
// Make newform 32.3.c.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_32_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_32_c_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_3_c_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_3_c_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, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 4]];
    Rf_basisdens := [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_c();"
function MakeCharacter_32_c()
    N := 32;
    order := 2;
    char_gens := [31, 5];
    v := [1, 2];
    // 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_c_Hecke();"
function MakeCharacter_32_c_Hecke(Kf)
    N := 32;
    order := 2;
    char_gens := [31, 5];
    char_values := [[-1, 0], [1, 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 := 3;
    raw_aps := [[0, 0], [0, 1], [2, 0], [0, -2], [0, -1], [-14, 0], [18, 0], [0, -3], [0, 10], [-14, 0], [0, -8], [-30, 0], [-14, 0], [0, 7], [0, 4], [66, 0], [0, -13], [82, 0], [0, 1], [0, 14], [66, 0], [0, -4], [0, -35], [-30, 0], [-14, 0], [-94, 0], [0, 38], [0, 39], [18, 0], [98, 0], [0, 0], [0, 17], [-206, 0], [0, -49], [226, 0], [0, -22], [-46, 0], [0, -39], [0, -10], [-142, 0], [0, 69], [-30, 0], [0, -80], [-206, 0], [322, 0], [0, -50], [0, -35], [0, -56], [0, 41], [2, 0], [2, 0], [0, 52], [-46, 0], [0, 3], [2, 0], [0, -34], [402, 0], [0, 108], [-382, 0], [-350, 0], [0, -85], [2, 0], [0, 21], [0, -62], [530, 0], [-590, 0], [0, 143], [98, 0], [0, 43], [434, 0], [130, 0], [0, 102], [0, -76], [-254, 0], [0, 67], [0, 32], [-94, 0], [-558, 0], [-110, 0], [-430, 0], [0, -119], [322, 0], [0, 132], [210, 0], [0, -94], [0, 67], [-462, 0], [-590, 0], [658, 0], [0, 28], [0, 45], [0, 8], [0, -90], [0, -73], [0, -235], [0, -14], [-110, 0], [178, 0], [0, 95], [-110, 0], [0, -151], [690, 0], [0, 85], [530, 0], [0, -29], [66, 0], [0, 175], [-222, 0], [0, 218], [994, 0], [0, 200], [-318, 0], [322, 0], [0, 71], [0, 210], [-718, 0], [0, 49], [0, -258], [530, 0], [0, 77], [290, 0], [-14, 0], [-670, 0], [0, -57], [0, 325], [-558, 0], [-478, 0], [0, -228], [0, -70], [754, 0], [0, 41], [0, -154], [0, 116], [1442, 0], [-110, 0], [434, 0], [-990, 0], [0, -307], [-686, 0], [-270, 0], [0, -105], [898, 0], [0, -62], [0, -189], [18, 0], [0, 238], [-1022, 0], [850, 0], [0, 235], [0, 360], [-1454, 0], [-1374, 0], [0, 357], [0, -206], [0, -209], [0, 140], [0, -150], [754, 0], [-894, 0], [1298, 0], [0, 437], [-1134, 0], [0, -434], [0, 31], [914, 0], [0, 314], [0, -216], [-1054, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_32_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_32_3_c_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_3_c_a(:prec:=2)
    chi := MakeCharacter_32_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_c_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_3_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_32_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;