
// Make newform 1638.2.c.i in Magma, downloaded from the LMFDB on 02 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_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_1638_2_c_i();".
// 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_1638_2_c_i();".
// 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 := [2, -10, 25, 8, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-62, 159, 28, 49, -18, 7], [-567, -22, 44, 77, -53, 11], [151, -803, -297, -44, 55, -31], [515, -2741, -746, -181, 183, -100]];
    Rf_basisdens := [1, 1, 173, 173, 173, 173];
    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_1638_c();"
function MakeCharacter_1638_c()
    N := 1638;
    order := 2;
    char_gens := [911, 703, 379];
    v := [2, 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_1638_c_Hecke();"
function MakeCharacter_1638_c_Hecke(Kf)
    N := 1638;
    order := 2;
    char_gens := [911, 703, 379];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 1, 1, 0, -2, -1], [-2, 1, 0, 0, 0, 1], [2, -2, 2, 0, 0, 0], [0, -1, -1, 0, -2, 0], [2, -1, 1, -1, 0, 0], [-2, 0, 0, -2, 0, 0], [0, -1, -1, 0, 2, 1], [0, 1, 1, 0, 4, 1], [0, 1, 1, 0, -4, 1], [4, -2, 2, 0, 0, 0], [0, -1, -1, 0, -6, 1], [6, 0, 0, 0, 0, 0], [0, -3, -3, 0, -6, -2], [2, 0, 0, -1, 0, 0], [0, -1, -1, 0, -6, -1], [0, 0, 0, 0, 0, 4], [0, -1, -1, 0, -4, 1], [0, -3, 3, -1, 0, 0], [0, -3, -3, 0, 2, 2], [0, 2, 2, 0, 10, 0], [0, 1, 1, 0, 8, 1], [6, 2, -2, 1, 0, 0], [8, -2, 2, 0, 0, 0], [4, -2, 2, -2, 0, 0], [0, 0, 0, 0, -2, 4], [2, -1, 1, -1, 0, 0], [-2, 1, -1, -3, 0, 0], [-2, -3, 3, 4, 0, 0], [0, 0, 0, 0, 0, -4], [2, -1, 1, 0, 0, 0], [0, -1, -1, 0, 0, -3], [0, 2, 2, 0, -2, 0], [6, 0, 0, -5, 0, 0], [0, -2, -2, 0, 4, -4], [0, 2, 2, 0, 8, -4], [-12, -1, 1, -4, 0, 0], [4, 4, -4, -4, 0, 0], [-10, 6, -6, 3, 0, 0], [-10, 0, 0, 2, 0, 0], [0, -2, -2, 0, 2, -2], [0, 3, 3, 0, 4, 1], [-4, -2, 2, 0, 0, 0], [16, -4, 4, -4, 0, 0], [0, -1, -1, 0, -2, 3], [0, 1, 1, 0, -6, 2], [0, -1, -1, 0, 0, -6], [-4, 1, -1, 5, 0, 0], [0, -6, -6, 0, -2, -4], [0, -4, -4, 0, -6, 4], [-4, 6, -6, 1, 0, 0], [-10, 0, 0, 4, 0, 0], [6, -4, 4, -2, 0, 0], [-6, -6, 6, 1, 0, 0], [0, 1, 1, 0, 10, 5], [-10, 0, 0, 4, 0, 0], [0, 0, 0, 0, -4, -4], [4, -2, 2, -5, 0, 0], [0, -3, -3, 0, 0, 0], [0, 3, 3, 0, -10, -6], [8, 4, -4, 0, 0, 0], [10, 0, 0, 2, 0, 0], [0, -1, -1, 0, -20, -3], [0, -3, -3, 0, 2, -1], [-10, -1, 1, -9, 0, 0], [-12, 2, -2, 4, 0, 0], [0, 7, 7, 0, 4, -4], [0, 3, 3, 0, -4, 1], [0, 8, 8, 0, 2, -2], [4, 0, 0, -4, 0, 0], [-22, 0, 0, -4, 0, 0], [0, -6, -6, 0, -20, 0], [0, -1, -1, 0, -10, -7], [-10, -6, 6, 0, 0, 0], [0, 7, 7, 0, 12, 4], [0, 2, 2, 0, 22, 2], [0, 4, 4, 0, -2, -4], [-4, -2, 2, -5, 0, 0], [0, -1, -1, 0, -16, 1], [0, 0, 0, 0, -6, 2], [2, -2, 2, 8, 0, 0], [-12, 4, -4, 8, 0, 0], [-16, 0, 0, -4, 0, 0], [0, -6, -6, 0, 0, -2], [0, -10, -10, 0, 6, 6], [0, 1, 1, 0, 24, 2], [0, -6, -6, 0, -4, 2], [-14, -5, 5, 0, 0, 0], [0, -2, -2, 0, 16, -4], [0, 0, 0, 0, 10, 2], [0, -10, 10, 2, 0, 0], [0, 5, 5, 0, 2, -3], [8, -6, 6, -2, 0, 0], [0, 5, 5, 0, 12, 0], [26, 0, 0, -2, 0, 0], [24, -4, 4, -3, 0, 0], [0, 10, 10, 0, 14, 4], [20, 0, 0, 2, 0, 0], [0, -3, -3, 0, 12, 3], [-24, 0, 0, -7, 0, 0], [-10, 9, -9, -3, 0, 0], [-12, -8, 8, 2, 0, 0], [0, 6, 6, 0, 6, -4], [0, -7, -7, 0, -6, -10], [0, -6, -6, 0, 6, 4], [12, 7, -7, 5, 0, 0], [2, 6, -6, 2, 0, 0], [-24, 6, -6, 2, 0, 0], [0, 9, 9, 0, -8, -3], [0, 2, 2, 0, 2, 2], [0, 3, 3, 0, -18, -4], [0, 4, 4, 0, 0, -8], [10, -5, 5, 7, 0, 0], [0, -5, -5, 0, 14, -2], [-12, -4, 4, 2, 0, 0], [-10, 12, -12, 6, 0, 0], [16, -2, 2, 0, 0, 0], [0, -1, -1, 0, -4, 10], [12, -5, 5, -5, 0, 0], [-2, -2, 2, 11, 0, 0], [0, 3, 3, 0, -10, 7], [0, -1, -1, 0, 30, 6], [-6, 2, -2, 8, 0, 0], [0, -5, -5, 0, 4, -5], [32, 2, -2, 4, 0, 0], [-24, 4, -4, 2, 0, 0], [0, 3, 3, 0, 12, 10], [0, -6, -6, 0, 28, 4], [0, -2, -2, 0, -18, -4], [6, -3, 3, 5, 0, 0], [-22, 2, -2, -4, 0, 0], [0, 7, 7, 0, 20, -5], [0, -1, -1, 0, 24, -1], [0, 1, 1, 0, -8, 4], [0, -1, -1, 0, 22, 4], [-22, 2, -2, 1, 0, 0], [-16, -6, 6, 6, 0, 0], [0, 11, 11, 0, 14, 8], [0, -2, -2, 0, -22, -4], [8, -3, 3, 3, 0, 0], [0, 0, 0, 0, -16, 12], [0, 9, -9, 4, 0, 0], [0, 1, 1, 0, -22, -1], [0, 5, 5, 0, -8, 10], [-10, 8, -8, 6, 0, 0], [-4, -2, 2, -5, 0, 0], [0, -2, -2, 0, 28, -6], [0, 5, 5, 0, 4, -11], [22, -2, 2, 6, 0, 0], [-12, -4, 4, 0, 0, 0], [8, 6, -6, 0, 0, 0], [20, 4, -4, -8, 0, 0], [0, -8, 8, -8, 0, 0], [-20, 5, -5, 7, 0, 0], [0, 9, 9, 0, 12, 13], [14, 4, -4, 4, 0, 0], [0, 3, 3, 0, -12, 2], [0, -14, -14, 0, -16, -4], [18, 4, -4, 4, 0, 0], [0, 8, 8, 0, 14, 14], [12, 2, -2, -11, 0, 0], [0, 10, 10, 0, -4, 6], [0, -10, -10, 0, 24, 8], [36, -3, 3, -1, 0, 0], [2, 2, -2, -1, 0, 0], [0, 0, 0, 0, 14, -8], [28, -11, 11, -8, 0, 0], [0, 4, 4, 0, 4, 8], [0, -7, -7, 0, -40, -6], [-32, -1, 1, 5, 0, 0], [0, 4, 4, 0, 4, 8], [-8, -14, 14, 8, 0, 0], [-14, -2, 2, 2, 0, 0], [0, -1, -1, 0, -22, 3], [0, 8, 8, 0, -2, 8], [-8, -12, 12, 0, 0, 0], [14, -4, 4, 5, 0, 0], [0, -10, -10, 0, 2, -4], [-20, 6, -6, 7, 0, 0], [2, -4, 4, 10, 0, 0], [0, -3, -3, 0, 4, 7], [0, -4, -4, 0, -14, 2], [-12, 13, -13, 12, 0, 0], [-30, -6, 6, -6, 0, 0], [0, -5, -5, 0, -10, -14], [0, 12, 12, 0, -8, -8], [0, 3, 3, 0, -18, 9], [10, 12, -12, 0, 0, 0], [0, -7, -7, 0, -14, -11], [36, -2, 2, 6, 0, 0], [0, -5, -5, 0, 36, 4], [8, -12, 12, -16, 0, 0], [-2, 0, 0, 8, 0, 0], [0, 6, 6, 0, 34, -2], [-2, 2, -2, -4, 0, 0], [0, -2, -2, 0, -2, -12], [-8, -6, 6, -2, 0, 0], [0, 7, 7, 0, -28, -3], [0, 6, -6, 8, 0, 0], [0, -1, -1, 0, -16, -12], [-14, 0, 0, 8, 0, 0], [0, 9, 9, 0, 18, 12], [24, -1, 1, -4, 0, 0], [0, -6, -6, 0, 4, 8], [-16, 18, -18, 4, 0, 0], [0, -8, -8, 0, -6, -12], [0, 8, -8, -1, 0, 0], [-6, 11, -11, 7, 0, 0], [22, 4, -4, -11, 0, 0], [6, 13, -13, -7, 0, 0], [0, -9, -9, 0, -10, -16], [0, -7, -7, 0, -2, -15], [0, 5, 5, 0, -28, 5], [12, -9, 9, 5, 0, 0], [-34, 4, -4, -10, 0, 0], [0, 8, 8, 0, 0, -8], [0, -7, -7, 0, 8, 1], [18, 14, -14, -12, 0, 0], [0, -17, -17, 0, -30, -5], [0, 18, 18, 0, 0, -2], [0, -12, -12, 0, -26, -6], [12, -18, 18, -5, 0, 0], [-26, 4, -4, -6, 0, 0], [6, 4, -4, 0, 0, 0], [34, 7, -7, -1, 0, 0], [8, 2, -2, -2, 0, 0], [0, -9, -9, 0, 6, -7], [-26, 10, -10, -4, 0, 0], [6, 5, -5, 4, 0, 0], [0, 6, 6, 0, 28, -2], [-8, -3, 3, 9, 0, 0], [-44, 2, -2, -5, 0, 0], [0, -1, -1, 0, 22, -13], [0, -9, -9, 0, 12, 15], [0, 17, 17, 0, 24, 7], [12, 12, -12, 4, 0, 0], [-52, 14, -14, 6, 0, 0], [0, -3, -3, 0, -34, -1], [12, -6, 6, 1, 0, 0], [-12, -10, 10, 14, 0, 0], [0, 2, 2, 0, -2, 12], [0, 7, 7, 0, -12, 5], [20, -6, 6, 8, 0, 0], [0, -1, -1, 0, -6, -13], [0, 7, 7, 0, -10, 6], [0, 17, 17, 0, 26, 3], [4, -7, 7, 19, 0, 0], [0, 1, 1, 0, 16, -5], [0, -6, -6, 0, 2, 16], [0, 8, 8, 0, 0, 4], [-2, 4, -4, -12, 0, 0], [-14, -8, 8, -1, 0, 0], [0, 0, 0, 0, 64, 4], [-14, 12, -12, 10, 0, 0], [0, -3, -3, 0, 10, -8], [-6, 8, -8, -17, 0, 0], [0, -15, -15, 0, -40, 1], [0, -8, 8, 4, 0, 0], [-12, -16, 16, 14, 0, 0], [0, -3, -3, 0, -32, -4], [18, 6, -6, -17, 0, 0], [0, -1, -1, 0, 16, -9], [18, -9, 9, -8, 0, 0], [0, -11, -11, 0, -48, -9], [0, -1, -1, 0, -48, -7], [0, 11, 11, 0, 38, 3], [2, -10, 10, -4, 0, 0], [2, -2, 2, -3, 0, 0], [0, -17, -17, 0, -50, -11], [0, 3, 3, 0, 24, -3], [-40, -8, 8, 0, 0, 0], [10, 10, -10, 10, 0, 0], [0, -12, -12, 0, -20, -8], [0, 2, 2, 0, -12, -4], [0, -1, -1, 0, 36, 1], [0, 12, 12, 0, 30, 0], [-2, 1, -1, 20, 0, 0], [28, 2, -2, -6, 0, 0], [0, 4, 4, 0, 0, 20], [-36, 12, -12, -2, 0, 0], [0, -13, -13, 0, -20, -10], [0, 9, 9, 0, 2, -20], [-54, -1, 1, 3, 0, 0], [-8, -5, 5, 7, 0, 0], [0, -6, -6, 0, 18, -8], [0, -1, -1, 0, -18, 9], [-6, -8, 8, 4, 0, 0], [38, 8, -8, -12, 0, 0], [-16, 12, -12, 3, 0, 0], [0, 18, 18, 0, 10, -6], [0, 1, 1, 0, 2, 6], [-10, -2, 2, 18, 0, 0], [4, 13, -13, -8, 0, 0], [24, 6, -6, -10, 0, 0], [34, -8, 8, -11, 0, 0], [-24, 4, -4, 1, 0, 0], [0, -5, -5, 0, -26, -2], [-42, -6, 6, -4, 0, 0], [0, 3, 3, 0, -4, -13], [-14, 0, 0, 10, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_c_i();".
// 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_1638_2_c_i(:prec:=6)
    chi := MakeCharacter_1638_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1638_2_c_i();".
// 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_1638_2_c_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, 0, 116, 0, 24, 0, 1]>,<11,R![7744, 0, 1296, 0, 65, 0, 1]>,<17,R![56, -44, -2, 1]>],Snew);
    return Vf;
end function;