
// Make newform 256.2.g.c in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_256_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_256_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_256_2_g_c();".
// 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_256_2_g_c();".
// 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, -37, 41, -27, 14, -4, 1], [-10, 30, -44, 43, -28, 14, -4, 1], [15, -66, 121, -146, 102, -59, 17, -5], [-19, 77, -133, 155, -105, 60, -17, 5], [20, -82, 145, -163, 113, -62, 18, -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_256_g();"
function MakeCharacter_256_g()
    N := 256;
    order := 8;
    char_gens := [255, 5];
    v := [8, 7];
    // 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_256_g_Hecke();"
function MakeCharacter_256_g_Hecke(Kf)
    N := 256;
    order := 8;
    char_gens := [255, 5];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 1, 0, 0, -1], [1, 1, 0, 1, 0, 0, 2, 1], [0, 0, 1, 0, 2, 0, 1, 1], [0, 0, 0, 0, 0, -2, -1, -1], [-2, 1, 1, -1, 3, -1, -1, -2], [1, 0, -1, 1, -1, -1, 0, 1], [1, 0, 1, 0, -1, 1, 1, -1], [0, 0, 2, 2, 0, 0, 3, -1], [-4, 0, 0, 0, 2, 0, 2, 0], [2, -2, 0, -2, -3, 2, -2, 3], [1, 0, 2, 0, -2, 2, 2, -1], [-2, 0, 1, 0, -2, 0, -1, -3], [0, -1, -1, -1, -1, -1, -1, -2], [-2, 2, 2, 0, -1, 2, 2, -3], [-2, -1, 0, -1, 3, 1, 2, -3], [-4, 2, -2, -2, -2, 0, -1, -1], [-3, -3, 0, 0, -6, 0, -3, 3], [3, -3, 0, -3, 0, 0, 2, 3], [-5, 1, -1, -1, 1, -1, -1, 3], [4, -1, -1, 3, 1, 3, -3, 6], [1, 0, 3, -3, -1, 3, 6, 3], [-3, 3, 1, 3, 1, -1, -1, -3], [4, 1, -1, 1, 3, -1, 3, 0], [2, 2, 0, 2, 5, -2, -2, -5], [-9, 1, 4, 1, 4, -4, 2, -9], [8, 0, -3, 0, -6, 0, 5, -3], [4, 0, -4, 4, -4, 0, 1, 5], [2, -2, -2, 0, 0, 0, -4, -4], [4, 4, -4, 0, -2, 0, -2, 0], [-1, 1, 0, 0, 0, 0, -5, 5], [3, 2, 2, -2, 6, 2, 2, -7], [-4, 4, 1, 0, 4, 4, 1, 3], [0, -2, 0, 0, 1, -2, -2, 1], [-3, 0, 1, 0, -13, 1, 1, 3], [0, 2, -2, -2, 2, 0, 7, -9], [9, -3, 0, 0, 6, 0, 3, -3], [5, -3, -4, -3, -4, 4, -10, 5], [-8, 0, 2, -2, 8, -2, -5, -7], [-5, 0, 3, -3, 5, -1, -8, -11], [-4, 0, 0, 0, 7, 0, -4, 7], [8, -4, 4, 0, -2, 0, -2, 0], [4, 3, -3, -1, -7, 1, -7, 0], [-4, -2, 0, -4, 3, 2, 2, -3], [1, 1, 4, 1, 4, -4, -2, 1], [1, 0, -5, 5, -1, -1, 2, 7], [-8, 0, 0, -8, -2, 8, -2, 0], [-1, 1, -4, -4, 0, 0, 1, -5], [0, -4, 0, 0, 1, 4, 4, -1], [-1, -5, 1, 5, -1, 1, 1, 11], [0, -1, -1, -1, -3, -1, 1, -18], [-6, 7, 7, 1, 1, 1, 13, -6], [4, -5, 0, -1, -3, 5, 0, 3], [0, 2, -2, -8, 0, 8, 0, 0], [-1, 5, -4, 5, -4, 4, 14, -1], [16, 0, -2, 2, -16, 4, -3, -1], [-4, -1, -1, -5, 11, -5, -13, -6], [6, 4, 6, 0, 13, 4, 16, 7], [3, -3, -3, -3, -3, 3, -5, 3], [10, -1, 0, -1, -3, 1, -10, 3], [-4, 2, 0, 8, -9, -2, 10, 9], [3, 0, -5, 5, -3, -1, 2, 7], [3, -4, 1, 4, 3, 1, 1, 5], [-7, 0, 6, 0, 6, -6, 6, -7], [0, -8, 0, 0, -8, 0, -1, 1], [2, 4, 5, 0, 4, 4, 11, -1], [4, -2, -2, 2, -2, 2, -2, 0], [-8, 3, 0, 3, -11, -3, 8, 11], [-4, 12, 4, 4, 8, 0, 7, -3], [-4, 2, -2, 8, 0, -8, 0, 0], [-5, 5, 4, 5, 4, -4, -6, -5], [0, 3, 3, 3, -9, 3, 15, 10], [6, 0, 2, 0, -3, 0, 8, -5], [-8, 3, 0, -5, 9, -3, 0, -9], [8, 8, -8, 4, 0, -4, 0, 0], [-2, 6, 0, -6, 1, -6, -10, -1], [4, 0, 2, -2, -4, 4, -11, -13], [6, -7, -7, -1, -9, -1, -5, 6], [1, 2, 0, 2, 0, 0, -4, 1], [-9, 5, 12, 12, -4, 0, 15, -3], [0, 2, 0, 8, -9, -2, 6, 9], [0, 3, 3, 3, 9, 3, -3, 2], [2, 1, 1, 3, -5, 3, 7, -10], [-9, 4, 5, -4, 3, 5, 5, 1], [4, 7, 0, 11, 7, -7, 0, -7], [-4, -5, 5, -1, -3, 1, -3, 0], [-7, 6, 4, -6, 12, 4, 4, -5], [-4, 0, 0, 0, 4, 4, 17, 17], [0, 3, 3, 3, 5, 3, 1, -14], [13, 0, -5, 5, -13, 3, -4, 1], [-4, -4, 4, 0, -10, 0, -10, 0], [-7, -3, 0, -3, 0, 0, 14, -7], [-2, -8, -3, 0, 2, -8, -13, 5], [-1, 0, 7, -7, 1, -5, -4, -11], [-11, -4, -3, 4, -5, -3, -3, 19], [4, 6, -4, -4, 10, 0, -17, 13], [-21, 4, 0, -4, 0, 0, 0, 13], [8, -12, -3, 0, -6, -12, -7, -3], [0, -10, -2, -2, -10, 0, 3, -5], [-3, -3, -4, -4, -6, 0, -15, 11], [-4, 0, -6, 6, 4, 2, -1, 5], [-3, 0, -5, 5, 3, 7, 8, 13], [3, -4, -8, -4, -8, 8, -8, 3], [4, -1, 0, 3, -9, 1, 0, 9], [-2, -6, 6, -2, 6, 2, 6, 0], [-14, 4, 1, 0, -12, 4, -9, -13], [4, -6, -6, -2, -6, -2, -6, 12], [7, 4, 5, -4, -9, 5, 5, -15], [9, 11, 1, 11, 1, -1, 23, 9], [12, 4, -4, 4, 0, -4, 0, 0], [16, -2, 0, -4, 11, 2, -18, -11], [13, -3, 1, 3, -21, 1, 1, -7], [-12, 8, 1, 0, 2, 8, -3, 1], [9, -4, 1, 4, 3, 1, 1, -1], [8, 1, -1, 5, -13, -5, -13, 0], [1, 5, -8, -8, 6, 0, -7, -1], [-9, 5, 0, 5, 0, 0, -2, -9], [-4, 0, 2, -2, 4, -12, -15, -17], [1, 0, -5, 5, -1, -13, -14, -9], [4, 0, -8, 0, 7, 0, -4, 15], [-8, 11, -11, 3, 13, -3, 13, 0], [10, -10, 0, -6, -19, 10, -6, 19], [4, 8, 13, 0, 12, 8, 25, -1], [21, 0, 3, -3, -21, 7, 10, 7], [-12, 2, -4, -4, -10, 0, 3, -7], [-26, 8, 0, -10, 23, -8, 8, -23], [0, -5, -5, -5, -15, -5, 5, 2], [-5, 4, 13, -4, 11, 13, 13, -3], [0, -4, 4, 4, -4, 0, -17, 21], [-11, 9, -4, -4, -2, 0, -9, 5], [25, 1, 0, 1, 0, 0, 10, 25], [-12, 3, 3, -9, 17, -9, -11, 10], [12, -2, -12, 0, -19, -2, -2, -7], [-19, -2, 12, -2, 12, -12, 0, -19], [8, -7, 7, -3, 11, 3, 11, 0], [26, 0, 0, 6, 3, 0, -20, -3], [-5, 0, 3, -3, 5, -1, -4, -7], [0, 14, 4, 4, 14, 0, 19, -15], [-7, 2, -4, -2, 12, -4, -4, 3], [-2, -8, 1, 0, 12, -8, -9, 11], [-22, 8, 10, 0, 1, 8, -4, -9], [-9, 4, -3, -4, -9, -3, -3, 1], [14, 3, 0, -1, 13, -3, -18, -13], [0, -10, 0, 0, -10, 0, -13, 13], [-15, 5, 0, 5, 0, 0, 14, -15], [10, 6, -6, 0, -9, 6, 10, -3], [7, -2, -6, -2, -6, 6, 6, 7], [-16, -1, 0, -5, 25, 1, 12, -25], [0, -4, 4, 0, -10, 0, -10, 0], [0, 0, 0, 0, 0, 4, 13, 13], [0, 8, 8, 8, 8, 8, 8, 4], [9, 0, -5, 5, -9, 3, 0, 5], [3, -8, 13, 8, 3, 13, 13, 13], [0, 8, -3, 0, -16, 8, 5, -13], [12, -9, -9, 3, -13, 3, -5, 22], [1, 0, -15, 0, 19, -15, -15, -1], [8, -9, 9, -9, 13, 9, 13, 0], [3, -1, 9, 1, -17, 9, 9, -1], [-20, 0, 8, -8, 20, 4, -3, -11], [-33, 0, 11, -11, 33, -9, 2, -9], [-9, -4, 8, -4, 8, -8, -16, -9], [1, -7, -4, -7, -4, 4, -34, 1], [-14, -8, -3, 0, -18, -8, -25, -15], [10, -1, -1, 9, 5, 9, -7, 10], [23, 0, -3, 0, -17, -3, -3, -23], [12, -8, 8, 12, -4, -12, -4, 0], [-14, 12, 0, 10, -1, -12, 12, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_256_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_256_2_g_c();".
// 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_256_2_g_c(:prec:=8)
    chi := MakeCharacter_256_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_256_2_g_c();".
// 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_256_2_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_256_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;