
// Make newform 231.2.j.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_231_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_231_j_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_231_2_j_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_231_2_j_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, -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_231_j();"
function MakeCharacter_231_j()
    N := 231;
    order := 5;
    char_gens := [155, 199, 211];
    v := [5, 5, 3];
    // 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_231_j_Hecke();"
function MakeCharacter_231_j_Hecke(Kf)
    N := 231;
    order := 5;
    char_gens := [155, 199, 211];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1]];
    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, 1, 0], [0, 0, -1, 0], [1, 0, 1, 0], [0, 0, 0, 1], [2, 0, 1, 2], [-3, 1, -1, 3], [4, -3, 4, 0], [0, 4, 0, 4], [2, 0, 6, -6], [-2, 2, 0, -1], [-3, 3, -3, 3], [0, 0, 0, -6], [0, -3, -5, -3], [-9, 0, -4, 4], [0, 3, -10, 3], [7, 2, -2, -7], [-1, 1, 0, 4], [8, -5, 8, 0], [12, 0, 2, -2], [-4, 11, -4, 0], [-1, 1, 0, -8], [3, 2, -2, -3], [-4, -2, -4, 0], [-3, 0, 7, -7], [9, -15, 15, -9], [-10, 8, -8, 10], [-4, 4, 0, 13], [0, -2, 9, -2], [-6, 0, 3, -3], [0, -10, 5, -10], [7, -4, 7, 0], [5, 0, 3, -3], [4, 4, 4, 0], [1, -1, 0, -2], [-1, 3, -1, 0], [0, 7, -5, 7], [0, -11, 11, -11], [14, -7, 7, -14], [4, -7, 7, -4], [0, 11, -7, 11], [0, -6, -9, -6], [-4, 7, -4, 0], [-2, 2, 0, 15], [-12, 2, -12, 0], [-4, 0, 14, -14], [-12, 0, -23, 23], [15, -9, 9, -15], [0, 8, 7, 8], [5, -5, 0, 4], [-4, -14, 14, 4], [17, -19, 19, -17], [0, -1, -20, -1], [3, 0, 12, -12], [9, -1, 1, -9], [16, -16, 0, -14], [-13, 0, 6, -6], [10, 13, 10, 0], [-18, 18, 0, 11], [-6, 2, -2, 6], [-3, -6, -3, 0], [0, 16, -13, 16], [-6, 6, 0, 4], [13, 0, -8, 8], [0, 11, -11, 11], [-5, -24, -5, 0], [25, -9, 9, -25], [-23, 0, 6, -6], [-2, 2, 0, 3], [4, -6, 4, 0], [0, -7, -6, -7], [7, 0, -18, 18], [-1, 1, 0, -11], [0, 0, 0, 5], [-24, 0, 6, -6], [-5, 10, -5, 0], [7, -14, 14, -7], [-2, 2, 0, 16], [-5, 0, -9, 9], [-19, 3, -19, 0], [-15, 26, -26, 15], [-35, 0, -1, 1], [0, -3, 23, -3], [9, -12, 12, -9], [-8, 8, 0, 2], [27, 0, 14, -14], [0, 14, 2, 14], [7, 15, -15, -7], [-19, 12, -19, 0], [11, 0, 28, -28], [-2, 0, -21, 21], [6, 17, 6, 0], [1, -32, 1, 0], [0, 1, -17, 1], [6, -6, 0, 12], [0, 0, 0, -7], [0, -34, 13, -34], [0, 18, -17, 18], [12, -12, 0, -1], [-30, 19, -30, 0], [19, -32, 32, -19], [0, -26, 23, -26], [-30, 30, 0, 2], [23, -15, 15, -23], [0, -6, -29, -6], [31, 0, 8, -8], [3, -42, 3, 0], [-36, 36, 0, 10], [10, 0, 0, 0], [9, 0, 9, 0], [12, -12, 0, 12], [4, 27, -27, -4], [0, 9, 10, 9], [3, 0, -7, 7], [0, -10, -6, -10], [6, -6, 0, 3], [0, 7, -2, 7], [-16, 20, -16, 0], [-5, -11, 11, 5], [-10, 10, 0, 25], [-11, 0, -16, 16], [5, 0, 22, -22], [23, -31, 31, -23], [7, -25, 7, 0], [-13, 0, -4, 4], [9, -22, 22, -9], [0, 8, 21, 8], [6, -4, 6, 0], [-25, 25, 0, 18], [6, 0, -13, 13], [-12, 12, 0, -25], [42, -45, 45, -42], [-5, -30, -5, 0], [0, 9, 36, 9], [-40, 22, -22, 40], [-29, 26, -26, 29], [-34, 0, -32, 32], [0, -8, -18, -8], [-9, 14, -9, 0], [3, -42, 3, 0], [33, 1, 33, 0], [0, 6, 38, 6], [10, -10, 0, 20], [-20, 16, -16, 20], [-37, 29, -29, 37], [7, -7, 0, 5], [0, 4, -21, 4], [-11, -7, -11, 0], [-27, 0, -8, 8], [-7, 0, 14, -14], [-36, 27, -36, 0], [0, 3, -17, 3], [-9, 0, 6, -6], [0, -5, -36, -5], [6, -6, 0, -35], [-17, 39, -17, 0], [35, -40, 40, -35], [0, 4, 0, 0], [16, -16, 16, 0], [11, 7, -7, -11], [-22, 38, -22, 0], [-8, 0, 21, -21], [-13, 13, 0, 2], [-40, 0, -14, 14], [0, -12, 9, -12], [-23, -9, 9, 23], [22, -22, 0, -37], [-28, 0, -1, 1], [-44, 44, 0, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_231_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_231_2_j_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_231_2_j_b(:prec:=4)
    chi := MakeCharacter_231_j();
    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_231_2_j_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_231_2_j_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_231_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 3, 4, 2, 1]>],Snew);
    return Vf;
end function;