
// Make newform 4560.2.a.br in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4560_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_4560_2_a_br();".
// 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_4560_2_a_br();".
// 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 := [3, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-4, 0, 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_4560_a();"
function MakeCharacter_4560_a()
    N := 4560;
    order := 1;
    char_gens := [1711, 1141, 3041, 2737, 1921];
    v := [1, 1, 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_4560_a_Hecke(Kf)
    return MakeCharacter_4560_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 := [[0, 0, 0], [-1, 0, 0], [1, 0, 0], [-1, 0, 1], [-2, -1, 1], [1, 0, 1], [1, -1, 2], [1, 0, 0], [-1, 1, 2], [0, -1, 1], [-1, -1, -2], [-1, 0, 3], [4, -1, 1], [3, 2, -1], [-3, -1, -2], [3, 3, -2], [0, -2, -2], [5, 1, 0], [4, 0, 4], [-4, 2, 2], [-4, -2, 0], [6, 2, 0], [-1, -1, 0], [0, -1, 1], [-3, 0, 1], [8, 0, 2], [2, -2, 0], [0, 0, 0], [-2, 2, 2], [-3, -1, -4], [4, 0, 0], [-6, 1, -1], [2, 0, 0], [12, -2, 2], [4, 2, 0], [5, 1, 6], [-6, 2, -6], [7, 0, -3], [7, 1, 0], [-7, -5, 4], [6, 0, 6], [-4, -4, 2], [-12, -1, -1], [-3, 2, 3], [3, -3, 6], [8, 2, 2], [6, -2, 0], [2, -2, 0], [15, 1, 4], [1, -3, 4], [-12, 2, 0], [8, -3, 9], [14, 0, 0], [-14, -1, -3], [-15, 3, 0], [9, 1, 4], [18, 1, -3], [6, -4, 2], [-12, -2, 4], [0, 5, -1], [13, 6, -3], [-13, 1, -4], [20, -2, 2], [-12, 3, -5], [12, 4, 2], [9, 5, -6], [9, 1, 6], [-5, -6, 5], [21, 1, 4], [26, 0, 4], [-18, 0, 4], [-8, -5, -1], [15, 6, -5], [15, 2, -3], [17, -3, -6], [6, 2, 4], [12, 2, 0], [-10, 4, -12], [6, -5, 3], [6, 6, -6], [-6, -1, -3], [10, 0, 0], [-18, 0, -2], [5, -6, 3], [4, 0, 12], [9, -1, -6], [6, -3, 5], [6, 2, -6], [2, -4, 0], [-21, 2, 3], [7, 1, -2], [-2, 1, -9], [-16, -2, 6], [-4, -1, 3], [4, 6, -6], [-35, 1, 0], [-16, -3, -1], [2, -1, -1], [-4, 2, 2], [6, -4, 8], [8, -6, 2], [0, -2, 4], [-1, -7, 4], [-6, 1, -11], [-2, 4, -2], [32, 0, 2], [-5, 1, 2], [-16, 2, -4], [-8, 4, 0], [12, 0, -2], [-6, 0, -6], [0, 4, 10], [-3, -5, 10], [-2, -8, -2], [-20, 0, -4], [-4, 3, 1], [1, 2, 1], [-31, 1, -8], [-19, 3, 2], [10, 2, -4], [-12, 0, -14], [1, 0, -3], [15, -1, -2], [-12, 0, -12], [-22, 0, 6], [-4, -2, 4], [13, -3, 4], [-14, -1, -7], [-21, -2, 3], [4, -2, -4], [28, 0, 0], [-18, -6, 4], [-1, -5, 2], [-2, -2, -14], [-12, 2, 4], [2, 0, -4], [1, 1, -10], [8, -6, -2], [1, 3, 12], [-12, -2, 8], [-24, -4, 8], [-14, -2, -6], [5, 0, -9], [5, -1, -12], [4, 0, 6], [-20, 2, 2], [-34, -2, 2], [9, 1, -10], [-2, 6, 8], [-8, 4, 0], [9, 0, 9], [6, 6, 2], [-37, 4, -3], [0, 8, 8], [14, 8, -6], [10, 6, 8], [-42, 2, -8], [-30, 2, -6], [-2, 2, 2], [-38, 1, 5], [-11, 1, -12], [-23, 3, -8], [-19, -4, 7], [-30, 2, -4], [-1, 3, -2], [-3, -5, 8], [32, 4, -4], [10, 10, 2], [-4, -2, 12], [-32, 2, 0], [24, 2, -2], [-26, 4, -4], [2, -7, 11], [6, -8, -4], [-2, -6, 0], [-6, 10, -14], [14, 4, 2], [-6, -2, -14], [-26, 0, -2], [10, -8, -4], [-27, 2, 1], [10, -2, -8], [-9, 6, -15], [11, 5, -4], [1, 7, -6], [30, 4, -4], [-23, 2, -5], [20, 2, 2], [0, -4, 2], [8, -3, 17], [17, 2, 7], [11, -7, -2], [-6, 6, -12], [-8, -5, 9], [-17, -3, -2], [-35, 3, 8], [-19, -7, 0], [-2, 4, 4], [39, -3, -6], [7, 5, 2], [-24, -1, -3], [34, 6, -8], [-9, -6, 5], [14, 0, -8], [-34, -7, 3], [15, -3, -2], [12, -8, 0], [1, -1, 16], [4, -8, -6], [8, -12, 0], [26, -6, 6], [-4, 8, 6], [-25, -10, 3], [51, 1, -4], [-4, 2, 6], [-32, -4, -2], [-1, 10, -17], [22, -1, 3], [-21, 5, 4], [-11, -1, -6], [8, -2, 4], [-9, -13, 10], [48, -1, 1], [-9, -12, -7], [-9, -7, -4], [14, 0, 12], [-1, 5, 8], [-18, -12, 2], [18, -2, -12], [-10, 1, -13], [-8, -8, -2], [27, 15, -10], [52, 8, -4], [-16, 7, -23], [1, -4, -17], [21, -5, -6], [49, 1, 8], [30, -4, 4], [2, 5, 3], [-24, 6, -2], [14, -2, 4], [-4, 12, 0], [-11, 10, -11], [8, 8, -18], [-31, -7, -6], [-18, 7, 13], [-33, 0, 5], [0, -8, -4], [67, -1, 6], [-23, 1, -8], [-10, -2, -6], [2, -4, -12], [19, -1, -8], [4, 4, -2], [1, -1, 14], [-24, 1, 9], [-9, -1, 8], [26, 6, -12], [33, -7, 2], [-6, 12, -4], [10, -2, 0], [4, 4, 12], [46, 0, -4], [-7, 4, -3], [15, -11, 18], [-15, -3, -14], [4, 11, -11], [18, -6, 10], [-44, 0, -8], [-3, -13, -2], [-10, -4, -20], [20, 0, 12], [-22, 2, -10], [8, 10, 6], [11, 14, -7], [-3, -16, 3], [3, -1, 4], [26, 6, -10], [10, 6, 10], [-14, 9, -9], [-28, -4, 12], [-10, 2, -16], [-5, 1, 10], [2, -12, 8], [-49, 2, 7], [-36, -1, -9], [30, 0, -4], [-41, 7, 10], [16, 16, 0], [50, 1, 13], [-6, 6, -14], [43, -1, 20], [-5, 3, -2], [-28, 6, 14], [35, 2, -7], [8, -4, -18], [-12, 0, 4], [7, 9, 6], [-18, -2, 0], [5, -4, 19], [-44, 2, -20], [15, 3, 18], [-18, -14, 4], [-15, -1, -8], [-12, -2, 2], [11, 2, -19], [28, -4, 12], [-32, 6, -4], [-12, 11, -5], [-22, -14, 2], [31, -11, -6], [-26, -8, 8], [10, -5, 15], [-20, -14, 6], [-15, -11, 4], [-22, -8, 4], [-10, -15, 11], [-8, 12, 4], [-2, -6, -22], [-40, 6, -12], [29, -7, 18], [12, -8, -2], [-4, -13, -3], [10, 4, 10], [-11, -13, 2], [42, -4, -8], [-2, 2, 4], [-4, 6, -2], [45, -1, 8], [39, 1, -6], [18, 16, -12], [-29, 3, -10], [-18, 8, -14], [9, -7, -16], [-20, 2, -6], [7, -9, -8], [-36, 10, 0], [7, -5, -2], [-15, 5, 0], [-7, -6, -15], [45, 12, -7], [30, -4, 0], [18, -9, 7], [-13, -1, -22], [15, -13, -14], [24, 16, 0], [-13, 3, 12], [-26, 6, 4], [12, 8, 8], [12, 10, -4], [11, -5, -14], [-29, 8, -5], [12, 16, 6], [26, -4, -10], [-8, 8, -2], [43, -5, 2], [-6, 11, 13], [-12, -7, -19], [27, 9, -2], [-28, 8, -16], [-30, 12, -8], [-46, 0, -16], [0, 0, 4], [-6, -4, 10], [23, 0, 21], [-29, 4, -1], [-16, -14, 8], [16, 10, -14], [20, -4, -6], [-16, -1, -17], [46, 6, 0], [33, 5, 4], [54, -1, 19], [18, -2, 12], [8, -4, 10], [10, 6, -20], [60, 1, -7], [26, 4, 20], [-30, 14, 6], [-5, -2, 1], [22, -5, 15], [27, 9, -6], [-29, -10, -5], [-12, 2, 8], [65, 1, 14], [51, 1, 8], [8, -6, -6], [-20, 6, -4], [-53, 0, -11], [-25, 5, 4], [-8, -16, 22], [-5, 7, 2], [-2, 3, 21], [11, -4, 17], [30, -8, -10], [1, -14, 15], [71, -5, -6], [-8, 17, -13], [-65, 3, -10], [-16, 2, -28], [-44, -6, 0], [7, 13, -14], [-48, -2, -6], [33, 3, -16], [2, -3, 5], [-22, 10, 16], [-10, -2, -6], [30, 9, 9], [20, -6, -18], [-10, -13, -3], [7, 14, -3], [27, 9, 10], [-8, 16, 4], [-6, 8, 14], [40, 6, -12], [40, 14, -16], [-50, 10, -12], [-14, -18, 12], [-6, 16, -8], [-14, -10, 20], [18, -13, 7], [-3, 0, 13], [47, 5, -2], [26, 6, 20], [-39, 6, -5], [-27, 11, -32], [-24, 8, -24], [30, -16, -20], [-18, 4, 2], [8, -3, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4560_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4560_2_a_br();".
// 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_4560_2_a_br(:prec:=3)
    chi := MakeCharacter_4560_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_4560_2_a_br();".
// 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_4560_2_a_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4560_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, -4, 4, 1]>,<11,R![-76, -12, 6, 1]>,<13,R![12, -8, -2, 1]>,<17,R![72, -44, -2, 1]>],Snew);
    return Vf;
end function;