
// Make newform 231.2.j.d in Magma, downloaded from the LMFDB on 25 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_d();".
// 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_d();".
// 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, 2, -1, 0], [0, 0, 0, -1], [-2, 4, -1, 2], [1, -1, 1, -1], [2, 1, 2, 0], [0, 0, 0, 0], [-2, 0, -2, 2], [6, -6, 0, -3], [9, -7, 7, -9], [8, -8, 0, -2], [0, 5, -3, 5], [-1, 0, 0, 0], [0, -7, 4, -7], [-7, 6, -6, 7], [3, -3, 0, 6], [-2, 9, -2, 0], [-8, 0, -2, 2], [4, 1, 4, 0], [9, -9, 0, -6], [-9, 6, -6, 9], [0, 6, 0, 0], [-7, 0, 1, -1], [17, -17, 17, -17], [-2, 12, -12, 2], [-6, 6, 0, 9], [0, -2, 19, -2], [-2, 0, -9, 9], [0, -14, 11, -14], [7, -4, 7, 0], [1, 0, -7, 7], [-8, 16, -8, 0], [-9, 9, 0, 12], [5, 5, 5, 0], [0, -5, -13, -5], [0, 7, -13, 7], [2, -15, 15, -2], [-10, 11, -11, 10], [0, -1, -13, -1], [0, -2, 1, -2], [2, -3, 2, 0], [-2, 2, 0, -11], [8, -18, 8, 0], [-8, 0, -2, 2], [-2, 0, -9, 9], [-9, 15, -15, 9], [0, -6, -3, -6], [21, -21, 0, -6], [8, -2, 2, -8], [-5, 3, -3, 5], [0, -3, 24, -3], [-21, 0, -6, 6], [-13, -9, 9, 13], [8, -8, 0, -22], [-3, 0, 6, -6], [-8, -11, -8, 0], [-12, 12, 0, -1], [18, -6, 6, -18], [-1, -14, -1, 0], [0, -6, 17, -6], [-2, 2, 0, 12], [13, 0, 10, -10], [0, 11, -1, 11], [11, -2, 11, 0], [-13, -7, 7, 13], [-11, 0, -6, 6], [-6, 6, 0, -5], [12, 6, 12, 0], [0, 29, -12, 29], [-23, 0, -4, 4], [-23, 23, 0, 19], [-6, 6, 0, -15], [24, 0, 10, -10], [3, 6, 3, 0], [11, -16, 16, -11], [14, -14, 0, -12], [9, 0, 27, -27], [7, -23, 7, 0], [27, -28, 28, -27], [-11, 0, -7, 7], [0, -15, 27, -15], [31, -20, 20, -31], [12, -12, 0, 10], [3, 0, -4, 4], [0, -14, -14, -14], [-7, -7, 7, 7], [-15, 12, -15, 0], [17, 0, 10, -10], [-18, 0, -29, 29], [-8, -19, -8, 0], [-17, 6, -17, 0], [0, 1, 15, 1], [6, -6, 0, 0], [0, 0, 0, 5], [0, -28, 13, -28], [0, 24, 3, 24], [-10, 10, 0, 13], [-28, 25, -28, 0], [-17, 12, -12, 17], [0, 22, -13, 22], [-2, 2, 0, -2], [29, -3, 3, -29], [0, 26, -3, 26], [7, 0, 0, 0], [-9, 24, -9, 0], [16, -16, 0, -26], [-14, 0, -16, 16], [-17, -4, -17, 0], [-24, 24, 0, 0], [18, -21, 21, -18], [0, 25, -6, 25], [-7, 0, -5, 5], [0, 18, 6, 18], [-10, 10, 0, 3], [0, 21, -6, 21], [0, 16, 0, 0], [-11, 25, -25, 11], [6, -6, 0, 3], [27, 0, 24, -24], [-3, 0, 0, 0], [15, -27, 27, -15], [-25, 17, -25, 0], [-15, 0, -28, 28], [-9, 0, 0, 9], [0, 0, 27, 0], [14, 28, 14, 0], [-1, 1, 0, 8], [24, 0, -3, 3], [18, -18, 0, 7], [-26, -1, 1, 26], [5, -34, 5, 0], [0, -3, -24, -3], [-44, 42, -42, 44], [3, -8, 8, -3], [42, 0, 4, -4], [0, -12, 30, -12], [3, 28, 3, 0], [25, -8, 25, 0], [-21, 3, -21, 0], [0, -18, 6, -18], [-10, 10, 0, -12], [4, 32, -32, -4], [-15, 31, -31, 15], [27, -27, 0, -21], [0, 26, -13, 26], [11, -27, 11, 0], [-53, 0, -2, 2], [-27, 0, -24, 24], [-20, 1, -20, 0], [0, 19, 11, 19], [-7, 0, -8, 8], [0, -13, -12, -13], [-4, 4, 0, 19], [-21, 15, -21, 0], [37, -44, 44, -37], [-12, 36, -12, 0], [32, -36, 32, 0], [7, -7, 7, -7], [10, 38, 10, 0], [0, 0, -1, 1], [-11, 11, 0, -26], [40, 0, -6, 6], [0, -6, -15, -6], [-33, 33, -33, 33], [-8, 8, 0, 25], [12, 0, 35, -35], [12, -12, 0, 6]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : 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;