
// Make newform 138.2.e.d in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_138_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_138_e_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_138_2_e_d();".
// 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_138_2_e_d();".
// 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, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_138_e();"
function MakeCharacter_138_e()
    N := 138;
    order := 11;
    char_gens := [47, 97];
    v := [11, 7];
    // 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_138_e_Hecke();"
function MakeCharacter_138_e_Hecke(Kf)
    N := 138;
    order := 11;
    char_gens := [47, 97];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 1, 0, 2, -2, 1, -1, 1, -2, 2], [1, 0, 1, -2, 1, -2, 1, 0, 1, 0], [-1, 0, 0, 0, -2, 2, 0, 0, 0, 1], [0, 1, -1, 0, 0, 1, -3, 3, -3, 1], [-1, 1, -4, 0, 0, 4, -1, 1, 0, -1], [1, -2, -1, -2, 1, 0, 1, -2, 2, -1], [-3, 0, 2, 0, -1, -1, 1, 0, 1, 2], [2, 0, -1, 1, -1, 1, 0, -2, 0, 5], [0, -1, -1, 0, -1, -1, 0, 0, 4, -4], [0, -2, 1, -3, 1, 3, 1, -3, 1, -2], [2, -2, 0, -4, 5, -3, 4, -3, 5, -4], [6, -2, 3, -3, 2, -6, 0, -3, 4, -3], [1, 0, 1, -2, -3, -1, 1, 3, 2, -1], [-2, 4, -2, 6, -3, 6, -2, 4, -2, 0], [-1, -3, 3, 1, 0, -4, -2, -2, -2, -4], [-2, 3, 1, 8, 1, 3, -2, 0, -1, 1], [-7, 5, -7, 0, -5, 6, -2, 2, -6, 5], [6, -2, 6, 0, 8, -5, 2, -2, 5, -8], [3, 2, 2, 2, 3, 0, -1, 6, -6, 1], [-6, 1, -1, 6, 0, 8, -7, 6, -7, 8], [0, 0, -4, -1, 2, -5, 2, -1, -4, 0], [2, -4, 2, -2, 4, -2, 0, -7, 3, -7], [3, -3, 0, 5, 0, -1, -4, -1, 0, 5], [0, -10, 8, -8, 5, -9, 5, -8, 8, -10], [2, -3, 3, -4, -5, 5, 4, -3, 3, -2], [4, 0, -2, -3, -2, 0, 4, 0, -2, 2], [0, 6, -6, 6, -6, 6, -6, 0, 0, 13], [0, -5, 0, 0, 3, -1, 6, -6, 1, -3], [0, -2, -5, -5, 4, -4, 5, 5, 2, 0], [0, 6, -11, 3, -1, 3, -11, 6, 0, 0], [0, 0, -9, 0, -3, 5, -5, 3, 0, 9], [-2, 0, -10, 0, -2, 3, -3, 2, 0, 10], [2, -6, 4, -5, 7, -7, 5, -4, 6, -2], [-3, -6, 6, 3, 0, -1, 3, -7, 3, -1], [-9, 4, -13, 4, -9, 0, 0, 11, -11, 0], [3, -1, 3, 0, 4, -5, -2, 2, 5, -4], [1, -1, 2, -7, 7, -2, 1, -1, 0, 3], [-7, 2, 6, -3, 2, -2, 3, -6, -2, 7], [-2, 2, 0, 6, -3, 3, 1, 3, -3, 6], [2, 1, -2, 2, -1, -2, 0, -3, -18, -3], [2, 6, 1, 10, -7, 10, 1, 6, 2, 0], [0, 3, 2, -3, 1, -8, 1, -3, 2, 3], [13, -7, 7, -13, 0, -5, 3, -4, 3, -5], [7, 2, -4, 6, -4, 2, 7, 0, 3, -3], [-6, 13, -7, 13, -6, 0, 1, 7, -7, -1], [6, -11, 11, -6, 10, -6, 11, -11, 6, 0], [2, -1, 5, -5, 1, -2, 0, -7, 4, -7], [-13, 0, 3, 2, -3, -2, 2, 3, -2, -3], [-13, 5, -4, 4, -5, 13, 0, 9, -5, 9], [0, -4, -9, 6, -2, 1, -2, 6, -9, -4], [17, -16, 20, -13, 19, -19, 13, -20, 16, -17], [-1, 6, -1, 0, 9, 4, -4, 4, -4, -9], [-17, 11, -9, 11, -17, 0, -8, 1, -1, 8], [-4, -6, 6, 4, 0, -1, 9, -2, 9, -1], [13, -8, 5, -13, 17, -13, 5, -8, 13, 0], [-11, 11, 0, 7, -1, 16, -12, 16, -1, 7], [-5, 0, -8, 1, -11, 3, -3, 11, -1, 8], [-2, 2, 0, -4, 1, -5, -11, -5, 1, -4], [-1, 1, -12, -1, -5, -1, -12, 1, -1, 0], [-3, -11, 11, -8, 8, -11, 11, 3, 0, -6], [-7, -7, -1, 0, -1, -7, -7, 0, 2, -2], [4, -4, 11, -5, 6, -6, 5, -11, 4, -4], [0, -1, 1, -2, 11, 1, 11, -2, 1, -1], [0, 0, 0, -8, 5, -4, -1, -4, 5, -8], [0, 10, -4, -3, 6, 8, 6, -3, -4, 10], [-6, 11, 0, 2, 0, 11, -6, 0, 2, -2], [10, 3, 10, 0, -5, -2, 7, -7, 2, 5], [-4, 1, 11, 1, -4, 0, -4, 0, 0, 4], [5, -3, -2, -6, -2, -3, 5, 0, 1, -1], [0, 6, -5, 4, -12, 10, -12, 4, -5, 6], [9, 0, 10, -8, 1, -8, 8, -1, 8, -10], [13, -13, 0, -16, 11, -6, 8, -6, 11, -16], [16, -10, 12, -13, 4, -4, 13, -12, 10, -16], [-10, -6, 1, -3, 1, -6, -10, 0, -13, 13], [3, 6, 3, 0, 7, 6, 2, -2, -6, -7], [-8, 10, -8, 9, -9, 8, -10, 8, 0, 12], [17, -21, 21, -17, 0, -18, 10, -1, 10, -18], [9, -9, 0, 12, 1, 5, -7, 5, 1, 12], [-4, 4, 0, -2, 3, -12, 3, -12, 3, -2], [-18, 2, -11, 14, -27, 14, -11, 2, -18, 0], [-6, 1, 4, -3, 3, -4, -1, 6, 0, 12], [6, -2, 4, -2, 6, 0, 10, -12, 12, -10], [1, 3, 1, 0, -6, 2, -16, 16, -2, 6], [-9, -2, 0, 12, -12, 0, 2, 9, 0, 3], [20, -12, 8, -16, 11, -11, 16, -8, 12, -20], [-2, 5, -3, 3, -5, 2, 0, 1, -7, 1], [16, 0, 1, 2, 2, 2, -2, -2, -2, -1], [14, -17, 19, -19, 17, -14, 0, -20, 18, -20], [-1, 1, -14, 7, -2, 7, -14, 1, -1, 0], [5, -1, 11, -1, 5, 0, -5, 0, 0, 5], [-14, 14, -15, 14, -14, 0, -7, 14, -14, 7], [0, 0, -13, -16, -13, 0, 0, 0, 0, 0], [-2, 0, 8, 5, 0, 5, 8, 0, -2, 0], [0, 7, -15, 15, -7, 0, 0, 9, 6, 9], [-3, -13, 4, -4, 13, 3, 0, 3, 12, 3], [5, 5, -2, 0, -2, 5, 5, 0, -7, 7], [7, -16, 6, -2, 2, -6, 16, -7, 0, -22], [-5, -11, -3, -15, 6, -6, 15, 3, 11, 5], [-15, 15, 0, 0, -7, 19, -11, 19, -7, 0], [-7, 7, 0, 7, 1, 9, -16, 9, 1, 7], [-18, 13, -17, 17, -5, 5, -17, 17, -13, 18], [-16, 13, -1, 9, -9, 1, -13, 16, 0, 5], [13, -5, 12, -5, 13, 0, -3, -3, 3, 3], [-2, -13, -2, 0, 16, -2, -5, 5, 2, -16], [4, 7, -7, -4, -8, 8, 4, 7, -7, -4], [-14, 14, 0, 10, -17, -2, -16, -2, -17, 10], [4, 0, 9, 0, 6, 11, -11, -6, 0, -9], [2, -10, -4, 4, 10, -2, 0, 7, -3, 7], [0, -5, 5, 3, 22, -4, 22, 3, 5, -5], [-11, 11, -6, 10, -6, 11, -11, 0, -2, 2], [8, 2, 22, 2, 8, 0, 4, -18, 18, -4], [-4, 5, -4, 0, -7, -15, -7, 7, 15, 7], [19, -11, 11, -19, 0, 2, 2, -15, 2, 2], [-7, 7, -9, 9, -7, 7, 0, 18, -12, 18], [11, 0, 14, 3, -3, -3, 3, 3, -3, -14], [4, 4, 12, -12, -4, -4, 0, -9, 17, -9], [0, 14, -6, 6, -1, 12, -1, 6, -6, 14], [-4, 4, 5, -2, 5, 4, -4, 0, 1, -1], [9, 5, 4, 2, -2, -4, -5, -9, 0, 5], [-3, -15, 8, 0, 0, -8, 15, 3, 0, -13], [6, 5, -5, -6, 0, -9, 4, -22, 4, -9], [-2, 6, 19, -12, 7, -12, 19, 6, -2, 0], [10, 0, 10, -1, 1, -25, 25, -1, 1, -10], [19, -12, 1, 5, 7, -7, -5, -1, 12, -19], [-17, 12, -15, 12, -17, 0, -6, 16, -16, 6], [3, 16, -8, 14, -14, 8, -16, -3, 0, 11], [0, 6, 3, 2, -11, -3, -11, 2, 3, 6], [-9, -8, 17, -17, 8, 9, 0, 10, 13, 10], [-9, 14, -16, 16, -14, 9, 0, 7, -27, 7], [0, -3, -11, -14, 14, -14, -11, -3, 0, 0], [-14, 13, -5, -8, -5, 13, -14, 0, -13, 13], [-4, 4, -4, 0, -8, 6, 4, -4, -6, 8], [-20, 14, -20, 0, -4, 13, -7, 7, -13, 4], [-8, -4, 4, 8, 0, -12, -1, -15, -1, -12], [0, 7, 19, 4, 9, -13, 9, 4, 19, 7], [-4, 4, 0, -10, 19, -2, 28, -2, 19, -10], [14, -16, 28, -19, 28, -16, 14, 0, 17, -17], [-13, -2, -2, -2, -13, 0, 2, 2, -2, -2], [1, -10, -14, 5, -5, 14, 10, -1, 0, 0], [-11, -15, -14, 6, -7, 6, -14, -15, -11, 0], [8, -8, 0, 13, 3, -11, -20, -11, 3, 13], [-20, 0, 7, 9, 7, -3, 3, -7, -9, -7], [-2, 0, -9, 1, -14, -2, 2, 14, -1, 9], [-12, -17, 16, -15, 13, -13, 15, -16, 17, 12], [-14, 9, -14, 0, -25, 2, 10, -10, -2, 25], [-5, 12, -17, 15, -15, 17, -12, 5, 0, 10], [12, -9, 8, -17, 8, -9, 12, 0, 21, -21], [6, 12, 0, 10, -1, 1, -10, 0, -12, -6], [5, -7, 4, -4, 7, -5, 0, -13, 29, -13], [12, -15, -5, -17, 0, -17, -5, -15, 12, 0], [0, 34, -27, 28, -16, 29, -16, 28, -27, 34], [-11, 26, -26, 11, 0, 5, -16, -1, -16, 5], [12, -4, 4, -12, 0, -8, -7, 7, -7, -8], [0, 4, -3, -12, 7, 11, 7, -12, -3, 4], [2, 0, -9, 19, 2, 3, -3, -2, -19, 9], [0, -9, -12, -20, 4, -3, 4, -20, -12, -9], [-24, 18, 6, 0, 0, -6, -18, 24, 0, 25], [15, 7, 15, 0, -17, 7, -10, 10, -7, 17], [20, -10, 23, -10, 20, 0, 13, 1, -1, -13], [6, 1, -1, -6, 0, -25, -7, 4, -7, -25], [7, 0, -11, 22, 2, 26, -26, -2, -22, 11], [-6, 6, 0, -3, -1, 3, 4, 3, -1, -3], [20, -21, 25, -8, 9, -9, 8, -25, 21, -20], [19, -15, 10, -6, 6, -10, 15, -19, 0, -7], [10, 22, 10, 0, -10, 15, -4, 4, -15, 10], [1, 4, -7, 38, -7, 4, 1, 0, -10, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_138_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_138_2_e_d();".
// 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_138_2_e_d(:prec:=10)
    chi := MakeCharacter_138_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_138_2_e_d();".
// 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_138_2_e_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_138_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![529, -138, 2324, -2185, 460, -76, 137, -41, 4, -2, 1]>],Snew);
    return Vf;
end function;