
// Make newform 8470.2.a.cg in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8470_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_8470_2_a_cg();".
// 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_8470_2_a_cg();".
// 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, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-3, 0, 1], [2, 2, -1]];
    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_8470_a();"
function MakeCharacter_8470_a()
    N := 8470;
    order := 1;
    char_gens := [6777, 6051, 7141];
    v := [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_8470_a_Hecke(Kf)
    return MakeCharacter_8470_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 := [[-1, 0, 0], [0, 1, 0], [-1, 0, 0], [-1, 0, 0], [0, 0, 0], [0, -1, 0], [-4, 0, -1], [-2, 1, -1], [-1, 2, 1], [0, 0, 2], [0, 0, 0], [0, 2, 2], [0, 2, 0], [-2, 4, 1], [2, -2, -2], [0, 0, -1], [2, 1, -1], [-4, -2, 1], [-4, 0, -3], [6, 2, 1], [-6, 2, 1], [3, -2, 0], [-2, -1, 0], [4, -2, 0], [-2, 2, 1], [4, 1, 0], [0, 2, -1], [8, -2, 1], [2, 2, 2], [7, 0, 3], [-9, 2, 1], [-4, 7, 2], [1, 6, 1], [-8, -1, -5], [0, 6, 0], [-3, 6, 1], [-4, -1, -2], [-8, 6, 3], [-4, -6, -3], [6, 4, -2], [14, 4, 2], [0, 3, -5], [9, 0, 1], [5, 0, 5], [-4, -8, -3], [0, 4, -2], [16, 0, 4], [8, 4, -1], [6, 2, 2], [12, -6, -1], [-9, 2, -1], [9, 10, -3], [-14, 2, -2], [-4, -4, 1], [0, 0, -3], [-9, 4, -3], [14, 1, -3], [-6, 6, 0], [10, 0, 0], [-3, -8, 2], [-2, 3, -4], [2, 7, 0], [-14, -2, 4], [-4, -10, 0], [-10, 0, 0], [4, 4, -5], [16, -2, -2], [9, 0, -1], [-26, 0, -1], [18, 5, -2], [-6, 6, 3], [10, 6, -3], [6, 8, -1], [-8, 6, 5], [14, 4, 10], [18, -4, 3], [2, -10, -2], [-2, 0, 6], [18, 4, 4], [10, 2, 2], [8, -1, -1], [-16, -8, -4], [27, 0, 4], [4, -10, -6], [-12, -2, -10], [10, 2, -6], [-7, 0, -4], [9, 8, 1], [0, -10, 5], [9, -12, -5], [2, -13, -8], [-14, 12, 2], [-3, 6, 1], [22, -2, 2], [2, -4, -2], [-4, 10, 1], [-20, -1, -7], [12, -4, -2], [2, -9, -4], [-8, 6, -6], [0, -10, -3], [16, 0, 2], [8, 5, 0], [25, -6, -2], [-10, 6, -4], [18, -4, 5], [14, 5, 2], [-14, -6, 5], [-1, 2, -4], [12, 0, -4], [16, -6, -3], [-2, 0, -1], [-2, -12, -4], [-18, -9, 4], [-2, -14, 1], [1, 2, 0], [-34, -2, -2], [14, 8, 7], [-2, 4, 9], [4, 8, 0], [24, 9, -4], [-7, -6, -3], [12, 3, 2], [-8, -4, 5], [4, -12, -3], [-6, -2, 2], [6, -8, -4], [0, -4, -6], [10, -2, 2], [2, 11, 8], [14, -2, -6], [0, -4, 2], [-1, -2, -5], [-34, 0, -1], [-10, -6, -2], [-16, -10, -2], [0, -15, -2], [-2, 2, 0], [14, 5, 6], [-16, -2, -7], [2, 6, 10], [32, 6, -4], [-40, 0, -2], [-6, -18, -3], [12, 9, -4], [24, 10, -2], [36, -5, 0], [-14, -2, 0], [0, -8, 3], [-16, -8, -2], [-8, 10, 3], [14, -8, -6], [10, 4, 1], [2, 8, -6], [-6, -10, 3], [-21, -4, 4], [24, 8, -4], [16, 12, 12], [2, -6, 8], [8, -6, 11], [14, -14, -2], [-21, 10, 5], [20, 4, 6], [8, 5, 7], [45, -2, 3], [12, 2, -4], [15, 2, 2], [-10, -9, 0], [-4, 22, 1], [-24, -1, -8], [0, 2, 4], [-10, -7, -3], [4, -4, 12], [7, -22, -9], [-20, -2, -12], [-8, 18, 8], [-24, -6, -6], [16, 12, -4], [10, 0, 1], [20, -6, -1], [-25, 14, 5], [-4, 12, 7], [-20, -8, 2], [-8, -4, -6], [12, 8, -4], [26, -7, 2], [10, -14, 4], [-6, 7, 12], [34, 4, 4], [14, 16, 6], [0, -16, 5], [-4, 4, -3], [40, 0, 8], [14, 9, -1], [12, -12, 3], [-6, -2, 5], [11, -14, -4], [-2, -10, -3], [30, -18, -4], [12, 4, 11], [10, 10, 2], [-4, -4, 4], [24, 9, 6], [-18, 2, -8], [-4, -4, -13], [-42, 7, 6], [-10, 12, -10], [24, 10, 0], [-15, -6, -16], [-6, 14, 2], [14, 0, -10], [-18, -5, -3], [-5, 10, 5], [16, 15, -4], [14, 14, 6], [-6, -10, -8], [-19, 4, -3], [-30, 10, 0], [57, -4, 1], [-10, 4, 9], [12, -8, 2], [-4, -10, 8], [-11, -16, -10], [-37, -6, -3], [-16, 15, 4], [22, 2, -10], [20, 2, -3], [26, 2, 3], [8, 6, -3], [-10, 12, -2], [-8, -10, 6], [10, 15, 13], [-39, 6, -4], [-23, -2, -5], [-54, 3, -2], [0, -10, -12], [20, 12, 12], [-12, -12, -9], [-10, 6, -6], [46, -10, -2], [2, 16, 11], [-36, 4, 1], [-44, 4, -3], [-8, -20, 4], [-18, -8, 0], [-30, -14, -4], [-4, 14, -3], [20, 13, -5], [12, 4, 10], [-23, 12, 5], [-20, -6, 3], [6, -6, -12], [5, -4, 13], [-26, 14, 0], [-10, -16, -16], [-38, 2, 0], [-8, 18, 4], [26, -10, -10], [46, -4, 4], [-36, 4, 6], [-32, -16, -6], [-42, -10, 1], [46, 4, -4], [-6, -25, -8], [-2, 10, 8], [30, -6, 6], [-20, -12, -4], [46, -2, -2], [6, -20, 3], [62, -2, 1], [-16, 27, 11], [-24, 2, -4], [-6, 12, -1], [-25, 12, 10], [28, -20, -2], [-36, -6, 3], [0, 0, -7], [-38, 10, 2], [-23, 10, 0], [-8, 8, 5], [-6, 8, 11], [53, -2, 2], [-10, 6, -14], [-2, -4, 0], [-20, -1, 12], [35, -2, -5], [-29, -14, -3], [6, -20, -13], [-2, -2, 10], [-30, 6, -2], [8, 10, -8], [32, -5, 0], [1, -4, -1], [22, -3, 1], [42, -10, -7], [16, -1, 19], [-30, -4, -8], [0, 7, 2], [-28, -4, -15], [18, -18, -2], [4, 10, 3], [4, -6, 3], [-38, 10, 5], [-38, 2, 1], [0, 16, 6], [-39, 10, 1], [10, -20, -11], [12, -3, 7], [7, -12, 8], [24, 4, -11], [4, -8, 11], [-6, -6, -4], [-32, -18, 9], [-4, -24, -4], [-11, -20, 3], [-10, 32, 6], [-14, -19, -4], [-13, -2, 2], [30, 12, -5], [36, 6, 15], [-22, -11, 8], [-10, -24, -12], [38, 11, -3], [7, 8, 17], [17, 16, 9], [66, 4, -4], [38, -18, -6], [36, 11, 12], [47, 10, -3], [6, 22, 4], [26, -2, -6], [-32, -4, 7], [-32, 12, 12], [6, -26, 2], [-12, -26, -6], [16, 11, 8], [-4, -22, 2], [-40, 18, 7], [-62, 4, 0], [68, 0, 4], [-10, -4, 11], [-17, -12, -9], [26, -15, 5], [-9, 6, -12], [-32, -2, 2], [28, -34, -12], [50, 13, 7], [-8, -26, 4], [-16, -16, -18], [0, -15, -10], [-20, 4, -3], [-1, -6, -17], [14, -4, -12], [-22, 24, 9], [42, 6, 8], [24, 16, -2], [22, 8, 20], [-54, 7, 0], [27, -10, -7], [28, -4, 6], [-8, -18, -3], [-8, -10, -10], [-9, -22, -3], [32, -6, -6], [12, -21, -12], [-22, -4, 2], [-16, 19, 8], [-3, 10, -13], [-2, 28, 8], [14, -22, -12], [-32, 9, 6], [-52, 12, 2], [4, 28, 10], [8, -32, -2], [22, 14, 1], [-18, 37, 15], [43, 4, -6], [24, 24, -1], [18, 8, -8], [40, 14, 9], [10, 21, 14], [-14, -12, -10], [32, 4, 24], [-19, -6, -9], [22, -1, -9], [-72, 0, -1], [-26, -2, -9], [26, 8, -10], [-28, 2, -4], [6, -10, 1], [-16, 18, -1], [36, -11, -4], [-34, 30, 14], [-30, -11, 2], [1, -8, -20], [-26, 0, 8], [-24, -10, -12], [-16, 24, 14], [-26, -6, -12], [54, -14, -8], [-10, 23, 10], [31, -22, -5], [-1, -24, -19], [46, 12, 16], [16, -2, 21], [-34, -8, 14], [6, -4, 5], [29, 0, -5], [-6, 14, -10], [24, -3, 14], [-40, -6, -7], [16, 20, -3], [-18, 14, 14], [8, 6, 12], [-29, -18, 13], [22, -15, 9], [11, -12, 3], [44, 14, 16], [-32, -28, -3], [-40, -16, 7], [22, 24, 14], [14, 28, 4], [-21, 12, -7], [-70, 5, 4], [-12, -24, 10], [24, -35, -16], [-14, -22, -9], [-44, 14, -9], [54, -10, -2], [6, 18, 2], [-22, 28, 2], [18, -16, 0], [-62, -4, -6], [42, 10, 18], [-30, 11, -10], [-26, 38, 14], [9, 32, 11], [22, 26, 12], [-10, -18, -2], [-8, -12, -10], [-26, 2, -14], [11, 6, 24], [40, -24, -6], [-3, -6, -20], [24, -4, 2], [-5, 0, 17], [-38, -10, -16], [-56, 0, 3], [42, -10, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8470_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8470_2_a_cg();".
// 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_8470_2_a_cg(:prec:=3)
    chi := MakeCharacter_8470_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(3169) - 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_8470_2_a_cg();".
// 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_8470_2_a_cg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -7, 0, 1]>,<13,R![-2, -7, 0, 1]>,<17,R![-32, 24, 11, 1]>,<19,R![-17, -21, 5, 1]>],Snew);
    return Vf;
end function;