
// Make newform 256.4.b.g 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_g();".
// 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_g();".
// 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, 2], [0, 1], [24, 0], [0, -22], [0, 11], [50, 0], [0, -22], [-56, 0], [0, 99], [160, 0], [0, 81], [198, 0], [0, 26], [-528, 0], [0, 121], [0, -334], [0, 275], [0, -94], [728, 0], [-154, 0], [656, 0], [0, -118], [-714, 0], [-478, 0], [0, -783], [-968, 0], [0, -390], [0, -997], [-942, 0], [-1408, 0], [0, 1346], [-1626, 0], [0, -342], [0, -151], [1352, 0], [0, 1571], [0, -1518], [-264, 0], [0, -1413], [0, -1542], [0, 1209], [960, 0], [2882, 0], [0, -543], [88, 0], [0, 1738], [-928, 0], [0, -78], [0, 817], [902, 0], [-1616, 0], [4818, 0], [0, -1070], [770, 0], [-7400, 0], [0, -1397], [-8624, 0], [0, 937], [-3338, 0], [0, 3586], [0, -2607], [0, -198], [-4056, 0], [-2154, 0], [0, -3693], [0, -566], [-3342, 0], [0, 1122], [0, -3261], [-11230, 0], [1848, 0], [-7120, 0], [0, -3175], [0, -3950], [-10368, 0], [0, -4415], [0, 4939], [-13134, 0], [-906, 0], [0, 2706], [0, 2321], [-656, 0], [9490, 0], [5544, 0], [0, 3826], [-446, 0], [-1562, 0], [0, 5291], [10768, 0], [0, 4938], [352, 0], [-15176, 0], [0, -4422], [0, -9702], [16488, 0], [0, -6477], [-10970, 0], [0, -8470], [0, 99], [0, 7634], [0, 10427], [0, 9658], [-7018, 0], [0, 12210], [23234, 0], [0, -5302], [-13838, 0], [-3960, 0], [5942, 0], [3040, 0], [0, 1265], [19206, 0], [0, 5498], [-6680, 0], [6274, 0], [0, -4542], [-23656, 0], [0, -3381], [0, -7638], [0, -5527], [-21278, 0], [0, -4463], [0, 4058], [0, 5882], [0, -2349], [0, -12319], [-16624, 0], [30216, 0], [0, -1661], [0, 7346], [28600, 0], [29616, 0], [0, -1447], [-14762, 0], [-7678, 0], [0, -13695], [0, -9878], [0, 19427], [14278, 0], [0, -358], [0, 11769], [-6616, 0], [0, 13618], [0, 6035], [-42024, 0], [0, -1207], [37686, 0], [0, 20322], [18656, 0], [0, -6501], [49490, 0], [0, -550], [-14104, 0], [0, -6358], [39632, 0], [5704, 0], [8162, 0], [55110, 0], [0, 8187], [0, -4230], [20502, 0], [-36520, 0], [0, 10122], [50034, 0], [37128, 0], [-27808, 0], [0, 14257]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_256_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![16, 0, 1]>,<7,R![-24, 1]>],Snew);
    return Vf;
end function;