
// Make newform 256.4.b.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_256_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_256_b_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_4_b_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_256_4_b_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 := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    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_256_b();"
function MakeCharacter_256_b()
    N := 256;
    order := 2;
    char_gens := [255, 5];
    v := [2, 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_256_b_Hecke();"
function MakeCharacter_256_b_Hecke(Kf)
    N := 256;
    order := 2;
    char_gens := [255, 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 := 4;
    raw_aps := [[0, 0], [0, 1], [0, -3], [-20, 0], [0, -7], [0, 27], [-66, 0], [0, 81], [-172, 0], [0, -1], [-128, 0], [0, -79], [-202, 0], [0, 149], [-408, 0], [0, 345], [0, 161], [0, -149], [0, 101], [700, 0], [418, 0], [744, 0], [0, -339], [82, 0], [-1122, 0], [0, -695], [-788, 0], [0, -807], [0, 1007], [-542, 0], [1712, 0], [0, -1059], [486, 0], [0, -643], [0, 1333], [-172, 0], [0, 419], [0, 673], [1052, 0], [0, 19], [0, -1395], [0, 1709], [-1968, 0], [-1058, 0], [0, -363], [-4116, 0], [0, 741], [-896, 0], [0, 1705], [0, -2251], [-2302, 0], [4024, 0], [-3586, 0], [0, 625], [-6638, 0], [1724, 0], [0, 2407], [-1640, 0], [0, -1991], [-4126, 0], [0, -1723], [0, 757], [0, 2745], [5556, 0], [2054, 0], [0, 1247], [0, 1457], [7186, 0], [0, -2223], [0, 215], [850, 0], [10988, 0], [872, 0], [0, -1727], [0, 745], [10240, 0], [0, 3729], [0, -2353], [5598, 0], [838, 0], [0, 13], [0, 7789], [-6792, 0], [-9314, 0], [3828, 0], [0, -7707], [-3650, 0], [15862, 0], [0, 39], [-4376, 0], [0, 6857], [19072, 0], [-19428, 0], [0, 1245], [0, 1413], [-2268, 0], [0, 5267], [9478, 0], [0, 2929], [0, 955], [0, 877], [0, 1147], [0, 8121], [15990, 0], [0, 10837], [-12542, 0], [0, 9289], [14514, 0], [3460, 0], [-20686, 0], [-9776, 0], [0, 4397], [-16398, 0], [0, -10687], [18916, 0], [-5250, 0], [0, -2251], [-11076, 0], [0, 15383], [0, -10259], [0, -35], [23070, 0], [0, -1311], [0, 7341], [0, -11635], [0, -15765], [0, -11007], [19016, 0], [-15996, 0], [0, 5311], [0, -4439], [21852, 0], [32024, 0], [0, 13301], [13958, 0], [-11970, 0], [0, -6159], [0, 13349], [0, -2397], [-39114, 0], [0, 545], [0, 4865], [-1100, 0], [0, 19037], [0, 12115], [-16820, 0], [0, 11081], [8790, 0], [0, -5279], [-7392, 0], [0, 3287], [47154, 0], [0, 321], [-23308, 0], [0, 10725], [-40904, 0], [27380, 0], [10302, 0], [-5054, 0], [0, 15231], [0, 16041], [-12970, 0], [-1652, 0], [0, 12325], [23646, 0], [-38108, 0], [-18640, 0], [0, -14511]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_256_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_256_4_b_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_256_4_b_b(:prec:=2)
    chi := MakeCharacter_256_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_b_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_256_4_b_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_256_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![4, 0, 1]>,<7,R![20, 1]>],Snew);
    return Vf;
end function;