
// Make newform 6160.2.a.bi in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6160_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_6160_2_a_bi();".
// 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_6160_2_a_bi();".
// 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 := [10, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-6, 1, 1], [5, 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_6160_a();"
function MakeCharacter_6160_a()
    N := 6160;
    order := 1;
    char_gens := [5391, 1541, 3697, 5281, 5601];
    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_6160_a_Hecke(Kf)
    return MakeCharacter_6160_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], [0, -1, 0], [-1, 0, 0], [1, 0, 0], [-1, 0, 0], [0, 2, 0], [-3, -1, -1], [1, 0, 1], [-1, 0, -1], [1, 2, -1], [-6, 0, 0], [1, -2, -1], [-6, -1, 0], [-3, 1, -1], [-1, 1, -1], [1, 2, -1], [-3, 3, 1], [7, -1, 1], [-8, 0, 0], [-2, 0, 2], [-1, -3, 1], [3, 2, 3], [8, -2, 0], [-2, -2, 0], [3, 0, 1], [1, 1, 3], [-7, 5, 1], [5, 1, -1], [9, -2, -1], [-14, 0, 0], [-2, 6, 2], [-3, 0, 1], [0, -2, -2], [3, -6, -1], [-3, 0, 3], [-5, -4, -1], [-8, 0, -2], [8, 0, 0], [-8, 0, 0], [-13, -1, 1], [3, 3, 1], [2, 4, 4], [-2, 2, 2], [-9, -1, -1], [2, -2, 0], [7, 1, -1], [-6, 2, 2], [9, -3, 1], [2, 2, -2], [-5, -1, -1], [4, -2, 2], [-3, 2, -3], [0, 3, -2], [-13, -3, -1], [-9, -6, -3], [0, -4, 0], [5, 1, -3], [0, 6, 0], [22, 0, 0], [-8, 0, 2], [-14, -2, -2], [-4, -2, 0], [-6, 4, 2], [-2, 4, 0], [-1, -6, -3], [3, -4, 1], [-3, 1, -1], [4, 2, 2], [-5, -5, 1], [-13, -3, 1], [-9, -6, -3], [-13, -6, 3], [-7, 5, 1], [8, 8, 2], [-10, -2, 2], [-21, -5, 3], [4, 0, -2], [-2, 6, 0], [5, 5, 1], [-4, 1, 2], [-3, -1, 1], [6, 0, -4], [-5, -4, -1], [7, -4, -3], [2, 0, -2], [-4, -8, 0], [3, -1, -1], [-4, -6, -2], [-11, 7, -1], [7, 8, -1], [8, 3, -4], [-2, -2, -2], [-5, -10, -1], [-2, -6, -2], [17, -3, -1], [-22, 10, 2], [14, 4, 8], [-10, -6, 4], [-12, -2, -8], [7, 0, -3], [-12, 4, 0], [-10, 2, -4], [-2, -2, 6], [0, 10, -2], [-6, 4, -2], [1, 4, -1], [0, -1, 4], [11, 5, 5], [-6, -4, -2], [-20, -1, 2], [11, -9, -3], [20, -4, 2], [-16, 8, 2], [19, -7, -1], [-18, 10, 2], [-2, 8, 0], [14, -5, 2], [-7, -3, 5], [7, -8, -3], [-16, -2, 0], [10, 8, 8], [38, 0, 0], [-17, -1, 5], [-14, -4, -2], [5, -3, -3], [25, -6, -1], [-8, -12, 2], [-3, 7, -3], [1, 3, 1], [-9, -5, -7], [10, -10, -6], [-4, -4, 4], [-34, 4, 2], [-15, 8, -1], [-32, 1, -2], [-8, -3, -2], [-16, -2, -2], [-22, -2, -6], [-10, 14, 0], [36, 0, 2], [-19, -4, -7], [-9, -4, 5], [17, -4, 5], [-12, 0, 0], [7, -1, -1], [-32, -2, 2], [-15, -11, 3], [19, -11, -3], [15, 5, 3], [5, 4, -7], [16, 4, 6], [14, 4, 0], [-12, -2, -4], [-14, -2, -6], [12, 10, 0], [-26, 12, 2], [-1, -4, -5], [16, -12, -6], [3, -13, -7], [-15, -7, -1], [-4, -14, -4], [25, 9, 5], [-8, 8, 0], [5, 15, 5], [-6, -8, 0], [-31, 5, 1], [8, 8, 4], [-1, -9, -3], [16, -16, -2], [20, -12, -2], [4, -6, -4], [-46, -4, 0], [-7, 4, -3], [-26, 8, 0], [14, 8, 0], [2, 8, -4], [-30, -6, -6], [-16, 2, 2], [-21, -1, -7], [-9, -1, 5], [15, -4, -5], [11, 2, -1], [-13, 0, -7], [-29, -1, -3], [23, -16, -5], [27, 7, -1], [36, -4, -2], [6, 5, 6], [-24, -10, -2], [16, 8, 0], [13, 4, 11], [1, 5, 3], [10, 10, -2], [0, 14, 2], [15, 3, -3], [13, 6, 11], [0, 0, 2], [19, 2, 9], [-5, -1, -7], [-13, 3, 1], [-7, -12, -1], [14, -6, 2], [-42, 0, 0], [-10, -3, 4], [-21, -1, -5], [36, -2, 6], [-21, -11, -5], [-15, -7, 3], [-8, 2, 2], [17, 5, 1], [-3, 5, 5], [45, 1, -3], [-11, 0, 1], [-2, 7, -2], [-2, 10, 6], [8, 0, -2], [28, -10, 0], [-30, 1, 0], [-17, -12, -1], [5, -8, 3], [-5, -6, -7], [18, -4, -2], [11, -17, -1], [-17, 12, 3], [20, 8, -4], [17, -14, -1], [-19, 20, 3], [-14, -8, -6], [25, -3, 3], [-4, -10, -4], [19, 2, 9], [21, -10, 5], [-1, -14, 7], [-12, 2, -2], [40, 5, 8], [-3, 1, -9], [2, 18, 0], [6, 12, 0], [-26, -6, -6], [-39, -9, -7], [54, 0, -2], [55, -8, -1], [11, 5, -9], [-60, 2, -2], [3, -5, -3], [-4, -6, -8], [25, -23, -3], [-7, -5, 5], [16, 16, -4], [52, 4, 4], [-32, -6, 6], [0, -15, 6], [-7, 6, 5], [-8, -10, -6], [6, 4, -10], [-46, -2, -6], [0, -8, -2], [-36, 2, 4], [-6, 0, -8], [-3, 1, 3], [14, -14, -6], [40, -12, -4], [-13, 3, 5], [5, -3, 7], [-7, 6, -1], [27, 9, -9], [-4, -4, 2], [-42, -10, 4], [12, 8, 0], [4, -10, -2], [30, -16, -4], [-10, 14, 10], [-41, -4, 5], [5, -4, -7], [-9, -1, -11], [27, -3, 3], [20, -20, -8], [23, 12, 5], [16, -2, -6], [1, 8, 1], [-4, 20, 4], [12, -6, 4], [23, -1, 5], [1, 17, 3], [2, 10, -6], [-28, 4, 8], [40, 4, -6], [4, -6, 2], [-33, 1, -1], [14, 7, 4], [-2, -4, 4], [-14, 11, 6], [-28, 22, 2], [-3, -10, -7], [10, 4, -4], [-21, -15, 1], [-56, 6, 2], [-38, -6, 8], [25, 4, 9], [-44, -10, -4], [7, 19, -3], [28, 0, -6], [-23, 0, -11], [-6, 0, 2], [42, 2, 6], [-46, 8, 4], [12, 22, -4], [5, 1, 1], [-2, 4, -2], [2, -22, -8], [-5, -21, -11], [28, -10, 2], [28, -6, -6], [-48, -2, -8], [44, 2, 0], [-4, 5, -6], [-17, 11, 7], [-39, 6, 1], [-31, 16, 7], [-42, -6, -12], [3, 18, -5], [32, -6, -6], [11, -1, -3], [-23, -10, 1], [-31, 1, -3], [34, -6, -4], [-23, -3, -9], [36, 7, 0], [10, -10, 6], [32, -4, -2], [29, 4, 3], [-3, -6, -5], [14, 12, 6], [32, -6, -8], [24, -10, 0], [-2, 5, -2], [12, -8, 2], [49, 1, 7], [-2, 22, 0], [11, -1, -7], [33, 4, -1], [16, 18, 6], [-21, -7, 5], [24, -24, 0], [-19, 0, 11], [38, -2, 6], [43, -1, 7], [-24, 4, 8], [-26, 16, 6], [-2, -4, 8], [-31, -10, -3], [10, -2, -8], [-52, 8, 6], [-2, 6, 6], [1, -28, -1], [11, 17, 1], [-38, -16, 4], [21, -10, -3], [13, -11, -7], [11, -8, -9], [52, 6, -6], [-22, -5, -16], [-45, -2, 3], [-50, -6, 6], [-42, -9, -6], [-42, -2, -4], [6, 24, 2], [-22, -12, 10], [-76, 0, 2], [10, -14, 6], [-7, -15, 9], [-42, 4, -2], [9, 20, 7], [-22, -20, 2], [-45, 6, 1], [57, 5, -3], [11, -10, 3], [-17, -13, -7], [-10, -21, 4], [-11, 17, 3], [-31, 17, 5], [26, -10, 8], [22, 20, 6], [-46, 0, 0], [53, -8, -3], [-19, 19, 1], [-4, -4, -12], [-12, 10, -10], [3, -25, 1], [61, -3, 5], [33, 5, -7], [-20, -6, 2], [-15, 1, 7], [-28, 9, 4], [-48, 4, 8], [-19, 5, 5], [18, 16, 6], [36, 8, 2], [-25, -9, 9], [-7, 2, 7], [37, 21, -5], [48, -12, -2], [0, -20, -8], [3, 10, 1], [13, 7, 3], [0, 6, 4], [49, -8, -1], [-38, 10, 4], [-4, 11, 4], [29, 19, 1], [-3, 5, -1], [-20, 2, 6], [26, 2, 10], [-32, 18, 0], [-12, 10, 10], [76, 6, 6], [-17, 12, 3], [-43, 9, -7], [-5, 26, 9], [-37, 15, 5], [44, -10, 2], [-56, 2, 4], [35, 16, -5], [-43, 6, 5], [17, -2, 7], [4, 18, 2], [-10, -14, -6], [-20, 20, 6], [7, -27, -1], [8, 16, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6160_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6160_2_a_bi();".
// 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_6160_2_a_bi(:prec:=3)
    chi := MakeCharacter_6160_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_6160_2_a_bi();".
// 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_6160_2_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -10, 0, 1]>,<13,R![-32, -40, 0, 1]>,<17,R![-128, -12, 8, 1]>,<19,R![56, -30, -2, 1]>,<23,R![-56, -30, 2, 1]>],Snew);
    return Vf;
end function;