
// Make newform 32.2.g.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_32_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_32_g_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_32_2_g_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_32_2_g_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 := [2, -12, 30, -44, 43, -28, 14, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-5, 24, -48, 59, -42, 24, -7, 2], [8, -29, 54, -61, 43, -24, 7, -2], [11, -42, 73, -87, 60, -35, 10, -3], [13, -53, 91, -102, 70, -38, 11, -3], [-19, 77, -133, 155, -105, 60, -17, 5], [23, -89, 152, -170, 115, -63, 18, -5], [31, -126, 223, -256, 175, -98, 28, -8]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    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_32_g();"
function MakeCharacter_32_g()
    N := 32;
    order := 8;
    char_gens := [31, 5];
    v := [8, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_32_g_Hecke();"
function MakeCharacter_32_g_Hecke(Kf)
    N := 32;
    order := 8;
    char_gens := [31, 5];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 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, -1, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, -1, -1], [-2, -1, 1, -1, 1, -2, 0, 1], [1, 0, -1, 0, 1, 0, 1, -1], [0, 2, 0, -2, 0, 1, 0, -1], [0, 0, 0, 2, 2, -2, 2, 2], [0, -1, 0, 1, -2, 1, 0, -1], [-2, -1, 1, 1, -1, 0, 2, -1], [-2, 0, 4, 0, 0, -1, 2, -1], [4, 0, 0, 0, 0, -2, -2, 0], [0, 0, -2, 4, -2, 2, 3, 3], [3, 2, -2, -2, 2, 0, -4, 1], [-1, 0, -1, 0, 1, -2, -3, 3], [2, 2, -2, 0, 0, 0, 0, -2], [-2, -4, 2, 0, -2, 0, 3, -3], [-3, 0, 1, -2, 1, 2, 3, 3], [4, -2, -4, -2, 0, -1, 0, -1], [-6, -3, 0, -3, 0, -3, -6, -3], [0, 3, -3, 3, -3, -2, 0, 3], [-5, 0, 2, 2, 0, 0, 2, -3], [-2, -2, 2, 4, 4, 2, -2, 6], [4, 3, 0, -3, 6, 3, -4, -3], [1, 2, -4, 4, -2, -2, 0, 3], [4, 0, 0, 2, -2, 4, 4, 0], [-4, 0, 2, -4, 2, 2, -5, -5], [4, 3, 5, -5, -3, 2, 0, -9], [5, 0, 3, 0, -3, 8, -3, 3], [0, 0, 0, 0, 8, -5, 0, 5], [-2, -2, 2, -2, -2, -2, 2, 4], [-4, -4, -4, -4, 4, -2, -2, 0], [0, 1, 0, 1, 0, -5, 0, -5], [7, 4, 0, 0, 4, 0, 4, 7], [1, 8, -1, 0, 1, 0, 3, -3], [2, 4, 0, 0, 0, 2, -1, 1], [2, -1, 1, 1, -1, 0, 14, 3], [0, -2, -4, -2, 0, -9, -4, -9], [6, -3, 0, -3, 0, 3, 6, 3], [2, -1, -7, 7, 1, 6, 0, 5], [6, 2, 0, -2, -4, 7, -6, -7], [-2, -1, 0, 1, 6, -11, 2, 11], [4, 0, 0, 0, 0, 4, -7, 7], [-8, 4, 4, 4, -4, 6, 6, 0], [4, 4, 4, 2, -2, -4, -4, 0], [6, 0, -4, 4, -4, -2, -3, -3], [-6, 3, 5, -5, -3, 6, 0, 1], [-4, -1, 0, 1, -10, 7, 4, -7], [8, -8, -8, 8, -8, 2, 2, 0], [-4, 1, 8, 1, 0, 5, 4, 5], [4, 0, 0, 8, 0, -4, -1, -1], [-5, -4, -6, -6, -4, 0, -2, -11], [2, 2, -2, 0, 0, -2, 2, -18], [8, 8, -8, 6, 6, 6, -6, 6], [-1, 0, 1, -10, 1, 0, -3, -3], [0, 10, 10, -6, 6, 2, 2, 0], [0, -9, 1, -1, 9, -18, 0, -1], [-14, -4, 0, 4, 4, 1, 14, -1], [6, 6, -6, -4, -4, 12, -12, -6], [-16, -8, 6, 0, -6, -10, -7, 7], [-3, 0, 6, -6, 0, -2, 0, -3], [9, 0, 1, -2, 1, -10, -3, -3], [2, 0, 8, -4, 8, -10, 9, 9], [-2, -1, 0, 1, -10, 7, 2, -7], [0, 3, 5, 5, 3, 0, -2, 5], [-1, -6, -6, 6, 6, 0, 0, 7], [8, 8, 0, 8, 0, 1, 8, 1], [11, 8, -5, 0, 5, 6, -1, 1], [0, 0, 0, -4, -4, 0, 0, 0], [-5, 0, -3, 6, -3, 8, -11, -11], [-12, -12, 8, -12, 0, -3, -4, -3], [-4, -6, -6, 10, -10, 2, 2, 0], [-4, -1, 9, -9, 1, 10, 0, -5], [-6, -6, 6, 0, 0, -12, 12, 10], [-8, 0, 2, 0, -2, -6, 5, -5], [-5, 0, 5, 6, 5, 0, 9, 9], [-8, -4, -4, -12, 12, -8, -8, 0], [-4, 0, -6, -12, -6, 10, -1, -1], [-6, -4, 0, 4, -4, 13, 6, -13], [-8, -8, 8, -6, -6, 2, -2, -6], [3, 2, -2, 2, -2, -4, 0, -1], [8, 5, -24, 5, 0, 3, -16, 3], [-2, 0, 8, -4, 8, -6, 9, 9], [-6, -6, 6, 0, 0, 6, -6, 2], [4, 4, -4, -2, -2, 6, -6, 10], [0, -9, 1, 1, -9, 0, 2, 1], [11, 0, -11, 14, -11, 0, 7, 7], [-4, -4, -4, -6, 6, -8, -8, 0], [3, 10, 2, 2, 10, 0, 8, 5], [4, -4, 0, 4, 0, -17, -4, 17], [-6, -6, 6, 0, 0, 2, -2, -14], [8, 3, 0, -3, -10, 1, -8, -1], [4, 4, 4, 4, -4, 14, 14, 0], [10, 3, -3, 3, -3, -14, 0, -7], [-13, -16, 3, 0, -3, -10, 5, -5], [6, -5, 0, 5, 14, -11, -6, 11], [18, 7, 1, 1, 7, 0, 2, 19], [-6, -6, -8, -6, 0, 13, -14, 13], [-17, 4, 4, 4, 4, 0, 0, -13], [-7, -24, 3, 0, -3, -4, -3, 3], [12, 10, -4, 10, 0, -5, 8, -5], [-10, -3, 8, -3, 0, -11, -2, -11], [10, -2, 0, 2, 12, -5, -10, 5], [-8, 7, 0, -7, -10, 13, 8, -13], [-9, 4, 12, -12, -4, 0, 0, -3], [3, 0, -3, -2, -3, 0, -9, -9], [-2, -4, -4, -8, 8, 0, 0, 0], [-9, 8, -1, 0, 1, -10, -13, 13], [-8, -8, 8, -4, -4, 0, 0, -12], [-16, -9, 1, 1, -9, 0, 14, -15], [21, 10, -12, 12, -10, 22, 0, -9], [-12, 0, 0, -8, 8, -4, -4, 0], [-14, 0, -4, 4, -4, 18, -11, -11], [11, -2, -4, -4, -2, 0, -22, 7], [-3, 16, -1, 0, 1, -4, 1, -1], [-6, 3, 5, 5, 3, 0, -2, -1], [8, -4, -4, 6, -6, -12, -12, 0], [-2, 5, 16, 5, 0, 1, 14, 1], [4, -5, 5, -5, 5, 2, 0, -9], [2, 12, 0, -12, -4, 17, -2, -17], [-4, -13, 0, 13, -10, -9, 4, 9], [4, 0, -8, 0, 8, -4, -15, 15], [-8, 8, 8, 14, -14, 24, 24, 0], [0, 0, -6, 20, -6, 6, 19, 19], [25, 16, -13, 0, 13, 12, -1, 1], [24, 7, 0, -7, 6, 7, -24, -7], [14, -2, -8, -2, 0, -7, 6, -7], [18, 0, -10, -16, -10, -8, -23, -23], [10, 10, -10, 0, 0, -10, 10, 2], [-12, -17, 9, 9, -17, 0, 2, -3], [0, 4, 8, 4, 0, 21, 8, 21], [-6, 9, 8, 9, 0, -5, 2, -5], [-26, -1, 1, -1, 1, -10, 0, 25], [6, 6, -6, -12, -12, 14, -14, 10], [2, 4, -12, 0, 12, -10, 7, -7], [-9, -14, -10, 10, 14, -12, 0, 19], [8, -4, -4, -10, 10, 4, 4, 0], [-26, 0, 6, 0, 6, 20, -3, -3], [-2, -1, 0, 1, 6, -7, 2, 7], [-18, -14, 8, -14, 0, -15, -10, -15], [-9, -2, 6, 6, -2, 0, 16, -3], [-9, -16, -1, 0, 1, -10, 11, -11], [4, -16, 10, 0, -10, 14, 9, -9], [8, -1, -7, -7, -1, 0, 6, 1], [17, 0, 1, 6, 1, -18, 13, 13], [10, 10, 0, 10, 0, 13, 10, 13], [10, -5, 5, -5, 5, -14, 0, -15], [-10, -12, -6, 0, 6, -16, 3, -3], [-1, 4, 8, -8, -4, 12, 0, -7], [-17, 0, 5, -2, 5, 12, 25, 25], [0, 4, 4, 4, -4, 14, 14, 0], [0, -4, 0, 4, 0, -13, 0, 13], [16, 16, -16, 0, 0, 0, 0, -4], [4, 3, 0, -3, -10, 5, -4, -5], [-8, -5, 21, 21, -5, 0, 10, 13], [5, 16, 3, 0, -3, 8, -13, 13], [6, 6, -6, 12, 12, -4, 4, 22], [14, 15, -15, -15, 15, 0, -34, -1], [8, 0, 0, -18, 18, 4, 4, 0], [11, 8, -10, -10, 8, 0, -26, 1], [12, -4, 0, 4, -16, 11, -12, -11], [-22, -9, 0, 9, 22, -9, 22, 9], [-5, -12, -4, 4, 12, -24, 0, 9], [10, 3, -11, 11, -3, 30, 0, 1], [-25, -16, 3, 0, -3, -22, -15, 15], [8, 8, -8, -10, -10, -6, 6, -10], [-20, 3, -3, -3, 3, 0, 14, -23], [-12, 20, 20, -4, 4, 12, 12, 0], [2, 0, 10, -24, 10, -12, 1, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_32_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_32_2_g_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_32_2_g_b(:prec:=8)
    chi := MakeCharacter_32_g();
    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_32_2_g_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_32_2_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_32_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -16, 96, -24, -32, 0, 8, 4, 1]>],Snew);
    return Vf;
end function;