
// Make newform 5070.2.a.bz in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5070_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_5070_2_a_bz();".
// 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_5070_2_a_bz();".
// 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 := [52, 20, -19, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-10, -1, 1, 0], [0, -14, -1, 1]];
    Rf_basisdens := [1, 1, 4, 4];
    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_5070_a();"
function MakeCharacter_5070_a()
    N := 5070;
    order := 1;
    char_gens := [1691, 4057, 1861];
    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_5070_a_Hecke(Kf)
    return MakeCharacter_5070_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, 0, 0], [1, 0, 0, 0], [0, -1, 0, 0], [1, 0, 0, 1], [0, 0, 0, 0], [4, 0, 0, 0], [0, -1, 0, -1], [2, -1, -1, 1], [2, -1, 1, -1], [0, -1, -3, 1], [2, 0, 3, -1], [-2, 2, 0, 0], [3, 1, -4, 0], [-3, 2, 2, 0], [2, 1, 2, 1], [1, 1, -2, 0], [4, 0, 2, 0], [-4, 2, -2, 0], [-2, -2, -2, 2], [-2, -2, 4, 0], [-1, -3, 0, 0], [-6, 2, 0, -2], [0, 1, 4, -1], [8, 2, -2, 0], [0, 0, 0, -2], [10, -1, -2, 1], [10, 0, 4, 0], [-6, 0, 0, -2], [-6, 3, 1, 1], [10, -1, 2, 1], [-8, 2, 1, -1], [7, 1, -6, 0], [8, -1, 0, 0], [1, 1, -2, -2], [-4, 4, 2, -2], [3, 2, -6, -1], [4, 1, 1, -3], [-11, 2, 6, 0], [4, 1, -2, -1], [-10, 1, -5, -1], [8, 2, -4, 0], [-2, -2, 6, 0], [10, 0, 2, 2], [-8, -1, 2, 0], [0, -2, -4, 0], [4, 1, -2, -2], [6, -1, -6, 2], [2, 0, -10, 0], [-10, 0, -4, 2], [-10, 1, -3, -3], [-2, 2, 2, 4], [8, 0, -1, 4], [4, -2, 5, 3], [6, -3, 3, 1], [6, 2, 5, 2], [-4, 0, 8, -4], [4, 1, 13, -1], [5, 0, 6, -1], [6, -2, 4, 0], [7, 1, 0, -2], [8, -1, -8, 2], [8, -4, 2, 0], [-2, 4, -2, 2], [-8, 6, -4, -2], [0, -1, 6, -4], [4, 6, 4, -2], [0, 4, 2, 4], [-8, -2, 6, 0], [2, 0, 10, 0], [18, 2, 4, -2], [12, 2, -4, 0], [-6, 2, -2, -4], [1, -3, -4, -4], [-6, 7, 2, 1], [11, -3, 2, 2], [-6, 5, 3, 3], [-6, 0, 1, -3], [16, -5, -2, 1], [8, -7, -2, 1], [-4, 2, 2, 2], [14, 4, -2, -2], [-14, 0, -2, -2], [6, -4, -10, 2], [-12, -6, -4, 2], [2, -6, 2, 2], [-4, 1, -14, -1], [4, -4, 6, -4], [3, 1, 10, 0], [18, -2, 4, 0], [-2, -2, 4, 2], [6, 4, -6, 0], [14, -1, -6, 2], [-2, 1, -10, 4], [-2, -2, -2, 0], [-16, -3, -4, 3], [-13, -1, 2, 0], [18, -1, 6, 0], [7, -3, 12, 0], [-8, 2, 0, 4], [8, -2, -16, 2], [0, -1, -8, 6], [2, 0, -10, 4], [-18, 3, 2, 0], [-4, -1, 2, -2], [6, 8, -4, 2], [-4, 4, -2, 2], [-3, 3, -2, -2], [-12, 8, -2, -6], [5, -2, -12, 4], [0, -3, 0, -3], [6, -2, -5, -5], [-7, 5, -2, -4], [4, 3, -12, -1], [-4, -2, 2, 2], [-4, 1, 4, -2], [32, -2, 2, -2], [4, -3, -8, -1], [-4, -1, 20, -3], [-22, 5, 11, 3], [-6, 2, -18, 0], [4, -4, 0, 4], [-2, 0, -6, -2], [4, -2, 0, 2], [-32, -1, 4, -1], [2, -9, 1, 3], [4, 0, 8, -2], [-16, 0, 16, 0], [-4, 5, 6, -3], [2, -3, 0, 0], [-18, 3, 4, 5], [17, -5, -2, -2], [-19, -3, -4, -2], [-6, -1, -10, 9], [-12, 7, -6, -3], [16, -5, -5, 3], [-10, -1, 8, -2], [-20, -1, 15, 1], [8, -12, -4, 4], [-14, 0, 0, -4], [8, -1, -14, 3], [-19, 1, 6, -4], [4, -7, 0, -3], [16, -4, 14, 2], [0, -2, 18, -4], [10, 4, 12, -2], [16, 3, -3, 1], [6, -1, 15, -5], [6, 1, -2, -2], [-21, 1, -6, -6], [20, 1, 3, 3], [-4, -1, 0, -2], [13, 3, -16, 2], [14, -8, -15, 0], [7, 5, -8, 0], [16, -2, -6, -2], [-4, 6, -8, -2], [24, -4, 8, 8], [-2, -4, -12, 4], [-14, 6, 4, 0], [-2, 6, 6, -8], [-22, 7, -9, -3], [-20, -5, 10, -6], [-10, 1, 10, 0], [-7, -1, -10, 8], [21, 2, 6, 4], [3, -3, 8, -6], [-26, 1, 2, -1], [-20, 6, 9, -2], [-2, 2, -8, 8], [20, 2, 12, 2], [-6, -2, 18, -2], [-14, 6, 14, -2], [28, 4, 14, -8], [-13, 5, 12, 0], [-12, -9, 10, -4], [-34, 0, -12, 2], [-30, 10, 4, 0], [24, 6, -12, 6], [-10, 8, -4, -6], [-2, 0, -10, -2], [8, 4, 9, -5], [23, 4, -14, 5], [9, -1, 14, 2], [-23, 0, -18, 0], [-18, -1, -17, 3], [-35, 2, 2, -7], [-32, 5, -3, 3], [4, -10, 5, 2], [-12, 2, -8, 4], [4, -8, -2, 6], [4, -6, -6, -4], [-28, 5, -18, -6], [5, 1, -18, -2], [-32, 10, 0, -2], [-24, -6, 8, -4], [-4, -9, -5, 3], [-15, -2, -18, 1], [-5, -3, 6, -2], [-6, 5, 17, -1], [22, -12, 0, 2], [1, -1, 4, 6], [-32, 2, 4, -4], [-8, -1, 16, -4], [12, 5, -14, 7], [2, 0, -2, -2], [-44, 6, 2, 2], [-8, 14, -2, -4], [2, 9, 2, -3], [6, -3, -2, -2], [-2, -4, -2, 4], [22, 9, -3, 3], [-4, 5, 28, 1], [-8, 2, -12, -4], [-28, -4, 10, -2], [-16, -6, -13, 10], [-6, 8, 16, -6], [10, 9, -6, 0], [6, -1, 12, -10], [-34, 3, -10, 4], [-20, 4, 8, 2], [-20, -8, -6, 2], [4, 9, -4, -1], [12, 12, -4, 4], [10, 12, 0, -2], [-10, 2, 16, -6], [-26, -9, 3, 1], [-24, -2, -4, 4], [-4, -8, -6, 8], [1, 0, -20, -3], [-1, -10, 14, 3], [8, 3, -8, 12], [-40, -2, 18, 0], [-8, -1, 2, -8], [13, -3, 0, 4], [18, 1, 0, 2], [-16, -1, 3, -5], [16, 4, 10, -2], [-22, -3, -2, -8], [4, 2, -24, 4], [40, -6, 8, 0], [-24, 2, -10, 2], [6, 1, 0, -1], [24, 8, -10, 2], [-10, 8, 8, 8], [-10, -6, -4, 2], [-28, 0, -4, 4], [-8, -1, 2, 1], [-50, -3, -4, -1], [10, -8, -1, -4], [4, 3, -11, 1], [-4, -9, -10, 5], [-31, 0, 6, -8], [-13, 13, 8, -8], [-26, -1, 4, -9], [20, 1, 14, 3], [8, 2, -12, -2], [0, 4, 22, 4], [14, -11, 6, 9], [-30, 4, -16, -8], [0, 13, -4, 1], [8, -6, -2, -2], [-26, 4, -10, 2], [10, 6, 16, -12], [13, -15, -2, -2], [10, 6, -12, 4], [14, -8, -8, -10], [44, -7, -16, -1], [16, -7, -7, -5], [-10, 12, -2, -2], [8, 10, 8, -8], [-4, 5, 21, -5], [24, 4, -16, -2], [-21, -5, -4, 2], [-32, 8, 4, 0], [-20, 4, -16, -4], [-2, -8, -16, 4], [-16, -4, 22, -4], [4, 1, -11, -3], [26, 3, 17, 1], [-8, 11, 4, -7], [-24, -1, 15, -3], [-14, 9, 9, 3], [4, -8, -12, 8], [-12, -5, -11, -1], [20, -2, 12, -2], [-14, -6, 2, 2], [8, -3, -10, 0], [-32, 3, -3, -9], [14, -3, 12, 6], [14, 3, -7, -3], [42, -10, -4, 0], [25, 3, 26, -6], [13, -6, -4, 3], [14, 10, 12, -4], [38, -1, 15, 3], [11, 5, -4, 16], [-12, -3, 0, 3], [-30, 1, -9, -5], [-8, 7, -5, 5], [-26, 2, 20, 0], [-22, 6, 6, 2], [39, -1, -12, 0], [-28, 2, 8, -2], [2, 0, 10, -6], [28, -2, -12, 4], [2, 6, -12, 4], [2, 2, -14, 0], [-28, -8, -12, 6], [22, -3, -2, 3], [6, -6, -5, -6], [31, -5, -2, 8], [4, 13, -12, 2], [37, -7, 16, 6], [2, 1, -8, -10], [4, -3, -32, -4], [45, 1, -10, 4], [-52, -4, -6, 0], [14, 6, -22, 6], [34, -8, 4, -4], [-6, 4, -10, 14], [36, 4, 8, -2], [10, -7, -17, -3], [14, 0, 34, -6], [18, 8, -8, 4], [31, 5, -12, -4], [-14, -4, 28, 2], [0, -11, 3, 7], [14, -16, -15, 4], [-34, -4, -14, 2], [-4, -12, -24, 8], [14, -9, 8, -5], [41, -11, -8, 4], [-22, -8, -2, 4], [-40, 6, 4, -2], [-8, 6, -14, 2], [-10, 2, -24, 8], [3, 5, 18, -4], [16, 8, 3, -4], [10, -3, 20, -1], [-30, 2, -3, -3], [40, -4, -4, 2], [15, -1, 34, 4], [5, 13, 4, -4], [34, -9, -4, -10], [-36, -6, -11, -1], [16, 2, -18, -2], [-12, -7, -1, -1], [-8, -20, 6, 2], [-18, 1, 2, -7], [-26, 3, -14, -7], [8, 4, 28, 6], [14, 8, 12, -10], [18, 1, -30, 3], [40, -6, 2, 10], [-14, 1, 33, 1], [-26, -4, 26, -4], [-11, 4, 12, -7], [30, 9, -1, -5], [5, 6, -18, 8], [34, -7, -24, 0], [8, -7, -16, -2], [10, 3, -9, 5], [4, -5, -22, 13], [8, 2, -16, 10], [-16, -6, -10, 6], [-24, 14, 24, 4], [8, -9, -16, -4], [-13, 5, 20, -8], [40, -6, 6, -10], [16, -3, 6, 2], [-12, -8, 4, -4], [4, 8, -24, -4], [-15, 9, 8, 4], [-6, -2, -28, 12], [0, 11, -14, -7], [2, 4, 0, 4], [44, -10, -8, 8], [-4, 23, 14, -8], [-6, 8, 14, 8], [-7, -7, -14, -10], [-22, -2, 4, -6], [6, 1, 12, 8], [17, -9, 14, 0], [-32, -5, 28, -5], [-34, 17, 28, 2], [36, -5, -7, 1], [-51, 9, 0, 4], [44, -1, 17, 7], [-20, 16, 6, 4], [16, -1, 26, 11], [-14, -1, 2, -8], [17, -4, -16, 9], [9, 17, -8, -4], [4, 0, 4, 0], [2, 4, 20, -2], [24, -11, -13, -3], [56, -2, 16, 4], [-10, -2, 12, -16], [-41, 11, -14, -4], [-10, 14, -2, 2], [-42, 3, -15, -1], [-4, 18, 6, -6], [-42, -2, 0, 0], [-42, 0, 8, 2], [21, 3, 12, 4], [-4, 18, -16, 0], [-32, -7, -24, 11], [-4, 9, -5, 7], [24, -3, -2, -1], [4, 0, -10, 0], [-8, 5, 6, 15], [38, 6, -8, 12], [10, -13, -10, -2], [-4, -6, 12, 0], [38, -1, -5, 9], [20, 10, 4, -2], [18, -1, 6, 1], [-1, 15, 26, -10], [0, -8, 30, -4], [44, -10, -8, -8], [2, -10, -1, -9], [6, 7, -8, 2], [32, 2, -5, 1], [8, -16, -4, 6], [2, 13, -2, 8], [8, -8, 0, -12], [4, -11, 6, -1], [-2, 1, -38, -5], [22, -6, -2, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5070_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5070_2_a_bz();".
// 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_5070_2_a_bz(:prec:=4)
    chi := MakeCharacter_5070_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_5070_2_a_bz();".
// 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_5070_2_a_bz( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5070_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![52, -20, -19, 2, 1]>,<11,R![181, 62, -34, -2, 1]>,<17,R![-4, 1]>,<31,R![-956, -556, -79, 4, 1]>],Snew);
    return Vf;
end function;