
// Make newform 8470.2.a.ci in Magma, downloaded from the LMFDB on 09 May 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_ci();".
// 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_ci();".
// 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, 1, 1], [-3, -1, 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], [1, 0, 1], [1, 0, 0], [1, 0, 0], [0, 0, 0], [0, -1, 1], [0, 0, 2], [3, -2, 1], [3, 0, -1], [1, 0, 3], [-4, 1, 1], [1, 0, -1], [-3, 2, -3], [-4, 3, -1], [0, -2, -2], [5, 0, 3], [0, -2, 2], [2, 2, 2], [0, 0, -4], [-10, 2, 0], [4, 0, -2], [-7, 2, -1], [2, 0, -2], [4, 2, -4], [-1, 2, 3], [2, 2, -6], [-2, 2, 0], [-4, -1, -5], [-3, 0, -1], [-2, 0, -4], [4, 2, -2], [11, 0, 3], [10, 0, 0], [-5, 4, -1], [7, -4, 5], [-1, 0, -5], [-12, 0, -2], [4, 4, 4], [8, 0, 0], [-2, 2, -2], [-4, 1, 5], [6, 2, -6], [-8, -2, -2], [-4, -3, -1], [-4, 4, -2], [-4, 4, 0], [-12, 2, 2], [2, 2, -4], [8, 2, 6], [16, -5, 1], [-18, -2, -2], [19, 2, 1], [-1, 2, -1], [-12, -2, -10], [5, 2, 1], [0, -4, 4], [8, 1, -1], [-2, 0, 6], [-2, -6, 6], [24, -2, 0], [-12, 2, -6], [12, 1, -1], [6, 0, 2], [-16, -1, 3], [21, -2, 5], [9, 4, 3], [0, -1, 7], [-6, -2, 2], [-8, 1, 1], [8, -4, 2], [13, 2, 9], [1, 2, -1], [-6, -6, -4], [-16, 2, -8], [-4, 8, -8], [6, 2, 0], [0, -4, -2], [0, 2, 4], [-4, -3, 11], [-7, -2, -11], [-8, -6, -2], [6, -2, 6], [-1, -8, 11], [-1, -2, -1], [-2, -6, 0], [16, 2, -2], [12, -3, 11], [-10, -4, -4], [12, -2, 0], [19, -4, 3], [9, 6, -9], [24, 4, -4], [-7, -6, -1], [8, -2, -6], [-20, 5, -7], [28, 0, 4], [-6, 2, -10], [0, 8, -10], [-2, -6, 4], [-19, -4, -5], [0, 0, 4], [-16, 0, -10], [0, 4, -8], [22, 0, 0], [-10, 4, -2], [9, 2, 5], [-11, 6, -5], [-8, 4, -2], [6, -10, 4], [-9, -6, -9], [-8, 3, -1], [24, 4, -6], [-8, 2, -16], [-8, -2, -6], [12, 2, -2], [-6, 6, 6], [11, -2, -7], [10, -4, -2], [-7, -4, -5], [18, 2, -8], [14, 4, 12], [6, -8, 0], [2, -2, -10], [2, 2, -8], [2, 0, -2], [17, 4, 7], [12, -4, -6], [-8, 0, 0], [-20, -6, 14], [42, 0, 0], [12, -2, -10], [44, -4, 0], [-14, 2, 4], [15, 0, 5], [13, -6, 5], [1, 2, -7], [14, 0, -8], [-12, 0, 8], [-12, -2, 4], [16, -4, -2], [-17, -4, 11], [1, 8, -13], [21, -2, -1], [-12, 4, 4], [8, -11, 3], [-12, 7, -1], [-10, 0, 4], [8, 4, -10], [0, -6, -2], [-31, 2, 7], [0, 8, -2], [30, 2, 6], [26, 0, 14], [-32, 8, 0], [10, 6, 4], [-34, 2, 0], [-13, -8, 7], [0, 10, -8], [-6, 10, -2], [18, 4, -4], [18, 0, 6], [28, 3, 5], [28, -16, 4], [32, -12, 0], [22, -4, 0], [-18, 2, 0], [0, 10, -14], [-6, -6, -6], [4, -4, 10], [-12, 10, -8], [-2, -10, 16], [2, 12, -16], [-11, -8, 1], [22, 0, 0], [28, -1, 7], [10, 2, -6], [-4, -2, 14], [18, -4, 8], [4, 5, -19], [18, 0, -8], [19, -6, 9], [11, -8, 11], [21, 4, -9], [-22, 8, 4], [23, 6, 9], [-8, 3, 1], [4, 10, -20], [13, 8, -11], [6, -12, 4], [-28, -4, 0], [-3, 10, -15], [20, 7, -5], [-12, 8, -8], [-2, -10, -2], [-28, -7, -3], [-1, -10, 15], [-20, 18, 0], [3, 12, -3], [2, 6, 10], [32, -7, -3], [-1, 4, -3], [-24, 10, -6], [-14, -6, 14], [-1, -10, 3], [-30, 8, -2], [46, 2, 10], [-36, 4, 8], [16, -13, -5], [-10, -10, -2], [10, 8, -10], [24, 9, 7], [32, 5, 3], [37, -10, 7], [19, 4, 7], [32, 0, -8], [-12, 4, -10], [-2, -2, -4], [-25, -2, 3], [31, -4, -9], [11, 0, 9], [-1, 0, -15], [34, -2, 8], [-28, -7, 15], [-21, -6, 17], [0, -4, 0], [9, 8, 11], [13, -6, -7], [-22, -4, -18], [12, -5, 19], [14, 10, 4], [27, -12, 13], [-31, 18, -5], [-19, -2, 15], [-26, -6, 2], [27, 0, -5], [44, 3, 3], [-32, -4, -14], [-6, -10, -2], [32, 0, 4], [4, -4, -16], [10, -8, 6], [5, 6, -21], [-8, 8, -16], [-18, 0, 0], [-14, 12, 8], [22, -6, 8], [-38, -4, -10], [8, -2, 2], [8, 0, 4], [-4, -12, 16], [6, 8, 12], [29, 2, 5], [31, -6, 13], [-34, -2, 10], [24, 7, -15], [-40, 2, -10], [-8, 6, 8], [10, 0, 14], [18, -8, 20], [42, -6, -2], [24, 2, 18], [-12, 4, -4], [-22, 10, -6], [26, -12, 4], [19, 2, -9], [-28, -2, -2], [-12, 18, -8], [8, -4, -14], [-20, -4, 20], [38, -2, 2], [-38, -4, 8], [-4, 6, -26], [-17, 2, -5], [1, 0, -19], [6, -6, 18], [28, -10, -6], [8, -10, -6], [33, -4, -1], [-6, 10, -10], [3, -14, 5], [-24, -8, -16], [38, 4, 10], [32, 9, -3], [-6, -6, 10], [20, 0, 0], [-8, -2, -2], [-12, 14, -16], [-14, -12, -4], [-12, 4, 8], [-55, 6, 5], [-10, -6, 26], [23, -10, -3], [14, -2, -18], [-3, -4, -15], [-30, -10, -2], [-8, 8, -18], [4, -9, 7], [4, 0, 6], [-17, -6, -15], [2, 8, -10], [-38, 2, -10], [-12, -8, 18], [-31, 0, 13], [22, 8, -14], [4, -16, 8], [-14, -14, -6], [-18, 8, 2], [-4, -17, 9], [-26, -12, 26], [-4, 16, -6], [68, 3, 11], [18, 10, -18], [-42, -4, -12], [-22, 16, -2], [-18, 10, 0], [-35, 10, -11], [-14, -12, 2], [-29, 14, -3], [-35, 6, -3], [28, -4, 18], [-9, -14, -7], [-38, 6, -18], [22, 2, 6], [19, 4, 11], [4, -13, 13], [-24, -10, -4], [-36, -11, 9], [1, -2, -1], [-20, -2, 2], [-36, 0, -6], [27, -12, -7], [41, 0, 15], [-26, 8, -10], [10, 0, 22], [-18, 14, -8], [5, -4, 25], [-12, 6, -20], [-22, 2, -18], [16, 10, -16], [-12, 13, -19], [-13, -12, -3], [2, -4, -12], [40, -6, 0], [4, 8, 20], [-5, -4, 5], [-40, 2, 6], [24, -9, -11], [-16, 0, 4], [-26, 8, -26], [46, -2, 14], [9, 0, -7], [28, 8, 18], [-24, 4, 6], [-20, 0, -20], [-25, -12, -3], [28, -6, 8], [40, -1, 19], [17, 6, -25], [-4, 7, 17], [-37, 2, 1], [18, 0, 8], [33, 2, 17], [-19, 12, -11], [32, -22, -2], [-25, -2, 5], [8, 4, -6], [-16, -15, -5], [-14, -4, 14], [0, -12, 22], [-12, 12, -4], [10, -12, -14], [6, 8, 22], [7, 0, -7], [-66, 0, -2], [7, 4, 9], [-14, -6, -16], [-19, 14, -17], [-2, 16, -24], [-35, 10, -19], [14, 2, 26], [-24, -1, 5], [56, -16, 6], [-18, -16, -2], [-2, 6, -2], [-41, -10, -19], [-40, 0, -8], [24, 16, -8], [34, -6, 14], [-20, 17, 5], [-22, -14, 20], [-36, -1, -11], [-18, 0, -8], [8, -3, 5], [19, 12, 7], [-4, 4, 8], [0, -3, 27], [26, 6, 16], [52, 0, 2], [-32, 7, -1], [-23, -4, 15], [-74, -4, -8], [8, 16, -2], [-12, 16, -12], [-7, -10, -15], [20, -8, 2], [-58, 2, -16], [-9, -12, 25], [52, 6, 28], [17, -8, 25], [32, 6, 2], [-38, 12, 0], [38, 2, 6], [-56, 0, -4], [-22, 10, 8], [10, -2, 10], [34, -6, -10], [35, 8, -17], [74, 4, 6], [57, -14, 5], [-16, -9, -13], [-34, 14, -10], [28, -11, -13], [-41, 12, 7], [-15, -6, -5], [-13, -10, 19], [-6, 0, -20], [-64, 2, 6], [-56, 14, 0], [4, -20, 12], [48, -6, 10], [-46, 20, -4], [-42, 16, -14], [16, 1, -15], [16, 0, 8], [58, -8, 4], [-14, -10, -2], [-4, -10, -4], [58, -6, 6], [-18, -14, -28], [4, 8, -28], [1, -6, -5], [-22, 0, -40], [-25, 24, -15], [45, -10, 7], [29, -6, -19], [48, -15, 7], [-12, -18, 6], [32, 7, 27], [42, 0, 24]];
    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_ci();".
// 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_ci(: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_ci();".
// 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_ci( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![8, -6, -2, 1]>,<13,R![-16, -16, 2, 1]>,<17,R![8, -28, 2, 1]>,<19,R![232, -46, -6, 1]>],Snew);
    return Vf;
end function;