
// Make newform 4620.2.a.w in Magma, downloaded from the LMFDB on 29 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4620_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_4620_2_a_w();".
// 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_4620_2_a_w();".
// 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, -15, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-11, 0, 1]];
    Rf_basisdens := [1, 1, 2];
    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_4620_a();"
function MakeCharacter_4620_a()
    N := 4620;
    order := 1;
    char_gens := [2311, 1541, 3697, 661, 2521];
    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_4620_a_Hecke(Kf)
    return MakeCharacter_4620_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, 0], [-1, 0, 0], [-1, 1, 0], [0, -1, -1], [0, 1, -1], [2, -1, 1], [1, -2, 1], [3, 1, 0], [3, 1, 0], [1, 1, 0], [3, 0, -1], [5, -1, 2], [0, 3, -1], [3, -2, -1], [2, -1, 3], [2, 0, 0], [9, 1, 0], [4, 0, 2], [1, 1, 0], [6, -3, 1], [3, -2, -1], [1, 0, -1], [4, -2, 2], [9, -2, 3], [1, 3, -2], [-7, -1, 0], [0, 1, -3], [5, 0, -3], [-1, 5, -2], [-2, -2, 2], [-4, -2, -2], [-4, 2, -4], [4, -2, -2], [3, 0, -3], [19, 1, 0], [-4, 2, 2], [-6, 2, -2], [5, 1, 4], [-2, 4, 0], [0, 0, 0], [2, 0, -2], [-2, -2, 4], [-6, 2, 2], [6, 4, -2], [5, -4, -3], [-4, 5, -1], [-5, 1, 0], [-3, -3, 4], [-7, -4, 1], [2, 2, 0], [-2, -6, 2], [-5, -5, 2], [-2, 2, 0], [9, 2, 1], [0, 1, -1], [-14, 2, 0], [-8, 0, -2], [19, -3, 2], [0, 1, 1], [-2, 0, 2], [12, -4, -2], [7, -2, -5], [-8, 0, 2], [-2, -5, 3], [7, -2, 5], [-10, -2, 4], [-6, -3, 5], [4, 2, 0], [3, 0, -1], [-5, 6, -3], [5, 2, -5], [-10, 1, 1], [-1, -5, -4], [-5, 1, -8], [-8, 4, -6], [-7, -3, -2], [10, -2, -4], [-7, 0, -1], [10, 1, 3], [-2, 2, 4], [16, -2, 0], [-22, 3, -1], [10, 0, 4], [-1, 7, 2], [11, -6, 5], [-26, -4, 2], [8, 2, 0], [10, -6, 0], [3, 1, 2], [8, 6, 0], [-7, -2, 3], [12, 1, -1], [-2, 1, 1], [7, -4, 5], [9, -4, 3], [18, 6, 0], [-7, -5, 0], [-11, 6, -1], [-6, 4, -6], [10, 0, -2], [-23, 0, -1], [7, -5, 0], [14, -8, 8], [-7, 1, 0], [-8, -8, 6], [-6, -4, -2], [-4, 11, -3], [10, -4, 6], [8, -2, 10], [-28, 4, -6], [10, 0, 0], [-2, 9, -3], [-10, 0, 0], [5, 0, 1], [2, 10, -4], [-12, 5, 1], [-13, 8, 3], [11, 5, 0], [5, -4, -3], [-16, 3, -5], [6, -6, -6], [6, -8, 8], [-29, 2, -1], [-6, 3, -7], [7, -10, 7], [-23, 6, 3], [-4, -2, 0], [14, -8, 2], [-6, -8, 6], [18, 3, -5], [-12, 2, -4], [-20, 2, 2], [-18, -3, 1], [20, -6, 0], [28, -4, 4], [-8, 6, 0], [8, 2, 8], [-20, 0, -4], [-13, 0, 5], [9, 3, -8], [-14, -6, -4], [4, -2, -8], [3, 0, -3], [19, -1, -6], [-6, 0, 0], [-23, -1, 0], [-4, 11, -1], [-19, -2, 3], [-39, 4, 3], [24, 0, 2], [28, 1, 7], [-29, 1, 0], [-14, -6, -4], [0, -10, 2], [8, 8, -6], [-3, -11, 2], [-27, 3, -6], [4, -3, 5], [-16, -6, -2], [-1, -12, 7], [1, 6, 1], [-36, -1, 3], [38, 0, 2], [-28, -1, -3], [-2, 4, 0], [-37, 3, -6], [-3, -7, 10], [19, 8, -5], [-25, -11, 4], [-37, 2, 5], [-21, -10, 5], [-20, 2, 0], [-16, 14, 4], [-1, 3, 0], [-6, -2, -6], [-14, 12, -2], [-10, -2, 4], [10, -2, -6], [-54, -2, 0], [20, -6, 12], [8, 7, 3], [28, -9, -1], [3, 8, -3], [-33, 2, 5], [-1, -4, -5], [31, 5, -4], [35, -1, 8], [18, 12, -4], [-28, -2, 2], [4, 6, 2], [-31, 4, -5], [15, -9, -8], [26, 6, -2], [-14, 14, 2], [15, -3, 0], [-4, -11, -3], [4, 18, -6], [-39, 0, 3], [-40, 5, -5], [29, 2, 1], [12, 4, 4], [-17, -12, 5], [14, -10, 2], [-1, -15, 8], [8, -6, -2], [-11, 2, 1], [29, 3, 0], [-3, 14, -1], [-19, 10, -3], [-24, 4, -2], [57, -3, 0], [36, -4, 0], [-48, -6, 4], [5, 0, 5], [2, -12, -2], [-2, 4, -4], [-22, 1, -13], [6, -10, -6], [-22, -4, -2], [-7, -15, 10], [25, -9, -4], [-18, -3, 9], [-31, 1, -10], [17, 7, 0], [27, -5, 0], [2, 18, -8], [15, -6, 13], [-21, 5, -8], [-2, -16, 2], [15, -9, 8], [9, -8, 7], [27, -2, 5], [16, -3, -1], [19, 7, -2], [-30, -2, -4], [-38, 8, 2], [-8, 4, -14], [4, -10, 4], [-14, 5, -7], [32, 2, -2], [16, -9, 15], [-18, 3, -3], [-7, -5, 2], [-24, 8, -8], [-7, 10, 5], [-37, 3, -12], [-22, 2, -4], [-28, 14, -2], [-13, 3, -8], [-2, 10, -2], [17, -1, 0], [22, -3, -1], [13, -2, 7], [-30, -13, 5], [-3, 17, -8], [-9, 5, 4], [44, -2, -4], [47, -2, 5], [20, -4, -4], [-14, 3, 7], [1, 5, 14], [-25, 3, -2], [-39, 1, -6], [6, 2, 6], [43, -4, 13], [-11, 8, 1], [3, -5, 4], [-6, -20, 8], [10, 12, -4], [39, -4, -3], [-27, -5, -6], [61, 3, -4], [-40, 2, 6], [0, 15, -9], [6, 14, -2], [1, 3, -8], [-16, 0, -16], [10, -9, 7], [4, -6, 6], [36, 8, -4], [-6, 3, -1], [27, -15, -2], [-15, 7, -10], [17, 9, 2], [4, 12, -8], [-18, 0, 4], [-11, 1, -16], [-22, 6, -2], [31, -5, 8], [-13, -5, -12], [-40, 8, 0], [-24, 2, -10], [-28, 8, -4], [12, 0, -10], [19, -9, 18], [36, -4, 0], [8, -12, 0], [48, -6, 8], [-15, 1, 8], [-29, -17, 4], [-14, 3, 13], [8, 7, -11], [7, 3, -6], [-2, -11, 7], [-9, 7, 10], [55, -5, 0], [59, -4, -1], [38, 4, 2], [27, -5, 10], [-45, -4, -3], [51, 3, -2], [-27, 15, 6], [4, -11, 7], [12, 0, -6], [31, -8, 19], [8, -18, 0], [25, -12, 11], [-24, 16, -4], [49, -7, 10], [-4, 2, -2], [9, -5, 12], [-10, -10, -8], [40, -12, 2], [18, 4, -6], [7, 8, -5], [37, -1, 2], [11, -9, 12], [11, 5, 12], [-14, -14, 2], [4, 15, -3], [4, 16, -8], [30, -12, -2], [16, -12, 16], [-34, 11, -15], [12, 4, -2], [49, -3, -6], [3, 3, 8], [-37, 3, -14], [-4, -5, -5], [13, -3, 4], [-46, 4, -2], [-12, 13, -15], [-35, 10, 3], [-17, 15, -10], [19, 11, 2], [8, 8, 12], [5, 19, -4], [0, 9, -1], [-24, -8, -2], [23, -3, -4], [-23, 7, 2], [38, 14, -2], [14, 3, -7], [-14, 0, -12], [11, -15, -2], [-26, -12, 2], [-32, 9, -11], [25, -10, 7], [72, -2, 6], [30, -12, 4], [7, 10, -3], [9, 0, 17], [42, 8, -4], [-58, -1, -7], [-5, -12, -1], [9, 11, -16], [-2, 14, -12], [-8, 12, 6], [2, 10, -12], [25, -8, 7], [58, -3, -3], [-21, -3, -8], [-2, -8, -4], [33, -4, -7], [-18, 14, -2], [51, -11, 10], [25, -5, -4], [17, 4, 3], [0, 3, -9], [-36, 0, -4], [-7, 3, 10], [24, 9, 7], [25, 7, -2], [-8, -3, 3], [-25, 12, -7], [-2, -18, 0], [38, 10, 4], [24, -17, 15], [56, -6, 4], [17, -15, 8], [-45, -6, 5], [28, -6, -10], [19, -14, 1], [-12, 14, -6], [-4, 22, -4], [50, -13, -9], [-1, -6, -5], [-20, 9, -9], [-8, 13, 1], [17, 0, 5], [77, 5, -2], [37, 3, 6], [8, 3, -7], [3, -13, -6], [82, 0, 0], [3, -14, -3], [-13, 2, 11], [18, -2, 6], [-57, -3, 6], [-88, 2, 0], [21, -19, 12], [-42, -8, 6], [-55, 4, -5], [-13, 16, -9], [-69, -2, 5], [31, -17, -2], [12, -18, 6], [-25, 19, -2], [32, 2, -4], [-58, 11, -7], [-9, -1, 16], [-33, 2, -17], [-5, -1, 10], [-12, -21, 3], [45, -8, -5], [-18, -12, 4], [29, -5, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4620_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4620_2_a_w();".
// 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_4620_2_a_w(:prec:=3)
    chi := MakeCharacter_4620_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_4620_2_a_w();".
// 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_4620_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4620_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-12, -14, 2, 1]>,<17,R![122, -51, 0, 1]>,<19,R![12, -23, -2, 1]>],Snew);
    return Vf;
end function;