
// Make newform 768.4.d.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_768_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_d_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_768_4_d_n();".
// 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_768_4_d_n();".
// 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;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_768_d();"
function MakeCharacter_768_d()
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    v := [2, 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_768_d_Hecke();"
function MakeCharacter_768_d_Hecke(Kf)
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    char_values := [[1, 0], [-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, 3], [0, 6], [16, 0], [0, 12], [0, -38], [-126, 0], [0, -20], [-168, 0], [0, -30], [-88, 0], [0, 254], [-42, 0], [0, -52], [-96, 0], [0, 198], [0, -660], [0, 538], [0, -884], [-792, 0], [-218, 0], [-520, 0], [0, 492], [-810, 0], [1154, 0], [0, -618], [-128, 0], [0, -1476], [0, -1190], [-462, 0], [-2536, 0], [0, -2292], [726, 0], [0, 380], [0, 1590], [-2432, 0], [0, -614], [0, 1852], [2136, 0], [0, -1758], [0, 540], [0, 1982], [-2688, 0], [-2302, 0], [0, 4374], [1600, 0], [0, -3332], [2648, 0], [0, -2244], [0, -5650], [-4698, 0], [-1200, 0], [-718, 0], [0, 6012], [-2046, 0], [6072, 0], [0, 6930], [1352, 0], [0, -1186], [-2442, 0], [0, 2828], [0, 4758], [0, 8476], [-4632, 0], [4822, 0], [0, 3426], [0, -2788], [434, 0], [0, 6684], [0, -2630], [-7422, 0], [10440, 0], [10424, 0], [0, 3278], [0, 6140], [-3072, 0], [0, 6150], [0, 106], [-1758, 0], [3670, 0], [0, 9660], [0, 8462], [9792, 0], [-7342, 0], [-10640, 0], [0, -17412], [-1710, 0], [646, 0], [0, 6018], [-6712, 0], [0, -5364], [9840, 0], [-1424, 0], [0, -4548], [0, -6500], [-12168, 0], [0, 21090], [5238, 0], [0, 8588], [0, -3062], [0, 8476], [0, 12546], [0, 12], [-19290, 0], [0, -12148], [-10366, 0], [0, 7644], [8658, 0], [-25800, 0], [-16202, 0], [-24136, 0], [0, -4642], [6726, 0], [0, -21220], [-29792, 0], [-10158, 0], [0, -29828], [-1944, 0], [0, -26718], [0, -4260], [0, 22862], [-32542, 0], [0, 14214], [0, -7092], [0, 13228], [0, -28062], [0, -27250], [-14400, 0], [-17984, 0], [0, -16598], [0, -1460], [30072, 0], [-18088, 0], [0, 24734], [22278, 0], [16130, 0], [0, 29718], [0, -9524], [0, 33906], [630, 0], [0, -20788], [0, -43098], [14272, 0], [0, 13644], [0, 2410], [-23160, 0], [0, 32078], [14406, 0], [0, 30620], [17568, 0], [0, 21706], [-14958, 0], [0, 32812], [38856, 0], [0, -28276], [8112, 0], [26080, 0], [49170, 0], [-48314, 0], [0, -34782], [0, 25116], [15462, 0], [736, 0], [0, -29268], [16674, 0], [31272, 0], [-15928, 0], [0, 42014]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_768_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_768_4_d_n();".
// 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_768_4_d_n(:prec:=2)
    chi := MakeCharacter_768_d();
    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_768_4_d_n();".
// 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_768_4_d_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_768_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![36, 0, 1]>,<7,R![-16, 1]>],Snew);
    return Vf;
end function;