
// Make newform 4368.2.a.bs in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4368_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_4368_2_a_bs();".
// 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_4368_2_a_bs();".
// 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 := [12, 2, -10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -8, -1, 1], [-5, 0, 1, 0], [5, 2, -1, 0]];
    Rf_basisdens := [1, 2, 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_4368_a();"
function MakeCharacter_4368_a()
    N := 4368;
    order := 1;
    char_gens := [3823, 1093, 1457, 1249, 2017];
    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_4368_a_Hecke(Kf)
    return MakeCharacter_4368_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, 0, 0], [1, 1, 0, 0], [1, 0, 0, 0], [1, 0, -1, 0], [1, 0, 0, 0], [1, 0, 0, -1], [1, -1, -1, -1], [1, -1, 1, 1], [1, 1, 1, 1], [-2, 1, 1, 0], [1, 0, 0, -1], [2, 0, -1, -1], [1, -1, 1, 1], [-4, -1, 0, -1], [4, 3, 1, 2], [-2, 0, 1, -1], [1, -2, 2, -1], [-2, -2, 2, 2], [2, 0, -1, 1], [7, -1, 1, 1], [0, 1, 1, 2], [-2, 3, 0, 1], [2, 1, 0, 1], [4, -3, -1, -2], [4, -4, 2, 0], [-1, -2, 0, 1], [6, 0, 0, -2], [11, -2, 0, -1], [-4, -3, 1, 0], [-2, 0, -2, -4], [-5, -2, 0, 1], [1, 0, -3, -4], [0, -6, 0, -2], [-2, 0, -3, 1], [-17, -2, 0, 1], [9, 0, 0, -1], [6, 0, -2, 0], [-1, 3, 0, 2], [-2, 2, 0, -2], [-6, -3, 1, 0], [2, -4, 0, 0], [3, 2, 0, -3], [12, 0, 0, 2], [-10, -4, -1, -1], [1, -2, -2, 1], [3, -1, -1, 1], [0, -1, -1, -2], [4, 0, -3, 1], [2, 4, 0, 0], [-14, -1, -1, -2], [-2, 4, -1, 1], [12, 5, -1, -2], [-5, -4, 2, 1], [-10, 4, -2, 2], [2, 3, 1, 2], [-2, -6, 0, -2], [-6, 4, 4, 2], [4, 1, 3, 2], [-4, 4, 1, 3], [8, 0, 0, 4], [-6, -5, 4, 3], [-2, 3, -3, 2], [6, -6, -4, -4], [12, 0, -4, -2], [-6, -2, 5, 3], [10, -2, 2, -2], [-9, -1, -3, -3], [16, 0, -2, -2], [-4, 1, 1, 0], [-4, 0, -1, 1], [-2, 4, -1, 1], [8, 6, 0, -2], [-8, 4, -4, 2], [20, 6, 0, -2], [-7, -2, 3, 2], [-2, 0, 0, 0], [0, 1, 1, 4], [-20, -2, 1, -3], [-7, 3, 5, 3], [5, 4, 2, 3], [2, -6, 0, -2], [16, 6, 3, 1], [4, 2, -6, -2], [-1, 0, 2, -3], [8, 5, 3, 0], [-7, 0, -1, -2], [0, 10, 4, 4], [13, 2, -5, 0], [-9, -4, 4, 3], [3, 0, -6, -3], [-7, -1, 6, 6], [0, -2, 4, 2], [2, 0, 0, 2], [12, 0, 0, 0], [-6, 2, -4, 0], [-13, 5, -4, 2], [15, -4, -6, -5], [20, 6, 0, -2], [7, 4, 2, 3], [10, -1, -7, -2], [2, 6, -1, 5], [-1, -10, -2, -5], [4, -5, -5, 0], [-10, 1, 1, -4], [30, 4, 2, 2], [14, 3, -8, -7], [-26, -5, 2, 1], [3, 3, -3, 3], [24, 4, 0, 2], [-1, 4, -2, 5], [-5, -4, -4, -7], [-5, 6, -5, 2], [3, 6, -2, -1], [-25, -4, 2, 1], [-12, -7, -1, 2], [-3, -2, 2, -3], [-24, -6, -2, 0], [-11, 2, 8, 5], [-8, -1, 7, 6], [-12, -3, 1, -4], [-13, -1, 1, 5], [2, -4, 0, -4], [11, 6, 1, 2], [-14, -1, 9, 6], [4, -3, 7, 2], [6, -8, -6, -6], [-10, -6, 4, 4], [3, -8, 4, 3], [14, 5, -1, -4], [-6, 6, -2, 2], [20, 4, -1, -1], [-2, 1, 3, 2], [4, 1, -1, 6], [-24, -5, 4, 5], [17, -3, 5, -3], [-7, -2, 1, -2], [-11, -9, -3, -7], [-20, -8, 0, 2], [-4, 9, 1, 4], [-5, 6, -4, -3], [-10, 0, 1, -3], [-4, 8, 0, 4], [-3, 2, 5, 4], [9, 2, -2, -1], [-26, 4, 1, -1], [-22, -1, 7, 2], [2, -8, 2, -2], [-8, 2, -6, 0], [-24, -8, 1, 1], [-2, 2, -2, 0], [17, 2, -4, 5], [35, 6, 0, 1], [-4, -4, 0, -8], [2, 9, -3, 4], [-5, 5, 1, 1], [10, 4, 6, 4], [-28, -1, 4, -3], [22, -2, 2, 4], [26, -3, 2, 3], [-15, 0, 5, 6], [-50, -3, 1, 2], [-39, 0, 6, 3], [-28, 6, 6, 4], [-21, -2, 1, 0], [1, 11, 2, 6], [-30, 0, 4, -2], [-18, -10, -2, -4], [-6, 4, -2, 2], [-20, -6, 0, 4], [17, 8, -3, -2], [-6, -12, -6, -6], [-2, 3, -7, -2], [-2, 2, 0, -2], [29, 2, 4, -1], [-18, -6, -2, -8], [-14, -12, 0, 2], [-6, 0, 9, 1], [24, -2, -4, 2], [-1, -2, -2, 1], [-23, 6, -6, -3], [-20, 6, 6, 4], [-30, 0, 0, 4], [-14, -9, 2, 1], [18, -2, -3, -3], [16, 4, 0, 10], [2, -1, 1, -4], [17, 7, 3, 3], [6, 4, -2, 6], [-13, -1, 0, 2], [-8, -2, -6, -2], [9, 0, -1, -8], [-10, 9, -5, -2], [-7, 9, -6, 2], [-16, 3, 3, -2], [9, 8, 2, 1], [28, 12, 0, -2], [-4, -3, 3, -10], [14, -7, -4, -7], [-20, -2, 6, 0], [34, -2, -7, -1], [-17, -10, 0, -7], [4, 5, 3, 6], [2, 2, 6, 0], [-12, 7, -6, -3], [22, 2, -2, -4], [-10, -11, 1, 4], [38, 5, -1, 2], [-13, 0, 3, -4], [-12, 2, -6, -4], [13, 9, -5, 3], [-12, -18, -6, -6], [-18, -4, -2, 8], [-8, 0, -5, -5], [-37, -3, -4, -4], [17, 1, 1, 5], [16, 9, 1, 2], [1, 4, 8, 11], [14, 4, 7, 1], [-2, 10, 3, 1], [18, -7, -1, -4], [-8, -8, 4, 4], [13, 4, -5, 2], [-4, -6, 4, 2], [-3, 0, 6, 3], [-28, 3, -3, -2], [-43, -4, -4, -5], [14, -1, -3, -2], [7, -12, 4, 3], [-25, 4, -3, 0], [22, 12, 4, 8], [30, 0, 2, -4], [1, 6, -2, 1], [8, 0, -8, -2], [-14, -6, 8, 8], [-1, -3, -2, -6], [-5, 10, -2, -3], [18, -2, -9, -3], [8, -1, -1, -2], [-16, 10, -6, -4], [34, -4, 3, -3], [0, -2, 10, -4], [31, 6, -10, -9], [42, 0, 4, 0], [2, 5, 8, 5], [8, 8, 6, -2], [8, 7, -3, -4], [-12, -15, -4, -7], [26, 4, -6, -4], [23, 1, -5, 3], [-19, -8, -6, -3], [-10, -4, 3, 3], [6, -18, -1, -5], [-38, 0, 0, 0], [20, -4, 2, -8], [-9, -14, -3, -10], [14, 7, -3, -4], [0, -8, 2, -2], [-10, -2, 6, 8], [15, -1, -5, 3], [18, -3, -7, 0], [14, -1, -1, 4], [-8, 17, -1, 2], [-3, -2, -2, -1], [30, 4, 1, -7], [-14, 2, -2, -10], [4, 0, -5, -3], [10, 5, -10, 3], [-12, 10, -2, 0], [-26, 4, -6, -2], [-26, 0, -4, -4], [14, 8, -4, -2], [-47, -6, -2, 3], [26, 3, 5, 2], [-8, 2, -10, -10], [-32, -8, -4, 4], [-13, 2, 7, 12], [17, 12, -4, -1], [-4, 2, 2, -2], [-40, 0, -2, 2], [28, 2, 6, 4], [-31, 4, 4, 5], [-4, 8, -4, -8], [-22, 5, -7, -6], [-33, 6, 6, 3], [10, -19, 1, -4], [23, -11, 7, -3], [12, -18, 7, -1], [8, 0, 6, 6], [-25, -4, -6, 3], [31, -3, 3, 5], [-19, -6, 6, 5], [-7, -2, -3, 10], [24, -11, 4, 1], [24, 5, 3, 2], [-4, 2, -2, -6], [-8, 10, -8, -6], [-26, 6, -6, -4], [9, -10, 10, 1], [-9, 7, 9, -1], [-8, 8, 4, 2], [26, 10, -3, -1], [44, 5, 3, 4], [42, 5, -3, -4], [15, -7, -9, -1], [4, 0, 6, 8], [24, 4, -10, 2], [-17, -4, 0, 1], [0, 10, 3, -3], [8, 3, 5, 2], [24, 8, 2, -6], [-22, 4, 11, -1], [0, 8, -2, -4], [-7, -3, -3, -5], [-8, 12, 3, 7], [8, 2, -10, -2], [-16, 9, -8, 1], [26, -2, 3, 3], [-7, 9, 11, 7], [40, 13, 3, -2], [-6, 8, 10, 8], [-24, -14, -2, 6], [4, 4, -8, -2], [-25, -2, 4, 5], [-7, 20, -9, 2], [22, 10, 6, 4], [4, 16, 2, 10], [5, 13, 3, 5], [11, 6, 4, -3], [-35, -2, 4, -3], [25, 10, 8, 9], [-9, 9, -3, 3], [28, 10, 2, 16], [2, -5, -4, 1], [4, 2, 4, 6], [26, 5, 2, 1], [9, -20, 4, -1], [-26, -8, -1, 3], [10, 12, -4, -8], [19, -12, -10, -7], [10, -2, 2, 4], [40, 8, -2, -4], [-37, -2, 5, 4], [-2, -12, -8, -6], [52, -2, -7, 1], [42, -4, 2, 0], [-5, 14, -6, 9], [-14, 4, -2, 0], [16, 17, 6, 3], [-46, 4, 0, 4], [-6, 13, -9, 2], [44, -4, 6, 4], [14, -14, 3, 1], [16, 2, -6, -4], [14, 8, 0, 8], [5, -14, -12, -7], [21, 1, -6, -8], [-6, 3, -6, -9], [50, 8, 2, 2], [-6, -22, 7, 3], [-29, 10, -2, 1], [-7, -2, -10, -5], [33, 11, -7, -11], [14, -12, -11, -5], [30, -14, -12, -8], [4, -5, -3, 2], [-17, 1, 4, 2], [21, 8, -10, -9], [-31, 13, 3, 3], [-42, 0, 0, -2], [-12, 10, 12, 6], [14, -16, 11, 1], [1, -7, 7, -9], [-19, -6, -6, -3], [10, -5, 13, 12], [30, 4, 7, 1], [-35, 6, 0, -5], [3, 10, -12, -5], [17, 12, -4, 7], [-2, -16, 0, -4], [-26, -17, -8, -1], [-1, -10, 7, -6], [1, -6, -6, -7], [-8, 6, 9, 1], [22, 11, 7, -4], [11, -11, -8, -8], [1, 14, 10, 5], [-3, -4, -4, -13], [0, 0, -10, 2], [26, -4, 2, 0], [2, -8, 6, 6], [46, 1, 2, 3], [22, 8, 7, 17], [4, 8, -4, -4], [10, 4, -5, 9], [12, -15, 3, -2], [6, -4, -8, 2], [49, 2, 6, -1], [-3, 1, 5, -5], [-36, 2, 9, 5], [-6, -1, 7, 10], [-8, -1, 1, -10], [-57, 0, -4, -1], [12, 5, -12, -5], [70, 7, 2, 9], [-17, -10, -10, -9], [-2, -10, -4, -6], [13, -20, -1, -2], [-51, -7, 3, -7], [-32, -5, 8, 3], [6, 14, -10, 8], [1, -21, -7, -13], [-6, -13, -1, -4], [-23, 2, -2, -7], [5, -3, 7, -5], [56, 14, -2, -6], [-40, -8, -9, -1], [1, -2, 6, 1], [4, -4, -7, -9], [4, -4, -6, -14], [35, -7, -7, -5], [34, -4, -18, -10], [6, -24, 7, -3], [-24, 4, -6, 2], [61, -6, -4, -3], [2, -17, 8, -5], [0, 21, -9, 4], [-28, -10, 5, 9], [-51, -15, -3, -5], [-36, -6, -2, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4368_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4368_2_a_bs();".
// 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_4368_2_a_bs(:prec:=4)
    chi := MakeCharacter_4368_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_4368_2_a_bs();".
// 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_4368_2_a_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 24, -15, -2, 1]>,<11,R![48, 112, -34, -3, 1]>,<17,R![16, 20, -36, -3, 1]>],Snew);
    return Vf;
end function;