
// Make newform 8470.2.a.bo in Magma, downloaded from the LMFDB on 20 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_bo();".
// 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_bo();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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], [-1, 0], [1, 0], [0, 0], [-2, 0], [-2, -2], [0, 1], [0, -1], [-6, 1], [0, -2], [6, -1], [-2, 1], [4, 2], [-8, 0], [-2, -3], [-8, 0], [-2, 4], [4, 4], [0, -2], [-2, -2], [8, -1], [8, -2], [-6, -2], [2, -1], [-10, 0], [0, -2], [-4, 2], [-6, -3], [2, 0], [-8, -4], [-16, -1], [-6, 4], [-16, -1], [10, -3], [-8, -1], [10, 2], [12, 0], [16, 0], [-10, 4], [12, -2], [-14, 0], [8, 4], [-10, -4], [18, -2], [0, -4], [-4, -4], [0, -10], [24, 0], [10, -4], [6, 0], [16, -3], [6, 1], [0, -4], [26, -1], [-8, 8], [2, 0], [0, -6], [2, 4], [-10, 6], [0, 8], [-2, 0], [-8, 6], [-16, 6], [-6, -1], [6, -1], [12, 6], [-18, -4], [-12, -6], [-10, -6], [-22, -1], [-8, -9], [0, 10], [2, -2], [4, 0], [-24, -2], [14, 6], [-6, -6], [10, 0], [-2, 5], [-8, -8], [6, 8], [0, 7], [18, 1], [-8, -6], [-20, 4], [-6, 0], [22, 0], [6, 2], [16, -9], [-8, 7], [-24, -4], [0, -13], [20, -4], [12, 6], [-8, 4], [-22, 4], [2, 14], [16, -2], [2, 11], [-12, -4], [10, 6], [0, 8], [6, -8], [-4, 14], [-6, 11], [-24, -5], [-26, 2], [16, 2], [-10, -1], [-24, -2], [-6, -6], [-38, 2], [40, 0], [0, -8], [-30, 0], [0, 5], [-16, 10], [-18, 9], [4, 2], [-6, 0], [14, 8], [14, 0], [28, -2], [-16, 6], [-38, -1], [-2, 10], [-8, 0], [16, -4], [46, 0], [4, -16], [16, 0], [0, 14], [6, -11], [-18, 9], [6, 3], [-22, 4], [-8, -8], [-22, 10], [6, -10], [-8, -5], [10, 7], [24, -1], [4, 16], [-6, 16], [-40, 2], [14, 4], [-10, 6], [16, -4], [0, -13], [-14, 6], [-14, 4], [36, 2], [0, 8], [12, 2], [-16, -2], [16, -13], [-22, -6], [-2, -8], [-2, -4], [-4, 14], [-2, 16], [-48, -4], [-8, -12], [2, -8], [0, 14], [-40, 0], [-42, 0], [-2, 2], [34, 2], [20, 10], [-6, 4], [-16, -3], [6, 8], [0, -14], [10, 4], [20, -8], [30, 0], [0, -2], [30, 4], [-8, -7], [-8, -9], [38, 5], [14, 8], [-8, 5], [-22, 12], [26, -2], [-24, -11], [6, -4], [-8, -8], [-38, -7], [-4, -6], [-28, -8], [-6, 16], [4, -2], [34, 9], [14, -2], [-2, 9], [-26, 12], [-32, -6], [-46, 3], [56, 4], [10, -12], [30, 1], [24, -6], [2, -12], [-24, 4], [12, -2], [-10, -8], [48, 2], [6, -16], [-14, 0], [-32, -9], [32, 11], [-8, -12], [-14, -2], [0, 2], [6, 9], [-56, 3], [14, -15], [18, -3], [-40, -2], [26, -16], [-24, 9], [-8, 12], [-46, -1], [34, -7], [16, -14], [16, -6], [-20, 2], [-18, 15], [-24, 11], [-24, -5], [6, 4], [-32, -5], [-8, -6], [-54, -2], [18, -4], [-4, -12], [24, 4], [-52, 2], [-8, 15], [8, 12], [-50, -4], [22, 8], [16, 14], [32, -2], [0, 0], [-4, 8], [-16, 20], [-6, 12], [-2, -1], [-40, -3], [-6, -20], [22, 8], [-36, 0], [-18, 14], [24, -6], [26, 16], [6, 20], [56, 0], [-12, -16], [-42, 0], [-18, -8], [18, 11], [-56, 8], [-2, 18], [58, -2], [20, 0], [-6, 12], [26, 12], [-20, 8], [2, 1], [16, 17], [22, -4], [-56, 8], [36, 0], [18, -15], [54, 0], [0, -7], [16, 8], [0, -2], [8, 10], [6, -4], [0, 12], [0, 0], [18, -14], [2, -28], [-16, 12], [14, 21], [-2, 16], [-8, -15], [-58, -8], [8, -15], [10, -20], [-10, -2], [40, -6], [2, -14], [48, -3], [16, -14], [38, 12], [-30, -6], [-24, 17], [-20, -2], [20, 0], [66, 4], [36, 2], [10, 16], [-8, -14], [18, 6], [-8, -14], [30, 16], [46, 0], [-4, 22], [24, -10], [6, -7], [24, 6], [8, 29], [-22, -15], [-18, 2], [24, 13], [-6, 8], [30, -4], [0, 15], [-2, -8], [-6, 18], [4, -2], [48, 15], [56, 4], [-54, -2], [-30, -9], [46, -9], [-8, -10], [-64, -10], [-60, 6], [-16, 21], [-58, -2], [6, -24], [-38, 6], [48, 6], [-50, 9], [-6, -12], [-50, -6], [16, -12], [-50, -9], [36, 4], [26, 20], [4, -20], [32, -14], [10, -20], [72, 5], [26, 22], [46, -6], [56, -4], [42, 5], [-2, -6], [16, -2], [-24, 11], [70, -8], [0, 9], [-30, 8], [-42, -3], [8, 1], [4, 24], [-56, -7], [-54, 10], [30, 4], [-64, -10], [-42, 2], [20, -16], [-32, -18], [16, 18], [74, -3], [24, 2], [-26, -5], [48, -6], [-32, 11], [-26, -16], [-42, 15], [6, 20], [-62, 4], [-22, 2], [24, -26], [-2, 24], [-40, -3], [-56, 4], [-8, 4], [-6, -16], [-4, -2], [-56, -6], [26, -12], [58, 8], [-28, 2], [16, 27], [-24, 20], [-82, 0], [-8, -10], [-46, 10], [52, -2], [-46, 7], [14, -20], [-14, 22], [-48, 12], [-6, -27], [-50, 14], [0, -18], [-26, 19], [-18, 6], [32, 1], [-88, 0], [70, 0], [-54, 12], [52, 4], [-68, -6], [-58, 12], [-22, -20], [-40, 3], [0, 22], [18, 27], [0, 6], [38, -20], [-34, -8], [-16, -21], [48, 11], [-38, -21], [-14, 16], [12, -12], [54, 2], [-64, 12], [-16, 16], [2, 8], [-44, -6], [-36, -14], [64, 8], [-14, -12], [58, 8], [-10, 10], [-34, -12], [-12, -10], [-32, 16], [8, -33], [2, -24], [-94, -5], [-56, 3], [14, 23], [38, 0], [0, 16], [48, -10], [-6, -20]];
    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_bo();".
// 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_bo(:prec:=2)
    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_bo();".
// 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_bo( : 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, 0, 1]>,<13,R![2, 1]>,<17,R![-28, 4, 1]>,<19,R![-8, 0, 1]>],Snew);
    return Vf;
end function;