -2, -18, 0], [-64, -2, 64, 4, 0, 0], [-17, 8, 0, -4, 0, 6], [-8, 10, 0, -5, 0, -9], [0, -2, 6, -2, 10, -10], [16, -8, 0, 4, 0, 4], [31, 5, -31, -10, 7, 0], [0, -3, 29, -3, 12, -12], [-26, 0, 0, 0, 0, 2], [0, -6, 1, -6, 0, 0], [21, 4, -21, -8, 2, 0], [-54, 2, 54, -4, 2, 0], [-12, 0, 0, 0, 0, 1], [12, 4, 0, -2, 0, -16], [-4, -5, 4, 10, -18, 0], [23, 10, 0, -5, 0, 9], [0, -7, -24, -7, 1, -1], [-16, 3, 16, -6, 9, 0], [-3, -4, 0, 2, 0, -22], [36, 4, -36, -8, -3, 0], [0, 7, 15, 7, 10, -10], [6, 6, 0, -3, 0, -17], [-30, 16, 0, -8, 0, 2], [-36, -6, 0, 3, 0, 6], [50, -3, -50, 6, -1, 0], [-2, 12, 0, -6, 0, -10], [0, -9, 5, -9, -13, 13], [-30, -6, 0, 3, 0, 14], [-36, -4, 36, 8, -10, 0], [0, 5, 26, 5, -4, 4], [0, 3, -10, 3, 14, -14], [-11, 2, 0, -1, 0, 6], [0, 3, -46, 3, 7, -7], [0, -8, -14, -8, 8, -8], [-53, -2, 53, 4, 7, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_2_s_x();".
// 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_s_x(:prec:=6)
    chi := MakeCharacter_2736_s();
    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_s_x();".
// 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_s_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, -96, 144, -16, 12, 0, 1]>,<7,R![-3, -9, 3, 1]>,<11,R![-136, -24, 6, 1]>,<13,R![1, 1, 1]>],Snew);
    return Vf;
end function;