
// Make newform 4830.2.a.bv in Magma, downloaded from the LMFDB on 25 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_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_4830_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 := [-8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [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], [1, 0], [-1, 0], [-1, 0], [-2, 0], [0, 0], [2, 0], [0, 0], [1, 0], [3, 1], [3, 1], [3, -1], [6, 0], [3, -1], [2, -2], [-1, -1], [9, -1], [-2, 0], [1, 1], [9, 1], [7, -1], [-1, 1], [-2, 2], [-5, 1], [-1, 1], [7, 1], [8, 0], [8, 0], [0, 2], [3, -1], [1, 1], [3, 1], [-15, 1], [-3, -1], [4, -2], [0, -4], [4, -2], [-2, 2], [4, 0], [9, 1], [10, -2], [-8, -2], [9, -1], [-8, 2], [4, 2], [-2, 2], [-6, 2], [19, -1], [-4, 0], [12, -2], [-10, 0], [-3, -3], [-2, 0], [-5, 3], [-17, -1], [-6, -2], [9, 3], [-13, 1], [-13, 1], [11, -3], [3, -1], [0, -2], [0, -4], [14, 0], [-5, 1], [-10, 4], [-4, 0], [-6, -2], [8, -4], [16, -2], [19, -1], [-1, 1], [-8, 0], [7, -5], [5, 3], [-12, 4], [6, 0], [-26, 2], [-13, 1], [16, -2], [-1, 3], [-12, 2], [-9, 1], [11, 5], [-13, -3], [-20, 0], [18, -4], [-30, 2], [11, -3], [-5, 3], [22, 2], [18, -2], [-5, 3], [12, 0], [10, 2], [0, 0], [5, 3], [3, -7], [-21, -1], [2, 0], [-8, -4], [-15, 1], [6, 2], [-13, 1], [-23, -1], [-9, -1], [0, -4], [5, -7], [-7, 5], [2, 4], [-15, 1], [-17, -1], [-3, 1], [8, 0], [-29, -3], [-21, 5], [19, -1], [0, 0], [-2, -4], [0, 6], [-2, 4], [-26, 0], [-8, 2], [-4, 0], [-11, -1], [-36, 2], [14, -4], [2, -8], [-2, -2], [14, 4], [14, -2], [40, 0], [-13, -3], [-13, -5], [-18, 4], [20, 6], [-2, 8], [-13, -1], [14, 4], [50, 0], [-3, -1], [3, -3], [1, 1], [0, 0], [38, 0], [0, -4], [34, -2], [-21, -1], [13, -1], [22, -2], [9, -5], [-15, 3], [6, -6], [24, 0], [-17, 3], [-19, -1], [-27, 3], [24, 6], [-29, -3], [12, -2], [8, -4], [1, 1], [-17, 7], [-21, -5], [-11, -7], [-48, 0], [-36, 0], [-4, 4], [-40, 2], [-29, -1], [36, -2], [16, -6], [9, 7], [10, -6], [-7, 7], [-17, -7], [-26, -2], [3, -3], [2, 2], [16, 2], [-5, 3], [3, 3], [25, 3], [-41, -1], [36, 4], [14, 0], [5, -1], [-31, 5], [8, -2], [48, 2], [-19, -3], [4, 0], [-11, 3], [-27, 3], [30, 6], [0, -10], [-34, 0], [-31, 1], [6, 0], [18, -2], [28, -6], [31, 1], [-46, 2], [10, -4], [-1, 7], [5, 5], [14, 2], [8, 4], [-38, 0], [-1, -11], [-22, 4], [57, -1], [-4, -4], [-10, 2], [46, 0], [-54, 0], [-15, 1], [-10, 4], [-22, -2], [-10, 4], [-26, 8], [12, -4], [-48, 2], [-4, 0], [6, -2], [8, -2], [52, 2], [63, -1], [4, -4], [24, -8], [27, 5], [20, -8], [-13, -3], [24, 6], [35, -1], [20, -4], [-40, 2], [-31, 5], [-4, 4], [24, -6], [8, -4], [-33, 5], [1, -3], [-46, 0], [17, 5], [-2, 4], [-27, 9], [49, 1], [35, 1], [10, -10], [-35, 1], [-33, -3], [54, 2], [38, -4], [-25, -1], [-6, 2], [-40, 2], [-37, -1], [-11, 1], [-43, -3], [20, 4], [-72, 0], [38, 4], [14, 0], [-9, 3], [-44, 0], [8, -2], [39, -3], [-7, 9], [4, -2], [-20, -10], [-29, -5], [5, 3], [37, 3], [-55, -3], [23, -1], [-44, 0], [10, 8], [26, -4], [21, -7], [0, 4], [-11, 3], [52, 4], [38, 0], [28, -4], [9, -7], [-10, 2], [-41, -1], [55, 1], [-60, 2], [30, -4], [24, 4], [30, 4], [27, 3], [-7, -1], [-2, -12], [0, 8], [-27, -3], [39, 1], [50, 2], [17, 3], [-9, -5], [-13, -3], [-32, 4], [-37, 1], [-62, -4], [44, 0], [40, -6], [15, -3], [-37, -3], [-24, 8], [30, 4], [-59, 3], [68, 0], [20, 4], [22, 0], [-35, 7], [17, -11], [1, -1], [60, -2], [-6, 14], [-10, 2], [19, 1], [25, 1], [-15, -3], [22, 8], [-35, -1], [28, 0], [-6, 6], [-7, 9], [52, 2], [-14, 8], [15, 1], [-6, -6], [-75, -1], [42, 2], [78, 0], [-6, 0], [26, -4], [-22, -6], [51, -3], [-37, 3], [-21, 5], [1, -9], [-75, -3], [-2, -14], [54, -4], [34, -2], [-36, 0], [-86, 0], [-5, 9], [-32, 2], [-33, -3], [38, -10], [-16, -10], [-57, 3], [4, 8], [63, 3], [-38, -4], [28, -4], [-53, 3], [-8, 6], [-40, -8], [-22, -8], [-72, -4], [48, -2], [-79, -3], [-22, -10], [-38, -4], [54, -2], [-1, -3], [60, 4], [-6, 12], [88, 0], [-53, 5], [13, 7], [-31, 3], [9, 3], [-33, -3], [37, 1], [44, -6], [-69, -1], [55, -1], [-58, -4], [6, 6], [-40, 8], [6, 14], [10, -2], [79, 3], [8, 0], [84, 2], [3, -5], [24, 12], [-26, 6], [23, 1], [20, 12], [10, -2], [-73, 1], [43, -3], [45, -1], [48, -6], [-26, 0], [62, -2], [18, 12], [-9, -7], [-13, 5], [29, 9], [20, 6], [-53, 7], [43, -3], [-43, 5], [-20, 10], [22, 6], [-5, -15], [-22, 6], [-51, -1], [-27, 5], [5, 1], [-54, -4], [40, 0], [-66, 0], [-14, 0], [16, 0], [-49, -7], [21, -3], [39, -5], [18, -2], [8, 2], [15, -3], [-58, -4]];
    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_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_4830_2_a_bv(:prec:=2)
    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_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_4830_2_a_bv( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4830_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2, 1]>,<13,R![0, 1]>,<17,R![-2, 1]>,<19,R![0, 1]>,<29,R![-24, -6, 1]>],Snew);
    return Vf;
end function;