
// Make newform 231.2.j.f in Magma, downloaded from the LMFDB on 24 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_f();".
// 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_f();".
// 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, 3, 5, 3, 4, -3, 5, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-7, -12, -7, -4, 0, -3, 2, -1], [9, 6, 19, -16, 20, -7, 0, 1], [-1, -10, 13, -16, 12, -9, 4, -1], [-5, -8, -13, -4, 8, -17, 10, -3], [-1, 6, 1, 16, -20, 23, -12, 3], [-7, -12, -11, -28, 32, -35, 18, -5]];
    Rf_basisdens := [1, 1, 8, 8, 8, 8, 8, 8];
    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_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, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, -1, 0, 1, 1, 0], [1, 0, 0, -1, 1, 0, 0, 1], [0, 2, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, -1, 0, 0, 0], [-2, 0, 0, 0, 2, 0, 0, 1], [0, 0, 0, 0, 0, -2, -2, 0], [0, 0, 4, -2, 2, 0, 0, 4], [2, 2, 2, -2, 4, -2, -2, 4], [-3, 0, 0, 3, 0, -4, 0, -3], [-1, 0, 4, 0, 1, -4, 2, -1], [2, 4, 0, 4, 2, -2, -2, 0], [-1, 0, -4, 0, -4, 4, 4, -1], [2, -6, -2, -2, -4, 6, 2, -4], [0, 0, 0, -5, 0, 6, 0, 5], [-2, 2, 4, 2, 4, -2, -4, 4], [-5, 2, 0, 2, -5, 0, 0, 0], [2, 0, -2, 0, -4, 2, 0, 2], [0, -4, -2, 2, -2, 0, 4, 0], [-8, 0, 0, 3, 0, -2, 0, -3], [0, -4, -4, 3, -3, 0, 4, -9], [0, 0, -2, 0, -2, 2, 0, 0], [1, 2, 0, 4, 1, 0, 0, 0], [0, 2, -6, 2, -2, 0, -2, 0], [-2, -6, -6, 4, 0, 2, 0, -4], [6, 0, 0, -8, 6, -4, -4, 0], [4, -2, 0, -6, 4, 0, 0, 0], [6, 0, 0, 0, 12, 0, -4, 6], [5, 8, 8, -5, 4, -8, -8, 4], [-3, 8, 8, -1, 0, -4, 0, 1], [2, 2, 2, -2, 5, -2, -2, 5], [0, 2, 6, -11, 11, 0, -2, -2], [-2, -2, -2, 6, 0, -6, 0, -6], [0, 2, -2, 5, -5, 0, -2, -14], [8, 0, 0, 0, 6, 0, -8, 8], [0, 4, 8, 4, -4, 0, -4, 10], [-1, -12, -10, 1, 2, 12, 10, 2], [4, -6, -4, -4, 4, 6, 4, 4], [1, -8, 0, -3, 1, 2, 2, 0], [-4, 0, 0, -2, -4, 4, 4, 0], [-12, 6, 2, 12, -10, -6, -2, -10], [11, 0, 0, -11, 8, 0, 0, 8], [0, 12, 2, -4, 4, 0, -12, 8], [7, 0, -4, 0, 1, 4, -12, 7], [0, 0, 4, 5, -5, 0, 0, -5], [1, -2, -2, -11, 0, 8, 0, 11], [-10, 14, 14, 2, 0, -12, 0, -2], [11, -12, 0, -1, 11, 2, 2, 0], [-10, 10, 4, 10, -6, -10, -4, -6], [-2, 0, 6, 0, 16, -6, -6, -2], [10, -10, 0, -8, 10, 4, 4, 0], [-8, -8, 0, 2, -8, 0, 0, 0], [-2, -12, -12, 2, -11, 12, 12, -11], [2, -12, -12, 4, 0, 6, 0, -4], [14, 8, 0, 4, 14, -4, -4, 0], [2, 0, 2, 0, 0, -2, 2, 2], [6, -4, -4, 11, 0, 0, 0, -11], [0, 6, -8, 8, -8, 0, -6, 6], [12, 0, 12, 0, 14, -12, -4, 12], [15, 8, 0, 1, 15, -8, -8, 0], [0, 4, 6, -11, 11, 0, -4, 13], [-20, 0, 2, 20, -14, 0, -2, -14], [-8, 0, -2, 0, -16, 2, -4, -8], [-4, -6, -6, -8, 0, 10, 0, 8], [14, 0, 8, -14, 6, 0, -8, 6], [0, 6, 10, -2, 2, 0, -6, 6], [15, -6, 0, -14, 15, 8, 8, 0], [-9, 2, 2, -13, 0, -4, 0, 13], [3, 0, 20, 0, 8, -20, -8, 3], [0, 0, -8, -7, 7, 0, 0, -7], [-12, -2, -8, 12, -6, 2, 8, -6], [-8, 0, 0, 4, 0, 8, 0, -4], [7, 0, -8, 0, -16, 8, 8, 7], [-18, 0, -14, 0, -8, 14, 14, -18], [6, 4, 4, 8, 0, -16, 0, -8], [0, 2, -10, 15, -15, 0, -2, -18], [-2, -6, 0, 6, -2, 0, 0, 0], [19, 0, -2, 0, 14, 2, -2, 19], [22, 0, 0, -6, 0, 4, 0, 6], [0, 2, 10, -7, 7, 0, -2, -6], [2, -10, 0, -6, 2, 4, 4, 0], [-10, -8, -8, 8, 0, -2, 0, -8], [-3, 4, -8, 3, -4, -4, 8, -4], [5, -16, 0, -9, 5, 8, 8, 0], [0, 0, 6, 0, 14, -6, -8, 0], [-2, 2, 2, 20, 0, -18, 0, -20], [-6, 4, -8, 6, -11, -4, 8, -11], [-13, 4, 0, 15, -13, -2, -2, 0], [0, -4, -4, 3, -3, 0, 4, -32], [-24, 2, 2, 18, 0, -8, 0, -18], [-4, 0, 0, -7, 0, -2, 0, 7], [0, 18, 4, -8, 8, 0, -18, 24], [0, 2, 12, -6, 6, 0, -2, 6], [-20, 14, 2, 20, -7, -14, -2, -7], [-8, 0, -4, 0, 8, 4, 4, -8], [-11, 0, 12, 0, -19, -12, 10, -11], [0, 14, 14, 0, 14, -14, -14, 14], [4, 22, 8, -4, 12, -22, -8, 12], [18, 0, 10, 0, 18, -10, -8, 18], [0, 6, -12, -2, 2, 0, -6, 10], [-1, 0, 0, 9, -1, -4, -4, 0], [-12, -12, -8, 12, -8, 12, 8, -8], [9, 0, 8, 0, 3, -8, -2, 9], [-14, 4, 0, 12, -14, -10, -10, 0], [-18, 4, 8, 18, -14, -4, -8, -14], [-1, 2, 2, -17, 0, 8, 0, 17], [0, -2, -18, 10, -10, 0, 2, -22], [12, 0, -8, 0, -10, 8, -6, 12], [-8, 12, 12, 10, 0, -16, 0, -10], [0, 0, 16, 1, -1, 0, 0, -12], [-14, 0, 2, 0, -12, -2, 26, -14], [10, -6, 0, -16, 10, 8, 8, 0], [-12, -16, -24, 12, -25, 16, 24, -25], [-2, 8, 8, 15, 0, -22, 0, -15], [34, -8, 4, -34, 30, 8, -4, 30], [5, 0, 18, 0, -2, -18, -10, 5], [2, 26, 26, -2, 19, -26, -26, 19], [0, 4, 12, -10, 10, 0, -4, 16], [-8, -18, 0, 4, -8, 2, 2, 0], [11, 0, 2, 0, 10, -2, -6, 11], [28, -4, -4, -24, 0, 8, 0, 24], [16, 14, 14, -24, 0, -2, 0, 24], [3, 8, 0, 16, 3, -4, -4, 0], [0, -6, 16, -6, 6, 0, 6, 14], [1, 0, 0, 3, 0, -12, 0, -3], [-32, 10, 0, 22, -32, 12, 12, 0], [17, -8, -6, -17, 22, 8, 6, 22], [0, 8, 16, -21, 21, 0, -8, 17], [-8, 0, 6, 0, -24, -6, -2, -8], [-4, -4, -4, -12, 0, 4, 0, 12], [-20, 0, -10, 0, -2, 10, 8, -20], [-15, 22, 0, -12, -15, -12, -12, 0], [0, 4, 0, 7, -7, 0, -4, -33], [17, -4, 14, -17, 26, 4, -14, 26], [15, -8, 0, 4, 15, -16, -16, 0], [22, -4, 0, -20, 22, 2, 2, 0], [-14, 2, 2, 14, 0, 4, 0, -14], [-8, -12, -12, 8, 8, 12, 12, 8], [0, -22, -14, 4, -4, 0, 22, -24], [0, -8, -6, 16, -16, 0, 8, -20], [0, -22, -22, 11, -11, 0, 22, -18], [-4, 2, 2, 4, 4, -2, -2, 4], [4, 0, 12, 0, 14, -12, 8, 4], [-5, 2, 0, 16, -5, 4, 4, 0], [4, 24, 0, 24, 4, -36, -36, 0], [-6, 0, -24, 0, -26, 24, 10, -6], [32, 0, -2, -32, 10, 0, 2, 10], [0, 12, -2, 16, -16, 0, -12, 14], [-18, 22, 22, -6, 0, -18, 0, 6], [2, -14, -14, -22, 0, 30, 0, 22], [0, 12, 0, -24, 24, 0, -12, 20], [8, -24, -8, -8, 3, 24, 8, 3], [-22, 2, 2, -2, 0, 12, 0, 2], [0, 0, -12, 0, -24, 0, 12, -24], [16, 0, 8, 0, 2, -8, 6, 16], [0, 0, 6, -9, 9, 0, 0, 5], [-4, 24, 0, -4, -4, -24, -24, 0], [0, -8, -26, 11, -11, 0, 8, -23], [0, -10, 4, -16, 16, 0, 10, -16], [8, 16, 0, -4, 8, 0, 0, 0], [0, -4, -8, 18, -18, 0, 4, -6], [-24, 16, 16, 8, 0, -12, 0, -8], [-7, 0, -14, 0, -22, 14, -14, -7], [-23, 10, 10, 25, 0, -24, 0, -25], [18, -22, -8, -18, 36, 22, 8, 36], [5, -8, 0, 9, 5, 10, 10, 0], [2, 0, -10, 0, -10, 10, -16, 2], [12, -4, -4, -8, 0, 16, 0, 8], [16, 0, 14, 0, 8, -14, -2, 16]];
    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_f();".
// 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_f(:prec:=8)
    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_f();".
// 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_f( : 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, -4, 39, -8, 9, 4, 1, -2, 1]>],Snew);
    return Vf;
end function;