
// Make newform 32.2.g.a in Magma, downloaded from the LMFDB on 25 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_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_32_2_g_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, 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_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, 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, -1, 0, -1], [0, 1, 1, 0], [-1, -1, -2, 2], [1, 0, 1, 0], [-2, 2, -1, -1], [1, 0, 0, -1], [0, -2, 0, -2], [-2, -3, 3, 2], [3, 0, -3, 4], [-1, 2, 2, -1], [-4, 0, 0, 0], [1, 1, 0, 0], [-3, 0, 3, 4], [4, -4, -3, -3], [0, 4, 6, 4], [1, -1, -4, -4], [-4, -4, 1, -1], [1, 0, 0, 1], [-2, 3, 3, -2], [-3, 4, -3, 0], [7, 0, -7, 0], [0, 0, -6, 0], [4, -1, 1, -4], [3, -8, 3, 0], [-10, 6, 0, -6], [5, 5, 4, -4], [1, -12, 1, 0], [-2, 2, -1, -1], [-5, 6, -6, 5], [0, 4, 12, 4], [-4, -12, 0, 12], [-10, -1, -1, -10], [-3, 0, 3, -8], [4, -4, 9, 9], [-5, 5, -10, -10], [7, 0, -7, -12], [1, 0, 0, 1], [4, 9, 9, 4], [9, -8, 9, 0], [1, -4, 4, -1], [-8, -1, 1, 8], [7, -7, 6, 6], [-12, 0, 0, 0], [-10, -6, 0, 6], [11, 11, -2, 2], [1, 24, 1, 0], [4, -9, 9, -4], [-4, 12, 0, -12], [8, -7, -7, 8], [-17, -17, 6, -6], [3, 0, -3, -8], [0, -8, -6, -8], [0, -6, 0, -6], [-4, -4, -11, 11], [6, 0, 0, 0], [-3, 4, -3, 0], [7, -10, 10, -7], [0, 0, -18, 0], [1, -1, 0, 0], [-9, 4, -9, 0], [-2, -2, -9, 9], [-1, -1, -14, 14], [-2, 9, -9, 2], [3, 0, -3, -8], [1, 12, 1, 0], [11, 2, 2, 11], [-2, 2, 3, 3], [0, -12, 0, -12], [8, 8, 1, -1], [7, -18, -18, 7], [6, 0, 0, 0], [-15, 4, -15, 0], [0, 0, 6, 0], [-11, 11, 12, 12], [4, 4, 21, -21], [0, 12, 0, -12], [-19, -19, 4, -4], [-17, 6, -6, 17], [0, -2, 0, -2], [13, -12, 13, 0], [2, -13, -13, 2], [7, 7, -6, 6], [0, 4, 18, 4], [0, 6, 24, 6], [-17, 0, 17, 0], [-10, -10, -5, 5], [6, 18, 0, -18], [1, 0, -1, -12], [7, 2, -2, -7], [0, 12, -6, 12], [10, 17, -17, -10], [12, -12, 0, 12], [-11, 0, -11, 0], [16, -16, 17, 17], [4, 3, -3, -4], [3, 0, -3, -20], [-7, 20, 20, -7], [9, 0, -9, -8], [10, -10, 3, 3], [-11, 12, 12, -11], [-14, -9, -9, -14], [13, -16, 16, -13], [10, -7, 7, -10], [9, -8, 9, 0], [-2, -2, 3, -3], [2, -12, 0, 12], [-2, 2, -13, -13], [0, -20, 0, -20], [-21, 0, 21, -8], [-5, 24, -5, 0], [-16, 12, 0, -12], [19, 19, 6, -6], [3, 0, -3, 28], [-14, 14, 3, 3], [7, 0, -7, 36], [-18, -18, 0, 18], [-2, -21, -21, -2], [9, 4, 9, 0], [-17, -10, 10, 17], [-8, 11, -11, 8], [-17, 17, -18, -18], [14, -6, 0, 6], [5, 5, -20, 20], [-8, 8, 5, 5], [-8, -21, 21, 8], [5, 8, 8, 5], [1, 1, -12, 12], [0, 4, -30, 4], [7, 0, -7, -24], [-17, 6, 6, -17], [4, 9, 9, 4], [9, -32, 9, 0], [0, -12, -6, -12], [-17, 17, -6, -6], [-27, 4, -27, 0], [14, 6, 0, -6], [-7, -7, -20, 20], [4, 3, -3, -4], [-7, 20, 20, -7], [-15, 0, 15, -20], [-26, 26, 3, 3], [-11, 11, -4, -4], [19, 0, -19, 24], [14, 14, -17, 17], [25, 12, 12, 25], [21, 16, 21, 0], [25, -25, 12, 12], [21, 16, 21, 0], [16, 16, 9, -9], [-12, -24, 0, 24], [-17, -6, 6, 17], [0, 16, 0, 16], [34, -15, 15, -34], [-9, 0, 9, 16], [-8, 8, -3, -3], [0, 28, 6, 28], [-17, 0, 17, 12], [-18, -6, 0, 6], [19, 0, -19, 0], [7, 2, -2, -7], [-14, -19, 19, 14], [9, -8, 9, 0], [-23, -24, -23, 0], [16, -16, -7, -7], [0, 10, 0, 10], [3, 0, -3, -32], [-16, 0, 0, 0], [-11, -11, 24, -24]];
    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_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_32_2_g_a(:prec:=4)
    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_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_32_2_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_32_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 4, 2, 0, 1]>],Snew);
    return Vf;
end function;