
// Make newform 211.2.a.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_211_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_211_2_a_b();".
// 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_211_2_a_b();".
// 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, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [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_211_a();"
function MakeCharacter_211_a()
    N := 211;
    order := 1;
    char_gens := [2];
    v := [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;

function MakeCharacter_211_a_Hecke(Kf)
    return MakeCharacter_211_a();
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 := [[-1, 0, -1], [0, -1, 0], [-3, 1, 0], [2, -1, 3], [-3, 3, -4], [-2, 2, -1], [1, 1, -2], [-2, -2, -1], [-6, 0, 1], [2, -7, 5], [0, -1, 4], [-1, -2, -3], [-4, -4, 2], [0, 1, -3], [4, 1, 2], [-5, 5, 0], [5, -7, 3], [-9, 4, 0], [2, -1, -2], [3, 5, -4], [0, -6, -4], [-2, 8, -6], [-10, 5, -4], [-10, -2, -1], [2, -8, 5], [-13, 1, -1], [2, 8, -3], [-4, 4, -7], [5, 4, 6], [4, 1, 5], [2, 1, 2], [-7, -2, 6], [15, -9, 17], [2, 2, 10], [-3, 7, 0], [10, -2, 2], [-5, 6, -8], [-10, 3, -6], [9, -1, 12], [-12, -1, -6], [19, -9, 11], [2, -9, -1], [-3, 7, 1], [-1, 9, -1], [3, 1, 6], [-1, -1, -15], [-1, 0, 0], [-13, -3, -6], [-2, 2, -1], [4, -2, 9], [2, -15, 0], [15, -5, -1], [-5, 9, -15], [1, -22, 9], [15, -7, 8], [10, -2, -7], [-6, 5, 8], [-10, -16, 5], [-11, -1, 7], [9, -3, 11], [14, -7, 5], [-2, -11, 5], [-3, 9, 6], [-12, 6, -6], [23, 1, 7], [13, 6, -3], [-8, 16, -6], [-10, -4, 6], [8, -12, -3], [-11, 6, -5], [-19, 15, -10], [10, -8, -5], [-15, 3, -3], [5, 19, -4], [-3, -2, -1], [4, 5, -12], [-22, 7, -21], [6, -5, 3], [-1, -2, 3], [-7, 8, 8], [-7, 6, 15], [14, -8, -11], [28, -9, 12], [-8, 15, -12], [-9, 12, 0], [12, -20, 2], [-22, 4, -8], [22, -17, 6], [-5, -3, -9], [18, 7, -2], [-4, -4, 2], [-14, 20, -4], [-17, -1, -17], [0, 17, 1], [-3, 11, -10], [0, -8, -1], [-6, 3, -12], [-32, 3, -8], [-8, 4, 8], [15, 1, 16], [10, -8, 22], [14, 11, -11], [3, -17, 1], [-28, 10, -20], [-14, -5, -5], [-5, 11, -11], [22, -1, -8], [2, 3, -5], [-10, -4, 3], [29, 3, 1], [-29, 12, 5], [2, -17, 16], [2, -5, -9], [11, -11, 9], [-4, 6, 3], [-35, 12, -3], [-14, 12, -6], [-12, 5, -12], [3, -10, 18], [-21, 1, -2], [-3, 14, -14], [-14, 24, -19], [-13, 0, 14], [33, -11, 19], [-2, 18, -1], [-17, -10, -8], [-8, -6, -1], [-2, -20, 19], [28, 4, -3], [-3, 6, 26], [17, -26, 9], [24, -21, 16], [-1, -17, -9], [-4, -15, 17], [23, 4, -11], [4, 9, 6], [-5, 21, -19], [-11, 5, -23], [-4, 12, -9], [-9, -10, -20], [-39, 10, -26], [25, -16, 21], [-5, -7, -10], [12, -7, 23], [3, -12, 2], [-16, -12, 4], [20, -11, 12], [3, -15, 2], [32, -20, 9], [11, -20, 32], [-1, 9, -6], [-20, -7, 17], [-22, 14, -9], [11, -10, 34], [18, -24, 17], [-19, -7, -9], [6, -1, -26], [-22, 10, -9], [-12, 27, -11], [-23, 11, -14], [-7, -26, 7], [-10, 34, -18], [-4, -18, -10], [-15, 12, 9], [12, -7, 23], [-27, 19, 14], [9, 26, -3], [-36, 2, -15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_211_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_211_2_a_b();".
// 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_211_2_a_b(:prec:=3)
    chi := MakeCharacter_211_a();
    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_211_2_a_b();".
// 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_211_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_211_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -1, 2, 1]>],Snew);
    return Vf;
end function;