
// Make newform 4560.2.a.bv in Magma, downloaded from the LMFDB on 24 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_bv();".
// 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_bv();".
// 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_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, -1, 0], [1, 0, 1], [1, -1, 2], [-2, -1, -1], [1, 0, 0], [4, 1, 1], [1, 0, -1], [0, 1, -3], [-3, 1, 0], [1, -2, 1], [5, 1, 2], [4, -1, -1], [2, 1, 1], [6, 0, 2], [-4, 3, -3], [0, 0, -4], [2, 2, -4], [-2, 2, -2], [0, 0, 0], [2, -3, -1], [-1, 2, -5], [-3, 1, -4], [0, 2, -4], [0, 4, -4], [8, -2, 2], [4, 0, 6], [-8, -1, 1], [12, -2, 2], [1, 4, -3], [-10, -2, 2], [-10, 0, 2], [2, 4, 0], [0, -3, 1], [-8, 2, -4], [-7, 1, -2], [10, -3, 3], [-4, -1, 1], [2, 2, 0], [0, 2, 4], [11, 0, 7], [-1, -3, 2], [-6, 1, 1], [-10, 0, -2], [0, 4, 0], [8, 0, 8], [18, 1, 3], [4, -3, -1], [-6, 6, -2], [-5, 0, -9], [-2, 4, 0], [-5, -4, -1], [-12, -1, -3], [10, -1, 5], [23, -2, 3], [14, -6, 0], [14, 0, 8], [-3, 2, 1], [-19, -1, 0], [8, -3, -5], [-6, 0, -2], [-7, -4, 1], [4, 2, -4], [-10, -5, -1], [-8, 1, -7], [-1, -5, 8], [10, 3, -3], [14, 2, -6], [2, 0, 0], [3, 0, 7], [-3, -1, 4], [1, -7, -4], [-4, -3, -7], [-4, 2, -10], [22, -6, 2], [-14, 2, -2], [11, 4, 9], [8, -8, -2], [5, 0, -3], [-14, 0, -12], [-10, 6, -4], [7, 5, 2], [-8, 4, 4], [4, 7, 3], [9, 4, -1], [20, 0, 2], [2, -6, 6], [-3, -5, 0], [-4, 1, 5], [9, 2, -9], [2, 6, 0], [-9, 10, 1], [-2, 4, 6], [-6, 3, 9], [19, 0, -7], [17, 0, 11], [-14, -4, -6], [-18, 4, 4], [-22, 4, 2], [6, 0, 8], [-6, 1, 11], [-7, 0, 3], [2, -6, 16], [-8, -2, 4], [-20, 1, -3], [-26, 0, 4], [16, 6, 6], [0, -10, 12], [14, 4, -6], [-16, 4, 6], [6, 3, 3], [-26, -2, 8], [8, 8, 0], [-17, 2, -13], [-21, 3, 2], [10, 3, -7], [10, 3, -13], [12, -6, -6], [0, 2, 4], [-5, 1, 2], [-22, 5, -3], [4, 2, 2], [6, 6, -8], [-14, 0, -12], [8, -1, 13], [15, -6, -3], [7, -5, -6], [-6, 8, -12], [0, -8, 4], [28, -10, 2], [0, -1, 3], [-32, 2, -4], [6, 0, -4], [-2, 4, 0], [2, -9, -1], [-6, 12, -6], [-24, 3, 1], [22, -2, 2], [-16, 4, -8], [-24, 0, -6], [-19, 1, -10], [-18, -5, -7], [20, 2, 0], [6, -10, 4], [-4, -10, 12], [-6, 11, -1], [-4, 10, 2], [8, -2, 14], [-21, 7, -4], [0, 0, -10], [-23, -7, 6], [12, 0, 4], [-18, -6, 12], [12, -6, 14], [36, 2, -2], [-36, -2, -8], [24, -4, 2], [23, 2, 7], [-18, -1, 5], [12, 3, 1], [7, -3, 4], [8, -2, -14], [-10, 9, -7], [-10, 3, -19], [20, 4, -8], [-8, 2, -12], [10, -10, 6], [18, -4, 8], [2, -10, -4], [-22, -8, 12], [27, -2, 5], [14, -8, 12], [12, -4, 8], [-20, -6, -4], [-6, -10, 4], [0, 8, 10], [-14, 12, -2], [-18, -2, 6], [-1, 13, 4], [-12, -2, 6], [-33, -3, 6], [-8, 1, 3], [-4, -9, -1], [22, 2, 10], [9, -5, -10], [-2, -12, -10], [-8, 14, -4], [-35, -4, -3], [-7, -7, 16], [-4, 9, 5], [16, -4, -8], [15, 0, 5], [-32, 5, -3], [12, 3, 1], [-40, -3, -9], [-34, -6, -14], [10, -7, -3], [-24, 1, -11], [11, -4, 13], [4, 0, 4], [-13, 13, -6], [-38, 0, -8], [5, 4, 9], [-22, 1, 9], [24, -8, 8], [-6, 7, -7], [-12, 14, -8], [40, -8, 4], [4, 0, 10], [-4, 0, -18], [-11, 7, 12], [-22, -3, 15], [18, -4, 14], [12, 10, -20], [1, 3, -4], [-35, 12, -5], [6, 13, -17], [18, 7, -17], [-22, -6, 14], [-32, -5, 7], [31, 2, 3], [-13, 11, -2], [34, -7, 11], [-10, 2, -6], [-16, -11, 15], [6, 2, 8], [4, -8, -12], [-19, 8, -11], [-28, 8, 6], [-28, -1, -13], [44, -4, 8], [39, -6, 11], [-13, 15, -2], [-4, 3, -5], [-4, 3, -7], [10, 10, 6], [-11, 4, -7], [-34, -4, -6], [-12, 10, 2], [20, 4, -4], [-31, 9, -14], [0, -10, 0], [-42, 3, -5], [29, -6, 11], [-17, 11, 10], [-4, 2, 18], [28, 7, 11], [42, -1, 13], [44, -2, -4], [-18, -8, 20], [-2, -15, 3], [52, 2, -4], [38, -5, 7], [11, -6, -3], [-8, -3, -13], [-8, -10, -6], [14, -5, -1], [22, -4, -8], [-4, 8, -12], [-16, 16, -4], [-18, -10, 6], [-57, -1, 0], [-6, 5, -23], [16, -5, -5], [19, 12, 5], [32, -2, -12], [32, 4, -8], [-42, -1, -13], [-10, -4, 12], [40, -12, 8], [32, -8, 6], [26, 12, -26], [-5, -1, -24], [3, -15, 20], [-30, -7, 3], [24, -8, 2], [16, 4, 2], [39, 0, -9], [4, 2, 14], [28, -10, 10], [-16, -11, 17], [30, -8, 8], [-1, -11, 8], [-59, -4, -11], [2, -4, -4], [6, 1, 21], [-8, -8, 16], [17, 4, 23], [40, -12, 6], [-14, 9, -5], [18, -7, -3], [14, 6, -8], [-35, 1, 8], [52, 0, -2], [-16, 12, 12], [-6, -7, 9], [-4, 0, -16], [-17, -5, 14], [30, 6, -14], [-30, -7, -15], [20, -10, -6], [-46, 11, -11], [-22, 12, -14], [31, 7, 4], [-48, 4, -8], [34, 10, -18], [41, 8, -3], [24, -10, 8], [36, 5, 13], [-22, 12, 0], [39, 4, 5], [14, -12, -10], [18, -9, 5], [18, -20, 4], [-27, -4, -7], [4, 0, 4], [36, -4, 6], [50, 4, 20], [16, -1, -17], [20, -10, 16], [43, 0, 1], [14, 0, 6], [-66, 3, -5], [-30, 8, 8], [8, 4, -8], [-10, -4, 14], [-26, 11, -3], [6, 5, 1], [-26, -10, -10], [26, 9, 9], [-26, -14, 16], [34, 7, 17], [-30, 8, 6], [6, -7, -9], [30, 2, 2], [-2, -11, 29], [56, -3, -1], [45, -7, 6], [3, 3, -8], [-2, 16, 4], [-17, 2, -25], [4, 5, 13], [42, -3, 13], [32, 4, 0], [-8, -13, 17], [-24, -14, 6], [-56, -2, -2], [6, -2, -26], [76, -3, 1], [11, -1, -16], [56, -12, 6], [26, 8, 10], [-56, 18, -4], [2, 13, 17], [29, -6, -13], [9, 6, 23], [-34, -3, 5], [-8, 2, 10], [22, 2, -6], [-26, -10, 10], [-20, 18, -10], [-6, -16, 22], [-23, 1, -18], [37, -1, -14], [-6, 16, -4], [46, 2, 24], [28, 14, -32], [21, 6, -21], [28, 8, 24], [22, -17, -7], [27, -14, 3], [28, 6, -6], [36, -4, 10], [36, 4, 4], [21, 12, 5], [-26, 4, -24], [-24, -6, 16], [-5, 19, -12], [-9, -14, -9], [42, 5, 17], [-17, 7, 14], [-6, -14, 34], [44, -9, 3], [26, 5, -5], [6, 8, -18], [-50, -4, -12], [-13, 1, 24], [-50, -7, -5], [20, -14, -12], [6, 13, 17], [-27, 0, -11], [-1, 9, 4], [-22, 2, 4], [25, -15, 0], [36, 1, 1], [-31, -2, 1], [0, 1, -15], [-42, -2, 6], [38, -6, 18], [18, -19, 9], [30, -2, -4], [-24, 15, -23], [9, 2, 9], [-8, 14, -18], [32, 6, 8], [-35, 8, -25], [-18, -4, 14], [45, 0, -11], [29, -17, 18], [-42, 1, -3], [16, 16, -12], [-14, -14, 16], [-22, -2, -26], [22, -16, 8], [-76, 0, -12], [-12, -12, 0], [-10, 4, 8], [20, 2, -10], [-39, -14, 17], [-17, 19, -12], [12, 9, 25], [8, -4, -24], [15, 5, 18], [44, -1, 1], [0, 10, -34], [-54, -4, -4], [46, -14, 4], [13, 8, -31]];
    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_bv();".
// 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_bv(: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_bv();".
// 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_bv( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4560_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![32, -14, -2, 1]>,<11,R![8, -6, -2, 1]>,<13,R![-76, -34, 0, 1]>,<17,R![-64, -16, 6, 1]>],Snew);
    return Vf;
end function;