
// Make newform 2736.2.s.v 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_v();".
// 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_v();".
// 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], [-1, 0, 0, -1], [-2, 0, 0, -1], [0, 0, 2, 0], [0, 0, 0, 0], [-4, 1, -2, 1], [0, 1, 1, 1], [0, 1, 1, 1], [3, 0, 0, 1], [3, 0, 0, -1], [5, 2, 5, 0], [6, -2, 6, 0], [0, 1, 7, 1], [0, -4, 2, -4], [0, -3, 0, 0], [0, -3, -7, -3], [0, 1, 2, 1], [8, 2, 8, 0], [-7, -2, -7, 0], [-4, 0, -4, 0], [0, 0, 0, 3], [0, 0, 0, 0], [9, 2, 9, 0], [0, -1, 11, -1], [8, 0, 0, 2], [10, 0, 0, 2], [-4, 4, -4, 0], [5, 0, 0, 4], [0, -2, 8, -2], [6, 3, 6, 0], [0, 4, -5, 4], [0, 1, -16, 1], [3, -3, 3, 0], [-5, 0, 0, -3], [0, -4, 0, 0], [4, 0, 0, -3], [0, 0, -12, 0], [-6, 0, -6, 0], [12, 0, 0, -3], [0, -5, 9, -5], [-4, 0, 0, 4], [-4, 4, -4, 0], [5, 0, 0, 1], [0, -6, 4, -6], [-8, 2, -8, 0], [5, 9, 5, 0], [-10, 0, 0, 1], [20, 0, 0, 0], [3, -6, 3, 0], [-12, 0, 0, 0], [0, 4, 3, 4], [0, -5, 16, -5], [0, 2, -7, 2], [3, -3, 3, 0], [0, 0, 0, 0], [15, 1, 15, 0], [-9, 0, 0, -7], [0, 10, 1, 10], [28, -1, 28, 0], [-21, 0, 0, -3], [-2, -1, -2, 0], [11, 0, 0, 1], [0, -6, -7, -6], [0, 0, 6, 0], [4, 0, 0, -9], [15, 2, 15, 0], [10, -5, 10, 0], [0, 0, 0, 8], [3, 0, 0, -6], [-3, -3, -3, 0], [0, 5, -3, 5], [-4, 0, 0, 0], [-16, 0, 0, 2], [13, 7, 13, 0], [0, 0, -12, 0], [-29, -3, -29, 0], [17, -4, 17, 0], [0, 4, 3, 4], [0, 0, 0, -12], [1, -9, 1, 0], [0, -6, 12, -6], [0, -6, 2, -6], [-13, -9, -13, 0], [0, -1, 8, -1], [19, 0, 0, 2], [17, 0, 0, 6], [-2, -8, -2, 0], [12, 0, 0, 10], [-22, 0, 0, 1], [0, -4, -4, -4], [9, 0, 0, -5], [-34, 2, -34, 0], [-18, -5, -18, 0], [0, 3, -33, 3], [0, -12, 0, -12], [17, 0, 0, -2], [0, -6, -14, -6], [0, 0, 8, 0], [0, 2, -6, 2], [0, 4, 16, 4], [18, 0, 0, -3], [4, 0, 0, -4], [16, 0, 0, -9], [11, 0, 0, 0], [30, 6, 30, 0], [0, -2, -35, -2], [0, 3, -9, 3], [21, 0, 0, -4], [1, 0, 0, -3], [-18, -4, -18, 0], [0, -6, 15, -6], [-18, 0, 0, 10], [0, -9, 1, -9], [-3, -6, -3, 0], [12, 7, 12, 0], [4, 0, 0, -10], [-24, 0, 0, 0], [0, 4, -8, 4], [0, -5, 3, -5], [2, 0, 0, -6], [22, 0, 0, -4], [-16, 0, 0, 4], [8, 0, 0, -4], [13, 1, 13, 0], [0, -1, 1, -1], [-8, -2, -8, 0], [12, 4, 12, 0], [13, 0, 0, -11], [-10, 9, -10, 0], [29, -7, 29, 0], [0, -6, -8, -6], [-6, -4, -6, 0], [-27, 0, 0, -6], [0, -2, 30, -2], [0, -3, -3, -3], [24, 0, 0, 7], [21, 0, 0, 9], [9, 0, 0, 0], [0, -2, 26, -2], [0, 0, 6, 0], [0, -6, 16, -6], [0, 1, -50, 1], [4, 0, 0, -8], [23, -7, 23, 0], [2, 4, 2, 0], [21, 0, 21, 0], [0, 5, 0, 5], [-39, 0, 0, 3], [39, -1, 39, 0], [21, 0, 0, 6], [0, -13, -6, -13], [0, -4, -28, -4], [32, -3, 32, 0], [9, 0, 0, 3], [4, 0, 0, 6], [1, 4, 1, 0], [0, 6, 23, 6], [0, -8, -38, -8], [-24, -12, -24, 0], [13, 10, 13, 0], [-8, 2, -8, 0], [20, -13, 20, 0], [-19, 0, 0, 10], [0, 12, 0, 0], [-24, -8, -24, 0], [0, -5, 3, -5], [-6, -10, -6, 0], [0, 0, -42, 0], [-30, 0, 0, -3], [37, 3, 37, 0], [-3, 21, -3, 0], [5, 0, 0, 0], [0, -17, 11, -17], [0, 6, -9, 6], [0, 9, 22, 9], [-24, 12, -24, 0], [0, 0, 0, -20], [13, -9, 13, 0], [0, 2, -36, 2], [14, 0, 0, -14], [0, 4, -36, 4], [-13, -10, -13, 0], [12, 0, 0, 12], [3, 0, 0, -15], [0, 0, -16, 0], [42, -2, 42, 0], [-25, 0, 0, 12], [-24, 0, 0, 6], [0, 10, 21, 10], [0, -14, -8, -14], [-24, 0, 0, 13], [36, -12, 36, 0], [0, 5, -16, 5], [0, 6, 18, 6], [0, 18, 5, 18], [11, -5, 11, 0], [35, 0, 0, 4], [-15, 0, 0, 15], [0, 0, 30, 0], [0, -4, 12, -4], [6, 20, 6, 0], [-35, 6, -35, 0], [2, 5, 2, 0], [0, 5, 11, 5], [0, -9, 7, -9], [0, 3, 36, 3], [-15, -18, -15, 0], [10, 0, 0, 15], [25, -12, 25, 0], [0, 23, 5, 23], [-16, 0, 0, -4], [0, 3, 42, 3], [-24, 0, 0, -6], [0, -16, -14, -16], [4, 2, 4, 0], [-18, 0, 0, 6], [59, 0, 0, 1], [27, -15, 27, 0], [0, 5, -41, 5], [28, 0, 0, 6], [-11, -2, -11, 0], [-4, -12, -4, 0], [2, 11, 2, 0], [0, -4, 34, -4], [-13, 0, 0, -2], [24, -12, 24, 0], [29, 3, 29, 0], [24, 0, 0, -12], [0, 5, -23, 5], [0, -3, -38, -3], [5, 0, 0, 11], [-5, 0, 0, 2], [-14, 0, 0, -3], [0, -6, 0, 0], [-29, 0, 0, -8], [-10, 0, 0, 16], [8, -19, 8, 0], [0, 3, -33, 3], [10, -17, 10, 0], [-6, 0, 0, 1], [0, 8, -24, 8], [0, -3, -13, -3], [54, -6, 54, 0], [27, 0, 0, -15], [0, 11, 33, 11], [0, -17, 10, -17], [32, 3, 32, 0], [0, 0, -36, 0], [9, 0, 0, 23], [-20, 4, -20, 0], [19, 0, 0, 17], [0, -12, 23, -12], [7, 1, 7, 0], [0, 14, -34, 14], [0, -16, -2, -16], [10, 0, 0, -15], [54, 0, 54, 0], [0, 16, 0, 16], [0, -22, -12, -22], [36, 3, 36, 0], [-7, 1, -7, 0], [23, 19, 23, 0], [0, 12, -3, 12], [-6, 0, 0, -11], [-72, 0, 0, 0], [-63, 0, 0, 6], [0, 6, 22, 6], [0, 0, 30, 0], [55, 0, 0, 7], [-12, 0, 0, 16], [-7, 16, -7, 0], [-17, 0, 0, 3], [0, 0, 2, 0], [1, 5, 1, 0], [24, 0, 0, -9], [-39, 5, -39, 0], [0, 4, 15, 4], [0, -12, 0, -12], [-45, 0, 0, -3], [9, 0, 0, -17], [0, 12, -12, 12], [-14, 0, 0, 16], [-48, -2, -48, 0], [0, 0, 36, 0], [19, 0, 0, -14], [0, 12, -48, 12], [0, 28, 0, 0], [-28, 0, 0, -20], [30, 0, 0, 12], [-24, 0, 0, -12], [0, -12, 27, -12], [-14, 0, 0, 14], [-41, -9, -41, 0], [-46, 0, 0, -8], [0, -18, 28, -18], [-23, -11, -23, 0], [-8, -2, -8, 0], [14, 0, 0, -7], [-15, -16, -15, 0], [31, 1, 31, 0], [0, 11, 9, 11]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : 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![-6, 2, 1]>,<11,R![-3, 4, 1]>,<13,R![4, 2, 1]>],Snew);
    return Vf;
end function;