
// Make newform 4830.2.a.ca in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4830_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_4830_2_a_ca();".
// 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_4830_2_a_ca();".
// 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], [-1, 2, 0], [-6, 0, 2]];
    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_4830_a();"
function MakeCharacter_4830_a()
    N := 4830;
    order := 1;
    char_gens := [3221, 967, 2761, 1891];
    v := [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_4830_a_Hecke(Kf)
    return MakeCharacter_4830_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 := [[1, 0, 0], [-1, 0, 0], [1, 0, 0], [-1, 0, 0], [0, 0, -1], [-1, -1, 1], [-3, 1, 0], [-1, 1, 0], [-1, 0, 0], [-4, -2, 1], [-5, 1, 0], [-2, 0, 0], [2, 0, 2], [0, 0, -1], [-5, 1, 0], [-2, 0, 0], [2, 2, -2], [-8, -2, 0], [-8, 0, -1], [2, 2, 1], [2, 0, -2], [2, -2, -2], [-5, -3, 2], [-5, -1, -1], [1, 1, 1], [-5, -5, 1], [-8, 0, 0], [-2, -2, 0], [2, 4, 0], [8, 2, 0], [-4, 0, 3], [-4, 0, 0], [6, 4, 0], [2, 2, -2], [-2, 0, 0], [-6, -2, 2], [-2, 0, -4], [-10, -2, 2], [-7, -5, 2], [-15, 1, 0], [0, -4, 2], [0, -2, 0], [-10, 2, -4], [-12, -2, 2], [12, 2, 0], [4, 4, -4], [6, -2, 2], [-13, -3, 1], [5, 3, -4], [14, 0, 2], [-2, 4, 2], [10, 2, 1], [9, 5, 0], [5, -1, -3], [-2, 4, -4], [-4, -4, 0], [7, -1, -1], [-5, 1, 4], [-10, -4, 3], [-6, 4, 1], [-13, 1, 3], [-12, 2, -2], [-8, 4, 4], [-17, 1, -1], [3, 7, -5], [0, -2, 2], [-8, 4, -4], [-14, -4, -1], [14, -2, 2], [5, -3, -4], [10, 0, 2], [2, -2, 4], [4, -4, -4], [-2, 0, 0], [2, -6, 4], [-8, 0, 2], [24, -2, 0], [-1, -1, 1], [-6, 4, -1], [-4, -2, 2], [-3, 7, 3], [22, 0, 2], [-14, -2, -2], [-3, 5, 1], [9, -5, -2], [14, -2, -2], [24, 2, 0], [-4, -6, -3], [3, -5, 1], [-14, -6, 1], [-3, 3, -6], [16, 0, 0], [-14, 2, -7], [8, -4, -2], [10, 10, -2], [10, -2, -2], [15, -1, 5], [11, -1, 7], [-13, 1, -5], [22, 0, 2], [10, 10, -6], [-6, -4, 0], [11, -3, 6], [2, 4, -1], [8, -4, 0], [0, 2, 2], [-16, 4, -4], [2, 0, -4], [20, 0, 3], [22, -4, 2], [1, 7, 1], [-28, 2, 0], [2, 0, -2], [5, -5, 8], [-14, 6, 6], [-14, 4, 1], [1, 3, -5], [-3, -1, 4], [-2, -8, -2], [18, -2, 3], [-2, 0, -2], [8, 2, -4], [-22, -4, 0], [2, -6, 6], [-28, 0, -2], [18, -4, 6], [6, 0, 2], [21, 3, 1], [18, 6, -2], [-4, -6, 0], [2, 2, 2], [6, 2, 6], [16, 0, -4], [24, -2, 4], [2, -8, -4], [-3, 9, -10], [26, 4, -2], [-27, -1, -1], [4, 2, -4], [36, 6, -2], [-10, -2, -6], [6, -4, 3], [12, -8, 7], [14, -2, 8], [-9, 3, 4], [10, -2, 2], [-17, 7, 5], [-18, 4, -2], [-8, 4, 6], [10, 6, 2], [24, 10, -5], [-19, -3, -5], [-18, 6, -2], [33, -5, 0], [-22, -2, 3], [-42, -6, 2], [2, 6, -2], [24, -6, 2], [-21, -9, 5], [10, 4, 4], [-2, 6, -6], [-4, -10, 0], [-22, -6, 1], [1, -13, 1], [38, 4, -2], [28, 4, -2], [26, 6, -2], [-17, -9, 5], [22, 12, -4], [11, 7, -8], [0, -8, 1], [5, 5, -2], [26, 6, -4], [-22, 4, -7], [25, 3, -6], [-17, -13, 5], [20, 8, -4], [4, 6, 7], [18, 6, -6], [12, 2, 2], [28, 8, 1], [-15, -5, 9], [4, 6, -5], [24, -6, 4], [4, -4, -8], [-2, 0, -6], [2, -8, 3], [-7, 3, 5], [-2, -4, 0], [-9, -7, 5], [-8, -14, 4], [22, 6, -6], [-2, -10, 8], [-15, 5, 1], [8, -4, -6], [-17, -9, 2], [-36, 6, -6], [4, 2, 6], [25, -3, 6], [25, 3, 4], [-8, 6, 4], [-13, -15, 2], [27, 3, 3], [25, 5, 0], [3, 9, -3], [-29, 7, -4], [4, -4, -4], [22, -2, -2], [22, 4, -8], [-6, -6, 6], [-18, -4, 2], [-29, 3, -7], [-28, 4, -4], [-11, 3, 10], [19, -3, 3], [-19, 1, -2], [6, -2, -1], [-6, -8, 4], [10, -2, 2], [-10, 0, -6], [20, -6, -8], [-22, -10, 0], [-38, -8, 4], [-36, 4, 4], [-24, 4, -8], [32, -2, 4], [23, -1, 10], [26, 10, -9], [44, -4, 4], [-8, -4, -2], [-40, 2, 1], [-15, -17, 6], [-10, -6, -2], [-8, 10, 0], [-21, -7, 3], [-18, 2, -2], [9, -3, 6], [14, 16, -4], [28, -8, 14], [-1, 9, -3], [0, 4, -2], [-14, 2, 6], [-7, -1, -13], [-14, 4, 6], [-38, 8, 4], [-1, -7, 5], [-23, 7, -1], [5, 15, -13], [-36, 0, 2], [18, -2, 6], [-64, -2, 0], [11, 7, 3], [-44, 4, 0], [-7, 5, 6], [17, -7, -10], [-60, 0, 0], [-22, 4, 0], [-7, 3, -9], [-1, -5, -6], [-30, 8, -8], [-40, -8, 0], [0, 12, -8], [-13, -9, 2], [6, 0, -10], [-22, 0, 6], [1, -1, -6], [6, -8, -2], [-45, -9, 3], [8, 8, -5], [6, 8, -6], [17, -7, -10], [-26, 2, 1], [21, -11, -7], [-48, 0, 0], [30, 12, -6], [-35, -5, 7], [32, -4, 0], [-42, 0, 6], [42, -8, 6], [19, 17, -7], [-13, 1, 2], [-12, 4, 0], [0, 0, -6], [16, 14, -8], [-6, 10, -2], [-34, -2, 1], [-28, 2, 7], [2, 4, -14], [1, 3, 2], [-36, 6, -2], [6, 0, -4], [-23, -1, 8], [42, -8, -6], [-3, 7, -15], [2, 0, 5], [-6, -12, 4], [14, -6, -8], [-49, 3, -4], [-2, 6, 2], [-36, -16, 8], [41, -7, -1], [-62, -4, -2], [8, 8, -12], [14, -10, 6], [-52, 0, -2], [-2, -4, 2], [12, 0, -8], [-50, -8, 2], [-6, 6, 4], [-6, -16, 5], [-39, -5, 0], [-16, -18, 10], [-14, 4, 1], [30, -2, 8], [-12, 4, 4], [-21, 3, 8], [-20, 0, 8], [-42, -18, 7], [13, -11, 13], [-2, 4, 0], [7, -7, 16], [2, -12, 11], [11, 11, -5], [26, 10, -1], [-12, -10, 10], [13, 1, -4], [-24, -4, 10], [24, -4, 4], [-10, 10, -10], [-42, 8, -4], [18, 12, 4], [14, 8, -6], [-17, -11, 10], [16, -4, 4], [-64, -4, 2], [12, -8, 8], [-10, 0, 0], [35, 3, 8], [36, 6, -2], [-38, 6, -2], [10, 16, -3], [18, 0, -6], [-35, 1, 5], [-8, -16, 4], [2, 12, -2], [8, 4, 6], [39, 3, -2], [-2, 6, 4], [-26, 10, -2], [14, 8, -8], [-18, -18, 6], [-6, 0, 4], [-2, 4, 1], [-6, 6, 2], [-18, 16, -4], [-22, 8, -10], [42, -10, 2], [-21, -7, -1], [18, -8, 14], [-20, 12, -4], [44, -6, 12], [-8, 18, 2], [-4, 4, 16], [26, 6, -3], [-40, 4, -12], [-20, -10, -4], [-33, 3, 10], [-4, -4, 0], [6, -4, -6], [-28, 8, -2], [-20, 0, -6], [24, -8, -12], [-34, 8, -2], [-60, 4, 0], [-2, 4, -7], [48, -4, -4], [46, 10, 0], [-17, 3, -17], [-45, -1, -1], [-6, 8, -18], [-6, -20, -2], [8, -6, -12], [-20, 0, -11], [14, 12, -14], [-31, -1, 6], [-13, 1, -16], [-22, -10, 6], [-9, -1, 5], [-28, 12, -5], [-28, 4, -2], [36, -2, -2], [-1, -5, -6], [20, -8, 0], [16, -4, 16], [-22, -10, 0], [-34, 8, 11], [50, 6, 0], [3, -1, -11], [18, -6, 12], [12, -10, 9], [13, 13, -6], [42, 16, -2], [40, -8, 0], [-1, 7, -2], [-15, 13, -15], [-19, 7, -14], [-2, 0, -10], [0, 2, -2], [-81, -3, 3], [26, 2, 10], [-4, -18, 4], [-22, -4, 4], [-28, 8, -2], [-18, -2, 14], [6, 0, 11], [31, -9, 3], [0, -12, -7], [11, 5, 11], [-41, 7, 6], [-10, 2, -10], [22, -8, -6], [-12, 10, 8], [28, -12, 12], [14, 6, 2], [-42, 4, -4], [5, 13, -12], [2, -14, 0], [14, -4, 8], [-78, 2, -2], [3, 5, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4830_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4830_2_a_ca();".
// 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_4830_2_a_ca(:prec:=3)
    chi := MakeCharacter_4830_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_4830_2_a_ca();".
// 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_4830_2_a_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4830_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-16, -28, 0, 1]>,<13,R![8, -28, 2, 1]>,<17,R![-16, 16, 10, 1]>,<19,R![-16, -12, 4, 1]>,<29,R![-352, -32, 10, 1]>],Snew);
    return Vf;
end function;