
// Make newform 2640.2.a.be in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2640_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2640_2_a_be();".
// 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_2640_2_a_be();".
// 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, -3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-4, -2, 2]];
    Rf_basisdens := [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_2640_a();"
function MakeCharacter_2640_a()
    N := 2640;
    order := 1;
    char_gens := [991, 661, 881, 1057, 1201];
    v := [1, 1, 1, 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;

function MakeCharacter_2640_a_Hecke(Kf)
    return MakeCharacter_2640_a();
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], [-1, 0, 0], [1, 0, 0], [0, 0, -1], [-1, 0, 0], [-1, -1, 0], [0, 2, 1], [-3, -1, 1], [0, 0, 2], [-3, 1, 1], [-2, 2, 0], [-2, 0, 0], [-5, -1, -1], [-2, -2, 1], [2, -2, 0], [-2, 0, 2], [-4, 0, -2], [-2, 0, -2], [2, 2, 0], [-2, 2, 0], [-5, -1, -2], [-3, 3, 1], [1, 3, 0], [-4, -2, -2], [8, 2, 0], [-9, -1, -3], [2, -2, -2], [-3, -1, 0], [-2, 0, 4], [-6, 0, 0], [10, -2, -1], [-2, -2, 0], [-2, -4, 0], [-13, 1, 1], [-7, -3, -1], [-3, -5, -3], [0, -2, -4], [2, 2, -2], [-1, 1, -4], [-4, 2, 1], [-2, -2, -6], [-4, 2, 2], [2, -2, 2], [1, 1, -4], [0, 6, 1], [-6, -2, 2], [-3, -1, -3], [0, 0, 2], [-13, 1, 0], [10, -4, 0], [-8, 2, -1], [-4, 4, -2], [6, -4, 2], [-22, 2, 0], [2, 0, 2], [13, 3, -4], [10, 4, 0], [-1, 1, 3], [-5, -5, -4], [-5, -1, -1], [-22, 2, 1], [-8, -2, -1], [20, 0, -1], [-8, 8, 2], [2, 0, 4], [18, 4, 0], [-8, -4, 2], [19, -1, -2], [-3, -1, 0], [14, 0, 2], [-2, -4, 2], [-10, 2, 0], [12, 4, 10], [7, -1, -6], [20, 0, 0], [2, -10, -2], [4, 2, 2], [-12, 2, 0], [2, 0, 0], [4, 6, 2], [-6, 2, 2], [16, -2, 2], [-10, -6, 6], [-4, -2, -4], [13, -5, -1], [2, -6, -2], [-8, -6, 6], [5, 5, 6], [1, -3, 7], [12, 4, -2], [-12, 0, -2], [16, 0, -2], [30, 2, 4], [-26, -2, -2], [-2, -2, -6], [-1, 1, 4], [6, 8, 8], [-10, 4, 4], [-20, 0, -3], [-18, 0, 0], [0, 4, 9], [-6, 4, 3], [-7, 11, 6], [-21, -1, -3], [-3, 7, 7], [8, 10, 0], [-24, 4, 2], [22, -4, -1], [12, -4, -4], [-8, 2, 0], [10, -2, -1], [35, 3, -2], [-2, 4, 0], [12, 8, 6], [10, -2, 6], [0, 2, -6], [-10, -2, 2], [2, -2, -10], [2, -4, -6], [28, 0, 4], [-10, 0, 4], [-11, -3, 0], [18, 4, 5], [2, 2, 8], [0, -4, -4], [-17, -1, 3], [12, 2, 6], [2, -2, 6], [-4, -4, -8], [-31, -3, 0], [21, 7, 3], [3, -3, 6], [-6, -10, -2], [8, -2, 0], [-9, 3, -3], [-12, -2, 4], [38, 0, 2], [-22, -6, -7], [2, -12, -8], [-23, 1, 1], [-5, -7, 1], [-27, 1, -7], [6, 10, 10], [-15, -5, 4], [-4, -6, -2], [4, -4, 2], [19, 3, 10], [4, -10, -7], [-20, -8, -10], [6, 2, -10], [7, 7, 2], [-4, 6, 2], [8, -12, -4], [-31, -1, 6], [12, 8, -10], [-6, 6, -2], [-3, 11, 11], [-12, -2, -6], [-9, 3, 4], [19, 11, 1], [10, 2, 4], [-22, 0, -3], [-14, -2, 7], [4, 0, -12], [6, 12, 4], [10, 6, 2], [22, 10, -8], [-5, 11, 6], [10, 8, 10], [18, -4, 2], [16, 4, 8], [-8, 14, 10], [-10, 2, -4], [-15, 9, -2], [-4, -12, -2], [-8, 2, 6], [-27, -1, -5], [26, -4, -8], [-10, -6, 7], [-4, 2, -2], [12, 4, 0], [-12, 8, 0], [-24, -2, -4], [-38, 0, -3], [46, 2, 0], [28, 10, -6], [5, -7, -4], [-20, 0, 2], [8, 2, -6], [-22, -10, -10], [-8, -6, 4], [-7, -13, -4], [2, 10, 2], [-4, 2, 2], [21, -1, 2], [-14, -8, 6], [-12, -2, 8], [-6, 4, 0], [-2, -12, 5], [-5, 5, 2], [-33, -1, -9], [-25, 9, 1], [-34, -8, 4], [-18, -12, 6], [18, 2, -6], [-6, -4, -12], [-28, -4, -4], [21, -1, -10], [-15, -7, 5], [-14, 2, 4], [51, -1, 4], [8, 14, -2], [-4, -4, 0], [16, 12, 2], [-6, -2, -8], [34, 8, -4], [2, -10, 5], [1, 1, 7], [24, 16, -4], [-30, -4, -2], [32, -10, 0], [-15, 7, 5], [44, 6, 2], [14, -6, -6], [-1, 5, 6], [8, -2, -6], [2, 8, 4], [-20, -4, 10], [-44, -4, -3], [16, -12, -6], [-56, -2, 0], [-9, -11, -7], [15, 17, 1], [-11, -11, 1], [18, -6, -4], [-5, -3, -6], [-20, 14, 10], [24, 14, -5], [-22, -14, 4], [36, 4, 4], [-2, -10, 6], [-5, -7, -9], [-16, 8, 4], [20, -14, -10], [20, -2, 3], [-34, -6, -12], [-22, -18, 8], [-34, -10, 4], [-15, 3, -5], [-13, 5, -2], [-45, -5, -4], [-15, -15, -1], [-4, -4, -8], [2, -8, -4], [-6, 4, -5], [10, -14, -10], [44, 2, 6], [-10, 6, -1], [18, -4, 0], [23, 3, 0], [-14, -10, -15], [-15, 3, -6], [-24, -2, 8], [-11, 9, 10], [18, -8, -10], [30, -2, 10], [-20, -14, 10], [0, 10, 14], [-20, 16, 7], [10, -4, 13], [-2, 0, -12], [32, -4, 10], [-20, 14, 8], [-59, -5, 3], [9, -7, -2], [-10, -6, 12], [-26, -2, -4], [2, 12, 6], [8, 18, 8], [-10, 6, -4], [-11, -13, 4], [30, 10, -4], [-13, -3, 10], [-20, 10, -6], [-4, 0, -3], [40, 8, 16], [36, 6, -4], [30, -8, 1], [30, 10, 0], [-19, 5, 3], [-8, -10, -10], [-18, -10, 10], [12, -2, -1], [22, -2, -8], [-3, -15, -6], [37, 9, -9], [38, -6, -10], [-14, -12, -14], [2, 18, 6], [-12, -16, -17], [-25, -13, 8], [30, 0, -9], [-17, 1, 9], [24, -4, -6], [22, -2, 12], [-10, 4, -4], [30, 6, 4], [10, 4, 0], [-52, -4, 4], [-13, 3, -6], [-17, 9, -4], [28, 10, -6], [-7, -7, -17], [-10, -2, -8], [-21, -11, -4], [-22, 0, -6], [24, 4, -8], [22, -6, -8], [-32, -14, -8], [-13, -1, -19], [-23, -13, -3], [56, 2, 4], [-37, 3, 3], [-18, 10, 8], [-32, -6, 7], [32, -6, -10], [14, 14, 12], [28, 8, -4], [13, -5, 8], [4, -14, 5], [-12, -6, 8], [10, 4, -4], [-7, -9, -11], [-29, 1, 0], [23, 13, -5], [6, -10, -18], [-18, 0, -4], [28, 6, -13], [48, 2, -2], [-40, 0, -1], [36, -6, -4], [-58, -4, 2], [9, -3, -13], [40, 0, 6], [-26, 8, 10], [40, 4, 10], [62, 8, 0], [-18, -18, 10], [-4, 4, 4], [-2, -8, -10], [19, 9, 13], [18, 8, 0], [24, 6, 6], [6, 2, -5], [38, 0, 14], [0, 10, -7], [-54, -2, 8], [24, 12, -2], [18, 16, 5], [8, -16, -4], [-27, 17, 10], [1, 9, -13], [20, 4, -6], [-4, -8, -2], [0, -4, -16], [12, 14, 4], [34, 12, 3], [16, 0, -5], [48, -6, 8], [28, -8, -10], [60, 8, 10], [35, -11, 2], [-41, -1, -3], [-27, 3, -5], [40, 6, -4], [30, -2, 10], [36, -4, -6], [19, -1, 0], [-5, 7, -15], [-11, -11, -22], [-20, -14, -6], [-30, -8, -2], [58, 6, 3], [-28, -2, -3], [3, 9, 19], [36, 4, 0], [6, -6, -6], [-32, -2, 0], [-4, 16, 5], [32, -16, 0], [-44, -10, -2], [30, 0, 18], [30, 14, 0], [-6, -6, 16], [-4, 4, -6], [25, 1, 18], [-21, -11, 17], [14, -4, -8], [0, -12, -4], [9, -19, 1], [-32, -2, 14], [38, -12, -8], [-52, 12, 3], [-22, 0, 10], [-11, 1, 7], [29, 11, 13], [16, 6, -16], [17, 1, -5], [6, 14, 0], [-18, 6, 16], [-45, 15, 8], [12, 10, -13], [-64, 4, 0], [-7, 3, -5], [15, -5, 6], [-22, -12, -12], [-26, -6, 2], [-10, 10, -6], [-22, -16, 10], [35, -3, 12], [32, -2, -14], [13, -7, -10], [-24, 8, 12], [24, 12, 2], [30, -4, -12], [26, -12, -6], [32, 12, 8], [-7, 1, 19], [24, -4, -12], [12, -4, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2640_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2640_2_a_be();".
// 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_2640_2_a_be(:prec:=3)
    chi := MakeCharacter_2640_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_2640_2_a_be();".
// 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_2640_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2640_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, -16, 0, 1]>,<13,R![-8, -12, 2, 1]>,<17,R![-184, -52, 2, 1]>,<19,R![-160, -16, 8, 1]>],Snew);
    return Vf;
end function;