
// Make newform 384.3.h.e in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_384_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_384_h_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_384_3_h_e();".
// 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_384_3_h_e();".
// 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, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [3, 4, 2, 2], [0, 16, 0, 4], [3, -4, 2, -2]];
    Rf_basisdens := [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_384_h();"
function MakeCharacter_384_h()
    N := 384;
    order := 2;
    char_gens := [127, 133, 257];
    v := [2, 1, 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_384_h_Hecke();"
function MakeCharacter_384_h_Hecke(Kf)
    N := 384;
    order := 2;
    char_gens := [127, 133, 257];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, 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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 1], [-4, 0, 0, 0], [0, -2, 0, -2], [0, -1, 0, -1], [0, 0, 2, 0], [0, 0, 2, 0], [0, 5, 0, -5], [0, -4, 0, 4], [-52, 0, 0, 0], [0, 6, 0, 6], [0, 0, 6, 0], [0, 0, -4, 0], [0, -9, 0, 9], [0, -16, 0, 16], [-20, 0, 0, 0], [0, -23, 0, -23], [0, 0, 2, 0], [0, 11, 0, -11], [0, -20, 0, 20], [-50, 0, 0, 0], [0, -18, 0, -18], [0, 23, 0, 23], [0, 0, 18, 0], [50, 0, 0, 0], [92, 0, 0, 0], [0, -10, 0, -10], [0, 9, 0, 9], [0, 0, -14, 0], [0, 0, 4, 0], [0, 6, 0, 6], [0, -15, 0, -15], [0, 0, -16, 0], [0, 65, 0, -65], [28, 0, 0, 0], [0, 54, 0, 54], [0, 0, -6, 0], [0, -31, 0, 31], [0, 4, 0, -4], [140, 0, 0, 0], [0, -15, 0, -15], [0, 0, 14, 0], [0, -40, 0, 40], [30, 0, 0, 0], [-180, 0, 0, 0], [0, -58, 0, -58], [0, 15, 0, -15], [0, -34, 0, -34], [0, -57, 0, -57], [0, 0, 22, 0], [0, 0, 18, 0], [0, 80, 0, -80], [318, 0, 0, 0], [0, -33, 0, -33], [0, 0, -12, 0], [0, -36, 0, 36], [-132, 0, 0, 0], [0, 62, 0, 62], [0, 0, 30, 0], [0, 0, 22, 0], [0, -19, 0, 19], [140, 0, 0, 0], [0, -61, 0, 61], [0, 100, 0, -100], [-290, 0, 0, 0], [-420, 0, 0, 0], [0, 85, 0, -85], [110, 0, 0, 0], [0, -33, 0, -33], [0, 0, 2, 0], [0, 0, 24, 0], [0, 140, 0, -140], [0, -138, 0, -138], [0, 0, -42, 0], [0, -65, 0, 65], [0, 136, 0, -136], [332, 0, 0, 0], [0, 0, -30, 0], [0, 0, -18, 0], [178, 0, 0, 0], [0, 55, 0, 55], [0, 0, 6, 0], [0, 80, 0, -80], [530, 0, 0, 0], [0, -106, 0, -106], [0, -113, 0, -113], [0, 0, -30, 0], [210, 0, 0, 0], [-372, 0, 0, 0], [0, 38, 0, 38], [0, 103, 0, 103], [0, -80, 0, 80], [0, 14, 0, 14], [0, -199, 0, -199], [0, -25, 0, 25], [0, 4, 0, -4], [332, 0, 0, 0], [0, 0, -24, 0], [0, 19, 0, -19], [0, 0, 114, 0], [0, 61, 0, -61], [60, 0, 0, 0], [0, -25, 0, -25], [0, 0, 96, 0], [0, -235, 0, 235], [-370, 0, 0, 0], [0, 105, 0, 105], [0, 0, -64, 0], [0, 140, 0, -140], [302, 0, 0, 0], [0, -10, 0, -10], [0, 0, -50, 0], [0, 0, -50, 0], [0, -195, 0, 195], [0, -18, 0, -18], [0, 0, -74, 0], [0, -211, 0, 211], [0, 4, 0, -4], [-660, 0, 0, 0], [0, -47, 0, -47], [0, 0, 86, 0], [-910, 0, 0, 0], [-260, 0, 0, 0], [0, -209, 0, -209], [0, 25, 0, -25], [-1028, 0, 0, 0], [0, 0, 102, 0], [0, 120, 0, -120], [0, 54, 0, 54], [0, 0, -126, 0], [0, 255, 0, -255], [0, -44, 0, 44], [0, -290, 0, -290], [0, 0, 110, 0], [0, 0, 12, 0], [-1378, 0, 0, 0], [1180, 0, 0, 0], [0, -311, 0, 311], [-580, 0, 0, 0], [0, 0, -28, 0], [0, -245, 0, 245], [-932, 0, 0, 0], [0, 350, 0, 350], [0, 345, 0, 345], [0, 0, -54, 0], [0, -340, 0, 340], [0, 0, -106, 0], [0, 0, -96, 0], [0, 335, 0, -335], [0, -56, 0, 56], [0, 0, 106, 0], [0, 0, 172, 0], [0, -291, 0, 291], [0, 84, 0, -84], [0, 191, 0, -191], [0, -320, 0, 320], [0, 190, 0, 190], [0, 0, -42, 0], [-510, 0, 0, 0], [812, 0, 0, 0], [0, -351, 0, -351], [0, 0, 120, 0], [0, 94, 0, 94], [0, -177, 0, -177], [0, 0, 74, 0], [0, -444, 0, 444], [0, -306, 0, -306], [0, 0, -138, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_384_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_384_3_h_e();".
// 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_384_3_h_e(:prec:=4)
    chi := MakeCharacter_384_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_384_3_h_e();".
// 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_384_3_h_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_384_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![4, 1]>,<11,R![-20, 0, 1]>],Snew);
    return Vf;
end function;