
// Make newform 2736.2.s.y in Magma, downloaded from the LMFDB on 24 April 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_y();".
// 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_y();".
// 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 := [4, -8, 18, 0, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-40, 8, -18, 25, -5, 1], [-121, 16, -36, 9, -10, 2], [72, -162, -25, -45, 9, -10], [-26, -454, -65, -117, 7, -26]];
    Rf_basisdens := [1, 1, 82, 41, 82, 82];
    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], [0, 0, 0, 0, -1, 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, 1, 0, 0, 0, 0], [1, 0, 1, -1, 0, 0], [1, 0, -1, 0, 0, 0], [0, 2, 2, -1, 1, 1], [3, 2, 0, 0, -3, -1], [-1, 1, -1, 1, 1, -2], [0, 1, 1, 0, 0, 0], [0, -3, -3, -2, -2, 2], [3, 0, 3, 3, 0, 0], [-5, 0, -3, -3, 0, 0], [-7, 2, 0, 0, 7, -2], [3, -4, 0, 0, -3, -1], [0, 5, 5, 2, -4, -2], [0, 2, 2, 1, -1, -1], [-4, -1, 0, 0, 4, 1], [0, -3, -3, -4, 0, 4], [0, 3, 3, -1, -2, 1], [1, 0, 0, 0, -1, -1], [3, 2, 0, 0, -3, 0], [-7, 2, 0, 0, 7, 1], [5, 0, 3, -2, 0, 0], [0, 6, 6, -1, 3, 1], [-1, 2, 0, 0, 1, 0], [0, -1, -1, -4, 0, 4], [-2, 0, 2, -4, 0, 0], [-6, 0, -2, -2, 0, 0], [1, -2, 0, 0, -1, -3], [10, 0, 2, 5, 0, 0], [0, 0, 0, 7, 1, -7], [-6, 5, 0, 0, 6, -1], [0, 0, 0, -2, -1, 2], [0, -1, -1, 1, -14, -1], [0, 9, 0, 0, 0, -6], [-5, 0, -1, -1, 0, 0], [-11, 6, 0, 0, 11, -3], [-5, 0, 1, -2, 0, 0], [0, -2, -2, -3, -3, 3], [-9, 2, 0, 0, 9, 1], [15, 0, 5, 2, 0, 0], [0, -5, -5, 0, 10, 0], [-4, 0, -4, -4, 0, 0], [5, 2, 0, 0, -5, 1], [5, 0, -5, -5, 0, 0], [0, 4, 4, -1, 9, 1], [7, 8, 0, 0, -7, -3], [-10, -3, 0, 0, 10, 2], [3, 0, 5, 0, 0, 0], [10, 0, 4, 6, 0, 0], [1, -10, 0, 0, -1, 0], [12, 0, -8, 0, 0, 0], [0, 0, 0, -4, 5, 4], [0, 9, 9, -5, 6, 5], [0, 14, 14, 8, 1, -8], [-8, -7, 0, 0, 8, 2], [-13, 6, 0, 0, 13, 7], [14, 1, 0, 0, -14, 0], [1, 0, -1, 5, 0, 0], [0, -10, -10, 0, -15, 0], [-8, 1, 0, 0, 8, 3], [9, 0, 11, -3, 0, 0], [16, -7, 0, 0, -16, 9], [-13, 0, 3, -5, 0, 0], [0, 2, 2, 6, -1, -6], [0, -2, -2, 1, -1, -1], [13, 0, -1, -2, 0, 0], [-1, -2, 0, 0, 1, -6], [18, 1, 0, 0, -18, 3], [10, 0, 4, 2, 0, 0], [12, 0, 4, 11, 0, 0], [16, -3, 0, 0, -16, 0], [0, -7, -7, -10, -2, 10], [6, 0, 4, -6, 0, 0], [-2, 0, 14, 4, 0, 0], [-2, -9, 0, 0, 2, 8], [0, -6, -6, -5, -13, 5], [6, -3, 0, 0, -6, 8], [-25, 0, 0, 0, 25, 2], [0, 0, 0, -8, 9, 8], [-16, 0, 4, 4, 0, 0], [-16, 3, 0, 0, 16, 6], [0, -16, -16, -3, -1, 3], [0, 0, 0, 11, -9, -11], [0, -1, 0, 0, 0, 4], [0, -3, -3, 3, -22, -3], [-8, 0, 8, 7, 0, 0], [20, 0, 0, 1, 0, 0], [-5, -6, 0, 0, 5, 9], [2, 0, -6, 4, 0, 0], [-9, 0, 1, 0, 0, 0], [0, -6, -6, 5, -15, -5], [5, 0, 17, 5, 0, 0], [9, -4, 0, 0, -9, 9], [-16, 1, 0, 0, 16, 5], [0, 7, 7, 10, 22, -10], [0, -2, -2, 3, -13, -3], [12, 0, -12, 1, 0, 0], [0, -8, -8, -3, -23, 3], [0, -2, -2, 5, 1, -5], [0, 0, 0, 9, 21, -9], [0, -2, -2, -1, 7, 1], [15, 0, -7, 4, 0, 0], [2, 0, 12, 10, 0, 0], [-3, 0, -1, 2, 0, 0], [18, 0, 10, 3, 0, 0], [5, -8, 0, 0, -5, 9], [0, -6, -6, 8, -23, -8], [0, 7, 7, 2, -34, -2], [-6, 0, -10, 5, 0, 0], [3, 0, -21, -7, 0, 0], [-7, 2, 0, 0, 7, -5], [0, 6, 6, -2, -7, 2], [10, 0, 6, 18, 0, 0], [0, 3, 3, 14, 4, -14], [-19, -6, 0, 0, 19, 10], [-12, 9, 0, 0, 12, -1], [-22, 0, -2, 8, 0, 0], [-2, 0, -4, -2, 0, 0], [0, 18, 18, 1, 17, -1], [0, -1, -1, -10, -18, 10], [-16, 0, 6, 4, 0, 0], [10, 0, -4, 0, 0, 0], [12, 0, 12, 0, 0, 0], [-16, 0, 4, 4, 0, 0], [-4, 17, 0, 0, 4, 0], [0, -9, -9, -8, -10, 8], [17, -12, 0, 0, -17, 11], [33, -2, 0, 0, -33, 5], [21, 0, -9, -9, 0, 0], [-6, -21, 0, 0, 6, 5], [12, -15, 0, 0, -12, 16], [0, 4, 4, -1, -19, 1], [5, 10, 0, 0, -5, -5], [12, 0, 16, 1, 0, 0], [0, -4, -4, -17, 3, 17], [0, -7, -7, -10, 14, 10], [21, 0, -1, -6, 0, 0], [5, 0, 11, -5, 0, 0], [18, 0, 10, -3, 0, 0], [0, 12, 12, 5, 13, -5], [0, -2, -2, -15, -1, 15], [0, -4, -4, 3, 1, -3], [0, -5, -5, -3, -8, 3], [6, 0, 12, 10, 0, 0], [12, 21, 0, 0, -12, -2], [-3, -6, 0, 0, 3, 11], [-1, 4, 0, 0, 1, 18], [0, -9, -9, -15, 6, 15], [9, 0, 1, -7, 0, 0], [-28, 15, 0, 0, 28, -12], [-4, 0, 0, 5, 0, 0], [0, 1, 1, 11, 20, -11], [0, 2, 2, -3, 33, 3], [-10, 3, 0, 0, 10, 11], [13, 0, 9, 13, 0, 0], [26, 0, 6, 8, 0, 0], [11, -24, 0, 0, -11, -2], [0, 18, 18, 16, 7, -16], [0, 14, 14, 9, -13, -9], [-1, 26, 0, 0, 1, -1], [-11, -6, 0, 0, 11, -6], [3, -4, 0, 0, -3, 1], [-14, -3, 0, 0, 14, 9], [-4, 0, 8, 1, 0, 0], [-5, 2, 0, 0, 5, -5], [-15, 10, 0, 0, 15, -11], [0, -9, -9, 2, -14, -2], [-5, 8, 0, 0, 5, -13], [0, -2, -2, -3, -13, 3], [27, 0, 5, 8, 0, 0], [32, -1, 0, 0, -32, -2], [0, 5, 0, 0, 0, -8], [14, 0, -6, 5, 0, 0], [0, -9, -9, -4, 0, 4], [0, 10, 10, 10, -7, -10], [0, 15, 15, -1, -6, 1], [31, 2, 0, 0, -31, 7], [12, 0, -4, 0, 0, 0], [-12, -13, 0, 0, 12, 14], [0, -20, -20, -17, -7, 17], [42, 0, 6, -6, 0, 0], [0, -14, -14, 1, 21, -1], [13, 18, 0, 0, -13, -8], [-12, 0, -4, 4, 0, 0], [-23, 0, 15, 1, 0, 0], [0, 6, 6, -11, -15, 11], [-1, 4, 0, 0, 1, -9], [2, 0, -22, -5, 0, 0], [18, 0, -10, -4, 0, 0], [0, 6, 6, 14, 11, -14], [0, 4, 4, 13, 11, -13], [-1, 0, 21, -2, 0, 0], [-1, 2, 0, 0, 1, -13], [0, 7, 7, -13, 0, 13], [0, 0, 0, 9, 25, -9], [0, -2, -2, -14, -25, 14], [28, -5, 0, 0, -28, -12], [-10, 0, -22, -5, 0, 0], [-23, 0, -19, -19, 0, 0], [0, -2, -2, 5, 19, -5], [0, -10, -10, -13, 7, 13], [-47, 2, 0, 0, 47, -5], [7, 18, 0, 0, -7, 0], [-24, 3, 0, 0, 24, 11], [0, -3, -3, 12, -34, -12], [0, -9, -9, -4, -12, 4], [0, 21, 21, 15, 6, -15], [45, 2, 0, 0, -45, -6], [5, 0, -1, -16, 0, 0], [-35, 0, 0, 0, 35, 0], [0, 3, 3, 6, -48, -6], [0, 0, -12, -12, 0, 0], [0, -7, -7, -11, -10, 11], [-2, 0, 10, 12, 0, 0], [0, 10, 10, 7, 41, -7], [7, 4, 0, 0, -7, 17], [-16, 0, -14, 12, 0, 0], [-33, 0, -5, -17, 0, 0], [-20, 13, 0, 0, 20, 2], [0, 17, 17, 6, -4, -6], [-30, 0, -18, -16, 0, 0], [1, -18, 0, 0, -1, -6], [1, -10, 0, 0, -1, 5], [-6, 9, 0, 0, 6, -21], [0, 14, 14, 3, 17, -3], [-16, 0, 8, 19, 0, 0], [-17, 2, 0, 0, 17, -9], [-36, -1, 0, 0, 36, -6], [-40, 0, -12, 4, 0, 0], [0, 29, 29, 8, -20, -8], [0, 23, 23, -5, 14, 5], [5, 0, 25, -7, 0, 0], [12, 0, -28, -13, 0, 0], [-25, 0, 5, -4, 0, 0], [-3, -8, 0, 0, 3, 11], [-2, 0, 14, 23, 0, 0], [14, 0, 0, 0, 0, 0], [16, -5, 0, 0, -16, 13], [0, -1, -1, 14, -6, -14], [-14, -3, 0, 0, 14, 15], [19, 0, 5, -4, 0, 0], [0, 18, 18, 11, 39, -11], [0, 9, 9, 14, -24, -14], [-15, 16, 0, 0, 15, -23], [15, 0, 11, -1, 0, 0], [0, 23, 23, -14, 20, 14], [0, 29, 29, 13, -14, -13], [8, -11, 0, 0, -8, 19], [0, -2, -2, -7, 17, 7], [11, 0, -15, -9, 0, 0], [-13, -18, 0, 0, 13, 7], [63, 0, 3, -1, 0, 0], [0, 20, 20, -8, 29, 8], [24, 5, 0, 0, -24, -6], [0, 8, 8, 7, 3, -7], [0, 18, 18, -5, 9, 5], [-25, 0, -15, 16, 0, 0], [-41, 2, 0, 0, 41, 5], [0, -14, -14, -3, -23, 3], [0, -20, -20, -9, -15, 9], [2, -23, 0, 0, -2, 1], [-6, 13, 0, 0, 6, -14], [0, -21, 0, 0, 0, 12], [0, -8, -8, -12, -37, 12], [-5, 0, -11, -8, 0, 0], [2, 0, 4, 10, 0, 0], [32, 0, -8, -11, 0, 0], [0, -4, -4, 21, -7, -21], [0, -26, -26, -11, 3, 11], [3, 0, 13, -7, 0, 0], [36, 0, 0, 8, 0, 0], [-3, 16, 0, 0, 3, -20], [13, 0, 5, -9, 0, 0], [0, 2, 2, -5, -35, 5], [6, -39, 0, 0, -6, 10], [-15, 0, -21, -26, 0, 0], [-26, 17, 0, 0, 26, 2], [0, -12, -12, 4, -47, -4], [0, 34, 34, 9, -15, -9], [-1, 0, 7, 11, 0, 0], [13, 0, 29, 25, 0, 0], [0, -22, -22, -11, -23, 11], [-14, 0, -16, 16, 0, 0], [3, -4, 0, 0, -3, -3], [0, -2, -2, -15, 1, 15], [8, 0, -36, -9, 0, 0], [0, 14, 14, -9, 15, 9], [-19, 6, 0, 0, 19, 13], [26, 0, -12, -14, 0, 0], [22, 0, 20, 28, 0, 0], [12, 0, 4, 8, 0, 0], [0, 12, 12, -22, 3, 22], [38, 0, -6, 6, 0, 0], [-24, 15, 0, 0, 24, 24], [-10, 0, -8, -4, 0, 0], [0, 0, 0, -1, 53, 1], [32, -11, 0, 0, -32, 4], [-3, -32, 0, 0, 3, 7], [-5, 0, -11, -16, 0, 0], [25, -4, 0, 0, -25, -8], [-16, 13, 0, 0, 16, -22], [0, -9, -9, 18, -28, -18]];
    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_y();".
// 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_y(: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_y();".
// 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_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, -8, 18, 0, 5, -1, 1]>,<7,R![32, -14, -2, 1]>,<11,R![4, 1, -4, 1]>,<13,R![5776, -2432, 1100, -120, 33, -1, 1]>],Snew);
    return Vf;
end function;