
// Make newform 2736.2.s.u in Magma, downloaded from the LMFDB on 25 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_u();".
// 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_u();".
// 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 := [49, 0, 7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 7, 7];
    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], [1, 0, 0, 0], [1, 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], [1, 1, 1, 0], [-2, 0, 0, 1], [3, 0, 0, 1], [0, -2, 1, -2], [2, -2, 2, 0], [-2, 0, -4, -1], [0, -1, -3, -1], [0, -2, -2, -2], [-2, 0, 0, -1], [-5, 0, 0, 0], [4, 0, 4, 0], [8, -1, 8, 0], [0, 2, 0, 2], [0, -3, 1, -3], [-5, -1, -5, 0], [0, 0, 5, 0], [0, -3, -6, -3], [10, 0, 10, 0], [1, 2, 1, 0], [6, -1, 6, 0], [10, 0, 0, -2], [0, -1, -11, -1], [0, -4, 0, 0], [0, 6, 4, 6], [0, 0, 0, 3], [14, 0, 0, 0], [14, -2, 14, 0], [3, 0, 0, -1], [0, 2, -2, 2], [-6, 0, -6, 0], [0, -2, -8, -2], [0, 1, 2, 1], [1, -1, 1, 0], [6, 0, 0, 2], [-5, -2, -5, 0], [-8, 0, 0, -3], [0, -3, 9, -3], [6, -6, 6, 0], [-13, 0, 0, -1], [0, -4, -16, -4], [7, 0, 0, 3], [-1, -8, -1, 0], [15, 0, 0, -1], [0, 3, 4, 3], [8, 1, 8, 0], [-8, 1, -8, 0], [-3, 0, 0, -7], [-15, 0, 0, -2], [12, -6, 12, 0], [-3, 0, 0, -3], [0, 6, -5, 6], [0, -2, -8, -2], [0, -1, -13, -1], [-18, 0, -18, 0], [-11, -1, -11, 0], [0, 4, 0, 0], [-8, 0, 0, 0], [0, 3, -1, 3], [-16, 4, -16, 0], [14, 0, 0, -4], [-12, -6, -12, 0], [25, 0, 0, 3], [0, -2, -14, -2], [0, 3, 3, 3], [8, 0, 0, -5], [15, 2, 15, 0], [5, -9, 5, 0], [-5, 0, 0, -6], [3, 0, 0, -3], [-24, 0, -24, 0], [0, 5, 0, 5], [-4, 0, 0, 0], [-10, 0, 0, -7], [-9, -1, -9, 0], [0, -5, 23, -5], [17, 2, 17, 0], [17, -1, 17, 0], [0, 2, -10, 2], [15, 0, 0, -3], [-6, 10, -6, 0], [0, -10, -8, -10], [0, -2, -5, -2], [0, -7, 0, 0], [0, 2, 14, 2], [-2, 0, 0, 2], [-19, 0, 0, 6], [-23, 7, -23, 0], [-20, 0, 0, 5], [12, 0, 0, 8], [0, 8, -22, 8], [-6, 0, 0, -2], [2, 8, 2, 0], [6, -11, 6, 0], [0, -2, 26, -2], [0, 16, 2, 16], [15, 0, 0, -1], [0, 13, 0, 13], [0, 6, 11, 6], [0, 5, -12, 5], [0, 6, 20, 6], [6, 0, 0, 0], [-20, 0, 0, 8], [4, 0, 0, 3], [-6, 0, 0, -2], [13, 5, 13, 0], [0, -3, -1, -3], [0, -1, -5, -1], [-25, 0, 0, -2], [28, 0, 0, -3], [-12, 8, -12, 0], [0, 3, -27, 3], [-16, 0, 0, 3], [0, 3, 10, 3], [-10, 4, -10, 0], [-38, -3, -38, 0], [22, 0, 0, 2], [24, 0, 0, -6], [0, 5, 21, 5], [0, 0, -2, 0], [23, 0, 0, 6], [-2, 0, 0, 8], [-13, 0, 0, -5], [-8, 0, 0, -8], [15, 11, 15, 0], [0, -4, 7, -4], [-7, 3, -7, 0], [6, 7, 6, 0], [8, 0, 0, 8], [-30, -1, -30, 0], [-7, 5, -7, 0], [0, 1, 0, 1], [-3, -4, -3, 0], [17, 0, 0, -7], [0, -4, -1, -4], [0, -12, -6, -12], [-4, 0, 0, -9], [-40, 0, 0, 2], [3, 0, 0, -3], [0, -10, 22, -10], [0, -6, 30, -6], [0, -2, -30, -2], [0, -2, 22, -2], [-41, 0, 0, 4], [-7, -1, -7, 0], [-3, -12, -3, 0], [42, 0, 42, 0], [0, -3, 14, -3], [-32, 0, 0, -2], [-3, 8, -3, 0], [9, 0, 0, 15], [0, 1, -20, 1], [0, 11, -13, 11], [-24, 8, -24, 0], [12, 0, 0, -12], [12, 0, 0, 5], [19, 7, 19, 0], [0, 0, -7, 0], [0, 12, 8, 12], [-14, -16, -14, 0], [25, 7, 25, 0], [-6, -3, -6, 0], [8, 14, 8, 0], [2, 0, 0, -20], [-51, -3, -51, 0], [4, 3, 4, 0], [0, -14, 9, -14], [-10, 12, -10, 0], [0, 0, 36, 0], [3, 0, 0, 21], [-21, 4, -21, 0], [18, 12, 18, 0], [21, 0, 0, -14], [0, 5, 40, 5], [0, 0, 30, 0], [0, -13, -22, -13], [-14, 8, -14, 0], [-2, 0, 0, -7], [51, -2, 51, 0], [0, 6, 32, 6], [10, 0, 0, 6], [0, 12, -11, 12], [9, 13, 9, 0], [34, 0, 0, 4], [6, 0, 0, 12], [0, 18, 14, 18], [-2, 15, -2, 0], [42, 0, 0, -2], [-17, 0, 0, 1], [0, -4, 41, -4], [0, 4, 4, 4], [14, 0, 0, 9], [6, -12, 6, 0], [0, 5, -1, 5], [0, 0, -24, 0], [0, -4, 48, -4], [37, 2, 37, 0], [-2, 0, 0, 18], [36, 0, 0, 6], [0, -9, -3, -9], [0, -6, -28, -6], [-11, -20, -11, 0], [15, -2, 15, 0], [29, -13, 29, 0], [0, 8, -40, 8], [0, -6, -14, -6], [0, 12, -12, 12], [0, 0, 0, 0], [4, 0, 0, 3], [-11, 6, -11, 0], [0, -18, -8, -18], [-36, 0, 0, 3], [0, -1, -41, -1], [-7, 0, 0, 5], [0, 6, 9, 6], [-2, -22, -2, 0], [9, 0, 0, 15], [10, 0, 0, 6], [13, -1, 13, 0], [0, 8, -17, 8], [6, 0, 0, -13], [3, -1, 3, 0], [-22, -3, -22, 0], [16, 0, 16, 0], [0, 14, -29, 14], [-10, 0, 0, 4], [-19, 13, -19, 0], [-26, 10, -26, 0], [12, 0, 0, -12], [0, -4, -59, -4], [0, 1, 66, 1], [8, 0, 0, 14], [-14, 0, 0, -22], [14, 0, 0, 10], [19, -1, 19, 0], [1, 0, 0, -2], [-4, 0, 0, 4], [36, -8, 36, 0], [0, -5, -43, -5], [11, 21, 11, 0], [2, 0, 0, -6], [0, -3, -34, -3], [0, -8, 43, -8], [-37, -11, -37, 0], [-26, 0, 0, -2], [0, 8, 24, 8], [0, -11, 1, -11], [20, 15, 20, 0], [0, -9, 27, -9], [25, 0, 0, 12], [-11, 1, -11, 0], [10, 0, 0, 2], [0, -4, -24, -4], [-7, 3, -7, 0], [0, 2, -34, 2], [0, 0, -33, 0], [4, 0, 0, -9], [36, 12, 36, 0], [0, 4, 9, 4], [0, 9, 8, 9], [51, -9, 51, 0], [-39, 0, -39, 0], [1, 8, 1, 0], [0, -7, -5, -7], [-30, 0, 0, 7], [2, 0, 0, 26], [-12, 0, 0, -12], [0, -7, -16, -7], [0, 12, -12, 12], [25, 0, 0, 16], [-8, 0, 0, 26], [-38, -4, -38, 0], [0, 0, 0, -28], [0, -12, -1, -12], [-32, 5, -32, 0], [44, 0, 0, 2], [39, -16, 39, 0], [0, -2, -27, -2], [0, 10, -40, 10], [-33, 0, 0, -9], [-6, 0, 0, 22], [0, 15, 45, 15], [-33, 0, 0, -12], [-32, -18, -32, 0], [0, -6, -6, -6], [-23, 0, 0, 4], [0, 1, 53, 1], [46, 9, 46, 0], [-44, 0, 0, -6], [42, 0, 0, -12], [-23, 0, 0, 13], [0, -2, 20, -2], [-44, 0, 0, 2], [39, -2, 39, 0], [-31, 0, 0, -11], [0, -2, -42, -2], [-7, -21, -7, 0], [-37, 3, -37, 0], [-28, 0, 0, -7], [-28, 12, -28, 0], [-22, -12, -22, 0], [0, 13, -2, 13]];
    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_u();".
// 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_u(:prec:=4)
    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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![36, 12, 10, -2, 1]>,<7,R![-3, 4, 1]>,<11,R![2, -6, 1]>,<13,R![729, -54, 31, 2, 1]>],Snew);
    return Vf;
end function;