
// Make newform 228.2.q.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_228_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_228_q_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_228_2_q_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_228_2_q_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, 0, 0, -1, 0, 0, 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_228_q();"
function MakeCharacter_228_q()
    N := 228;
    order := 9;
    char_gens := [115, 77, 97];
    v := [9, 9, 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_228_q_Hecke();"
function MakeCharacter_228_q_Hecke(Kf)
    N := 228;
    order := 9;
    char_gens := [115, 77, 97];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 1, 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, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [1, 1, 0, 0, -1, -1], [1, 2, -1, -1, -1, -1], [0, 1, -3, 1, -3, 1], [0, -2, 0, 2, 2, 3], [1, 1, 3, 2, 0, -2], [2, 2, 1, -2, 2, -2], [-3, 1, -4, 1, -3, 0], [-4, -4, -1, 4, 0, 1], [-2, -3, 0, 2, 3, 3], [-2, 2, 2, 0, 1, -3], [-3, 0, 3, 0, -6, -3], [-2, -7, 2, -2, 7, 2], [0, 0, 5, -1, 1, -5], [-3, -2, 2, -2, -3, 0], [5, -3, 1, -4, 0, 4], [-1, -1, 5, -1, -1, 0], [0, 0, 2, -2, 2, -2], [-6, 3, -6, 6, -3, 6], [3, 0, -3, 1, -2, 4], [2, 0, -2, -2, -9, 0], [5, 3, -7, -5, 4, 4], [-1, -6, 0, 7, 7, -4], [5, -2, -2, -7, 0, 7], [-2, 0, 0, 2, 2, 7], [0, -1, -5, 5, -5, -1], [8, -4, -2, -8, 6, 6], [4, -1, -1, 1, 1, -4], [4, 6, 6, 0, 4, -10], [1, -5, 0, 4, 4, -11], [9, 4, 1, -8, 0, 8], [4, -10, -1, -10, 4, 0], [-6, 3, 0, 3, 3, 7], [1, 0, -1, 5, 14, 6], [-1, 7, 7, 0, -11, 4], [13, -1, 8, -8, 1, -13], [0, -1, 4, 17, 4, -1], [-4, -4, 6, -4, -4, 0], [-6, -2, -5, 1, 0, -1], [0, -3, 10, -10, 10, -3], [-2, -2, -11, 7, -5, 11], [-10, 1, 1, 0, -6, 5], [-7, 0, 7, -6, -8, -13], [5, 16, -12, -5, -4, -4], [-6, -6, 12, 5, 1, -12], [12, -6, 2, -10, 0, 10], [2, -1, 13, -13, 1, -2], [-6, 8, 8, 0, 2, -10], [1, -8, -8, 0, 12, -4], [12, 1, 7, -7, -1, -12], [0, 0, -14, 5, -14, 0], [4, 4, 0, -8, -8, -2], [-7, -5, 10, -5, -7, 0], [0, 0, 2, 8, -8, -2], [14, 0, -14, -12, -4, 2], [-10, 0, 10, -2, 4, -12], [-3, -4, -9, 9, 4, 3], [0, -1, 2, -11, 2, -1], [-8, 11, 17, 11, -8, 0], [5, 4, -2, -7, 0, 7], [0, 16, -21, -5, -21, 16], [15, 0, -15, -15, -1, 0], [1, 2, -3, -1, 1, 1], [-3, -3, 1, -6, 9, -1], [1, -3, 0, 2, 2, -17], [0, 9, 4, 12, 4, 9], [6, -10, -3, 3, 10, -6], [1, -6, -7, 7, 6, -1], [14, -8, 8, -14, 0, 0], [0, 2, -7, 12, -7, 2], [-15, 11, -10, 5, 0, -5], [10, 4, 0, -14, -14, 7], [10, 6, 6, -10, -12, -12], [-1, 2, 2, 0, 11, -13], [-4, 0, 4, 5, 15, 1], [0, 0, -28, 2, -2, 28], [-8, 1, 5, 13, 0, -13], [-17, 1, 6, 23, 0, -23], [12, 12, 13, -9, -3, -13], [-20, 17, 17, 0, -6, -11], [-6, 0, 6, 15, -4, 9], [0, -10, 0, 10, 10, 10], [-2, 1, 14, 1, -2, 0], [11, 6, -3, -14, 0, 14], [4, -13, 0, 9, 9, 5], [-5, -14, 5, 5, 9, 9], [1, -14, -14, 0, 3, 11], [-6, 12, 12, -12, -12, 6], [-12, -11, 3, 12, 8, 8], [0, -2, -2, 3, -2, -2], [-18, 3, -9, 3, -18, 0], [-5, 5, -1, 5, -4, -4], [-3, 0, 3, -19, -5, -22], [1, 9, -3, 3, -9, -1], [7, 7, 4, -13, 6, -4], [-2, -13, 8, -13, -2, 0], [0, 3, -1, -23, -1, 3], [-10, -10, 7, -10, 20, -7], [1, 1, 17, 5, -6, -17], [-1, -1, -25, -1, -1, 0], [2, -18, 0, 16, 16, 14], [-1, 6, -13, 1, 7, 7], [31, -6, -6, 0, 4, 2], [7, -3, -3, 0, -10, 13], [-23, 4, -5, 23, 1, 1], [17, 1, 8, -9, 0, 9], [17, 7, -2, 7, 17, 0], [-13, -13, -3, -10, 23, 3], [13, 13, 4, -13, -17, -17], [14, 5, 5, 0, -7, 2], [-13, 29, -8, 8, -29, 13], [22, 22, 3, -17, -5, -3], [0, 0, 3, 13, 3, 0], [18, 1, -14, 1, 18, 0], [-10, -9, -2, 2, 9, 10], [-9, 0, 9, 17, -6, 8], [1, 29, 29, 0, -27, -2], [11, 15, -16, -11, 1, 1], [-4, 0, 0, 4, 4, 14], [13, -14, -25, -14, 13, 0], [0, 0, -11, 9, -11, 0], [-14, 5, -24, 14, 19, 19], [14, -25, -25, 0, 10, 15], [-5, -8, 16, 5, -8, -8], [-3, -11, 4, 7, 0, -7], [0, -2, 0, 2, 2, -33], [-5, 0, 5, 4, 42, -1], [-11, -24, -3, 3, 24, 11], [0, -17, 13, -20, 13, -17], [-12, -16, -8, 4, 0, -4], [-6, 6, 15, 21, 0, -21], [-2, -2, 14, -13, 15, -14], [-8, 0, 8, 13, -13, 5], [-3, -17, -17, 0, -8, 25], [-6, -6, -21, -13, 19, 21], [-11, 20, 0, -9, -9, -10], [0, 2, -13, 13, -13, 2], [-32, -8, -8, 0, 14, -6], [0, -4, 15, -16, 15, -4], [23, -1, 0, -22, -22, 16], [18, -18, 6, -18, 18, 0], [10, 1, 0, -11, -11, 4], [-19, -19, 4, 15, 4, -4], [-3, -23, 3, 3, 20, 20], [-17, 0, 17, 14, 8, -3], [6, -4, 19, 13, 0, -13], [-23, -11, -15, 8, 0, -8], [-22, 33, 7, 33, -22, 0], [0, -17, 8, -1, 8, -17], [-6, 0, 6, -17, 16, -23], [-20, -11, 11, 20, 0, 0], [-13, -13, 6, 19, -6, -6], [-12, 11, 0, 1, 1, 31], [-16, -4, -2, 2, 4, 16], [0, 0, 0, 0, 9, -9], [-3, -19, 31, 3, -12, -12], [-1, -34, 6, 7, 0, -7], [-10, 10, 0, 0, 0, 23], [-14, -14, 20, 3, 11, -20], [-15, 0, 15, 20, -8, 5], [-15, 0, 15, 18, 21, 3], [12, 11, 7, -5, 0, 5], [-26, 1, -3, 23, 0, -23], [0, 8, -3, -10, -3, 8], [-13, 19, 2, -2, -19, 13], [0, 0, 0, -18, 29, -18], [20, 20, -18, -20, 0, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_228_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_228_2_q_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_228_2_q_a(:prec:=6)
    chi := MakeCharacter_228_q();
    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_228_2_q_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_228_2_q_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_228_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, -27, 36, -30, 18, -6, 1]>],Snew);
    return Vf;
end function;