
// Make newform 211.2.a.a 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_a();".
// 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_a();".
// 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];
        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_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 := [[0, 1], [1, 1], [2, -2], [1, -1], [-3, 0], [5, -2], [6, -1], [-1, -3], [3, 2], [-1, 2], [-8, 5], [-6, 8], [-3, 0], [9, 0], [1, -1], [6, 1], [-3, 6], [-3, 0], [-12, 0], [2, -10], [-1, -5], [-8, 6], [2, 4], [9, -3], [-1, 3], [-3, 5], [-7, 12], [-3, -3], [3, -1], [15, -2], [11, -6], [7, -10], [-4, 4], [-16, -3], [-4, 8], [-13, -5], [13, 0], [7, 9], [6, 9], [9, 0], [-15, 15], [-3, 0], [7, -20], [12, -21], [-7, -10], [0, -15], [1, 0], [-11, -10], [11, -16], [7, 6], [-4, 1], [-9, 18], [-18, 10], [-3, -15], [-13, 22], [-7, 2], [10, 5], [2, 0], [5, -9], [-8, 10], [0, 18], [11, -14], [-6, 3], [12, 5], [6, 11], [6, -11], [-8, -10], [5, 6], [12, -13], [-5, 15], [-15, 18], [10, -20], [-24, 4], [11, -14], [15, 0], [2, 14], [-5, -10], [-4, 24], [12, 10], [-11, -13], [13, 4], [-18, 20], [-8, -5], [2, -6], [9, -8], [22, -11], [29, 2], [-15, -9], [27, 5], [7, 9], [19, -32], [-33, 6], [10, -9], [-13, 10], [11, -17], [17, -1], [7, 16], [12, -15], [29, -5], [7, -15], [25, 6], [30, -14], [-8, 4], [-29, -2], [-3, 20], [12, -3], [17, 17], [-16, 5], [-15, 0], [27, 0], [33, -10], [0, -2], [6, 4], [30, 0], [12, 0], [-23, 10], [6, 1], [13, -20], [30, 13], [-25, 20], [12, 5], [-6, 15], [-9, -6], [-11, -20], [12, -20], [-3, 5], [-4, 3], [-9, 3], [28, -15], [3, -18], [34, -3], [-6, -10], [-8, 35], [-3, -8], [7, 10], [-3, 6], [-9, 21], [2, -18], [-24, 24], [-9, -27], [-13, 0], [-23, -5], [17, -11], [6, -21], [-15, -20], [-13, 11], [8, 27], [21, -31], [-20, -15], [0, -12], [28, 15], [-13, 25], [-18, 9], [13, -35], [33, -20], [-23, 35], [-13, -24], [-24, 33], [-11, 18], [32, -5], [5, -4], [16, -14], [33, 0], [-38, 20], [-36, 33], [10, -32], [-8, -25], [-27, -5]];
    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_a();".
// 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_a(:prec:=2)
    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_a();".
// 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_a( : 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, 1]>],Snew);
    return Vf;
end function;