
// Make newform 8470.2.a.by 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_by();".
// 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_by();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 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], [-1, 0], [-1, 1], [0, 1], [-6, 1], [0, -2], [-3, -1], [0, 4], [5, 1], [-1, -1], [1, 3], [1, -3], [-2, 3], [8, -2], [-5, -1], [-6, 2], [5, 1], [-8, -1], [-2, 1], [-15, -1], [8, -2], [1, 0], [3, 5], [-2, 2], [-3, 3], [-1, 0], [-4, -1], [-8, 5], [9, 2], [-2, -1], [-4, 0], [-10, -1], [13, -2], [-6, 0], [5, -3], [-2, -8], [-9, -1], [7, 0], [14, 5], [1, -4], [12, -2], [0, -2], [-8, 2], [3, -5], [-6, 6], [-2, 4], [3, -6], [-4, -3], [0, 10], [-12, -8], [-7, 1], [8, -7], [-1, -12], [6, -6], [1, -11], [1, -12], [-24, -1], [5, 0], [2, -6], [-7, -9], [24, -2], [-15, 7], [-15, -3], [-15, 8], [6, 0], [13, -6], [-7, 1], [-16, 0], [-3, 5], [10, -2], [-2, 6], [-15, -1], [-25, 3], [6, 0], [-2, -12], [-16, 2], [-26, -1], [-12, 4], [18, 7], [3, -7], [5, -9], [-2, -10], [-3, 12], [23, 6], [6, 8], [4, 3], [-14, 5], [3, 11], [0, 1], [-29, -1], [0, 12], [-1, -11], [11, 8], [-10, 4], [20, 7], [4, 14], [9, -7], [11, 9], [12, 11], [-3, -2], [-14, -10], [3, -17], [6, -13], [-28, -4], [-2, 13], [-11, -7], [-30, 4], [-21, 9], [-26, -8], [-14, -15], [6, -10], [3, -12], [24, -8], [-7, -1], [33, -3], [23, -1], [-9, 6], [19, -4], [19, 12], [16, -10], [-40, 0], [-7, 11], [28, -4], [-20, 12], [-8, -14], [35, 6], [11, -13], [-28, 0], [0, -5], [-27, -7], [-3, 15], [12, 0], [-19, -4], [-4, 16], [11, 8], [24, -10], [2, -2], [-22, 4], [-6, 10], [-1, -5], [-3, -4], [11, -11], [1, -2], [16, -12], [-8, 4], [-12, -20], [5, 21], [4, -4], [-18, -2], [-3, -1], [18, 10], [-28, -7], [-4, -4], [-16, -12], [-16, -2], [38, 8], [-19, 13], [-7, -8], [-6, -22], [4, -9], [-37, -4], [-6, 22], [26, -3], [33, 12], [20, 14], [43, -8], [7, -11], [21, -10], [-2, 18], [15, 14], [6, 2], [22, 2], [7, -17], [-57, -3], [-9, -5], [24, -14], [46, 1], [8, -12], [29, 1], [13, 17], [40, -8], [-19, 18], [-2, 10], [6, 5], [-36, 2], [29, -17], [4, -2], [-2, 6], [-10, 8], [-3, 2], [-7, 13], [-11, -9], [-47, 4], [-14, 6], [2, 0], [26, -6], [16, 6], [-44, -2], [-23, 12], [21, 19], [-36, 0], [5, 0], [36, 2], [-9, 1], [37, -2], [24, 4], [-18, 8], [-5, -6], [46, -3], [-22, -5], [35, 9], [16, 14], [6, -19], [51, -3], [28, -9], [-16, 18], [21, 9], [-5, -11], [5, 4], [12, 15], [2, 15], [7, 23], [-14, 10], [-56, 8], [38, 6], [-34, -10], [-21, -9], [30, -19], [0, 13], [-57, 4], [10, -7], [-7, -15], [0, 26], [-33, 5], [14, -12], [-9, 5], [5, 13], [-46, -6], [14, -18], [73, 1], [-7, -19], [26, -4], [-49, 11], [-42, -3], [37, 11], [-44, 11], [54, 0], [14, 8], [2, 9], [-42, 8], [2, 8], [-9, -15], [-18, -14], [-24, 0], [8, 2], [-24, 14], [-8, 12], [-75, -1], [66, 0], [-33, -2], [0, -8], [-2, -14], [-23, 15], [25, 17], [10, 24], [-3, -15], [-13, -24], [10, -32], [-40, -16], [-34, 11], [11, 17], [58, -8], [-30, 24], [-15, -27], [15, -12], [-18, 26], [-22, 8], [30, 11], [-32, -8], [-30, -16], [24, 21], [-6, -3], [21, 8], [-38, 14], [-34, -12], [-5, -21], [-2, 2], [-50, 3], [49, -8], [-20, 21], [20, 4], [1, 10], [-23, 21], [-1, -4], [36, 0], [38, 10], [-32, 0], [-57, 3], [-36, 20], [19, -1], [17, -3], [15, 20], [63, 1], [-5, 2], [10, 23], [39, -21], [-52, -14], [27, 11], [20, 18], [-7, -1], [-18, -31], [-21, -19], [0, -27], [-2, -19], [23, -9], [-54, 8], [-6, 31], [-6, 4], [-57, 14], [54, 15], [5, -8], [6, -24], [-17, 11], [36, 9], [-32, -17], [-21, 7], [-42, -18], [-32, 24], [5, -7], [52, 8], [12, 30], [-58, 3], [-37, -13], [36, 12], [-17, 15], [12, 18], [31, -3], [-9, -36], [35, 0], [-2, -15], [-68, 6], [15, 13], [-31, 10], [-56, -16], [10, -20], [67, -10], [72, -4], [41, 20], [-56, 0], [22, 2], [9, -1], [-61, 9], [28, -22], [18, -19], [-25, 10], [-24, -24], [27, -25], [-3, -21], [-76, -1], [-34, -12], [46, 19], [24, -10], [51, 16], [-26, 11], [-48, -22], [2, -8], [4, -21], [-4, 36], [25, 33], [-81, -1], [-41, -19], [-18, 3], [-4, -7], [-1, -13], [-54, -18], [-78, 8], [1, 2], [-46, -20], [36, 4], [-31, 18], [-20, -33], [-68, -4], [-14, 26], [30, -20], [-44, -8], [22, -18], [-28, -6], [-35, 4], [35, -11], [22, -9], [18, -23], [-14, 0], [-76, -4], [-13, 5], [-5, 11], [46, -8], [67, -2], [-53, 14], [-36, -27], [51, -15], [-70, -4], [48, -6], [22, -4], [-57, 6], [-82, -10], [38, 21], [56, -18], [45, 15], [-32, -28], [38, -6], [1, -36], [5, 2], [40, -15], [-39, -7], [-57, -15], [15, -3], [-19, -7], [40, 10], [-16, -15], [24, -7], [-8, -6], [37, -28], [54, -12], [-24, 26], [20, 0], [10, -38], [17, 13], [-43, 3], [-45, 11], [16, -14], [-41, 18], [19, 1], [19, -2], [-40, 24], [27, -17], [-1, 13], [-64, 12], [-49, -18], [-26, -8], [-40, -3], [-44, 8], [-17, 12], [-12, 28], [33, 5], [21, 5]];
    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_by();".
// 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_by(: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_by();".
// 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_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-5, 0, 1]>,<13,R![1, 1]>,<17,R![-4, 2, 1]>,<19,R![-5, 0, 1]>],Snew);
    return Vf;
end function;