
// Make newform 256.5.c.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_256_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_256_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_256_5_c_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_5_c_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 := [6, 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_256_c();"
function MakeCharacter_256_c()
    N := 256;
    order := 2;
    char_gens := [255, 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_256_c_Hecke();"
function MakeCharacter_256_c_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 := 5;
    raw_aps := [[0, 0], [0, 1], [-8, 0], [0, 8], [0, 11], [216, 0], [-162, 0], [0, 45], [0, -72], [1304, 0], [0, 64], [-1512, 0], [-1890, 0], [0, -297], [0, -144], [1976, 0], [0, 231], [-2376, 0], [0, -171], [0, -792], [-2750, 0], [0, 816], [0, 953], [-2430, 0], [7454, 0], [1496, 0], [0, 264], [0, -109], [14904, 0], [-702, 0], [0, 352], [0, -979], [32670, 0], [0, 135], [-872, 0], [0, -968], [28728, 0], [0, 3177], [0, 5544], [-25256, 0], [0, 913], [-36072, 0], [0, -3168], [41374, 0], [67640, 0], [0, -2552], [0, 2673], [0, 2112], [0, 7953], [-14472, 0], [2754, 0], [0, 9360], [-97570, 0], [0, -6921], [80514, 0], [0, 1512], [-44872, 0], [0, -9328], [-90504, 0], [-23166, 0], [0, 7227], [-37768, 0], [0, 3933], [0, -8712], [-93602, 0], [-49192, 0], [0, 1683], [-93758, 0], [0, 22495], [48600, 0], [108162, 0], [0, -5112], [0, 6256], [54648, 0], [0, -10593], [0, -12672], [129112, 0], [-232200, 0], [151902, 0], [-28450, 0], [0, -29739], [-119880, 0], [0, 17424], [215518, 0], [0, 7672], [0, -18909], [177822, 0], [203294, 0], [286648, 0], [0, 33968], [0, -5995], [0, 19008], [0, -37080], [0, 10975], [0, -39303], [0, 21528], [171512, 0], [184734, 0], [0, 42867], [230904, 0], [0, -4059], [-23144, 0], [0, 50125], [127710, 0], [0, 3987], [-270718, 0], [0, 27899], [-62910, 0], [0, 26712], [-203902, 0], [0, 37728], [222264, 0], [-363582, 0], [0, 32427], [0, -35560], [67230, 0], [0, 7605], [0, -38808], [-89288, 0], [0, -22775], [560088, 0], [-810722, 0], [655768, 0], [0, 19311], [0, 58113], [519224, 0], [919512, 0], [0, -66960], [0, 49880], [-706536, 0], [0, -17631], [0, -25560], [0, 54736], [470232, 0], [691038, 0], [304030, 0], [533720, 0], [0, 119889], [552376, 0], [-803682, 0], [0, -7029], [-867592, 0], [0, -3080], [0, 88251], [-449928, 0], [0, -81720], [-121608, 0], [1264734, 0], [0, 98703], [0, -43776], [-519048, 0], [434754, 0], [0, -48411], [0, 30456], [0, -70785], [0, -115632], [0, -53768], [-371682, 0], [532418, 0], [-614600, 0], [0, -59603], [-604962, 0], [0, 142280], [0, 67583], [-815778, 0], [0, -125928], [0, -42592], [1802520, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_256_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_256_5_c_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_5_c_c(:prec:=2)
    chi := MakeCharacter_256_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_5_c_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_5_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_256_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<3,R![96, 0, 1]>,<5,R![8, 1]>],Snew);
    return Vf;
end function;