
// Make newform 4830.2.a.ce 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_ce();".
// 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_ce();".
// 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 := [1, -1, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [-5, 0, 2, 0], [-1, -10, 0, 2]];
    Rf_basisdens := [1, 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, 0], [1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 1, 0, 0], [1, -1, 0, -1], [-1, 0, 0, 1], [1, 0, 0, 1], [1, 0, 0, 0], [4, 0, 1, -1], [-1, -1, 1, 0], [2, 1, -1, -1], [0, 0, -2, 0], [2, 2, -1, 1], [3, -2, 0, -1], [2, -1, 1, 1], [4, -1, -1, -1], [0, 2, 0, 2], [-2, 0, 1, -1], [4, 0, -1, -1], [-2, 1, 1, 1], [0, 1, -1, -1], [1, -2, 0, 1], [-1, -4, 1, -2], [3, 0, 1, -2], [-5, 2, 1, 2], [-2, 2, 0, 2], [6, 0, 2, -4], [-8, 0, -2, 4], [0, 1, 1, 3], [0, -2, 1, 1], [4, -3, 1, -3], [2, 3, -1, 3], [-2, -5, 1, -1], [-2, -2, 0, 0], [2, 2, 0, 2], [-8, 4, -2, 4], [-10, 0, -2, 0], [1, 2, -2, 3], [5, -1, -1, -2], [0, -4, 2, -2], [-4, 0, -2, -4], [-6, 1, -3, 3], [2, 4, 0, -4], [0, 2, -2, 0], [0, 2, 0, 0], [4, -4, -2, 2], [-1, 0, 5, 0], [-5, 2, 2, -3], [-10, -2, 0, 0], [0, 0, 2, -4], [-4, 0, -1, -1], [7, -2, 2, -5], [3, -2, 3, -2], [10, 5, -1, 3], [-6, 0, 0, -2], [-9, 6, -1, 4], [11, -5, 1, -4], [8, -2, -1, -3], [8, -2, 3, -3], [-5, -2, -1, -2], [-8, 0, 0, 2], [-14, -2, 0, 2], [13, 1, -2, -1], [-15, 2, -3, 0], [4, 2, 0, 6], [2, 2, -2, 0], [-2, -1, 0, 0], [-6, 0, 2, 0], [-7, 6, 2, 1], [-10, 3, 1, 1], [-6, 3, 3, 1], [2, 2, -4, 2], [-10, -7, 3, -1], [-2, -3, 3, 1], [0, 8, -2, 6], [2, -8, 0, 0], [1, -3, 6, -3], [4, -4, -1, 5], [-20, -4, 0, 2], [-9, 8, -1, 6], [-2, 8, 0, 8], [4, 5, -3, 1], [-9, 0, -3, -2], [-7, 1, 1, 2], [6, 0, -2, 0], [-6, 0, -2, -6], [0, -3, 4, -2], [-9, 2, 1, -2], [-2, -2, -1, 5], [19, -6, 0, -5], [-6, 0, 2, 0], [6, -2, 3, 1], [-2, 2, -2, 4], [-20, 0, -2, 2], [-2, -8, 0, -10], [-1, -4, -1, 4], [-13, -4, 5, -2], [-13, 6, -1, 6], [-2, 2, 4, 4], [-4, 2, -6, -2], [4, 1, -5, 1], [-15, -6, 4, -3], [0, 0, -1, -7], [8, 1, -1, -1], [-10, -5, 1, -3], [0, 4, -2, 10], [14, -1, 1, 1], [-6, -2, -5, 1], [-14, -2, -2, 6], [5, 10, -1, 0], [6, -3, -3, -3], [10, -3, 3, -1], [3, 8, 0, -5], [4, -7, 3, -9], [-28, 2, -1, -3], [-23, -2, -1, 4], [1, 4, 2, -1], [-10, -2, -4, 4], [2, 3, -6, -4], [2, 8, -2, 2], [-6, 4, -2, -6], [-2, 2, 2, 2], [-6, 0, -2, 4], [12, 3, 3, -1], [6, -8, 0, 0], [-20, -2, 6, 0], [3, 3, 2, 1], [-14, 4, 2, 4], [-16, -8, 4, -2], [6, -2, 4, -2], [16, -2, 4, 0], [8, -5, -1, 3], [26, 1, -3, 1], [-10, 4, -6, 6], [-29, -2, 0, -3], [-10, 2, -6, -6], [-3, -2, -1, -8], [-4, -4, 8, -2], [-6, 2, -6, -2], [-14, 7, -1, 9], [6, 2, 1, -7], [8, 2, 1, -7], [-2, 4, 2, 4], [-17, 4, 0, 5], [-16, 8, 0, -4], [-15, 1, 2, 1], [-10, -5, -3, 5], [-8, -5, -1, -9], [-2, 0, -2, 4], [18, -4, -3, -3], [29, -6, -5, -2], [-2, 0, 6, 0], [29, 0, -2, -1], [8, -6, -1, -9], [-2, 3, -3, 7], [-14, -3, 3, -7], [-4, 8, -4, -2], [-7, -8, 5, -8], [-6, 2, -6, -2], [8, 2, 4, 4], [4, 11, -1, 13], [10, -6, -1, 1], [-9, -6, 3, 2], [-6, -3, 3, -9], [-16, -8, 2, 2], [-6, -2, 4, -2], [-23, -5, 0, -1], [-30, 2, -2, -2], [19, -7, 5, -6], [8, -7, 2, 2], [9, 0, 2, -7], [10, 1, 5, -5], [-10, 1, 0, 8], [-15, -9, -1, 8], [-21, -2, 1, 2], [24, -10, 6, -14], [-8, 6, 5, -1], [-20, 2, -2, -2], [-12, -2, 0, -2], [-8, 2, -9, -1], [-1, 6, -5, 10], [-6, -4, -3, 5], [40, 1, 3, -3], [4, 4, -4, 0], [-12, 4, 0, -6], [12, 8, -3, 7], [-11, -6, 1, -6], [-20, 6, 2, 8], [21, 5, 4, -11], [-4, 1, 3, -7], [44, 2, -4, 0], [10, -3, 3, 5], [-3, 4, -1, 2], [16, 0, 4, -8], [-19, 4, 2, 1], [-24, 6, -4, 10], [-2, -7, -1, -1], [-9, -2, 0, 1], [5, -10, 0, 1], [-30, 10, 2, 2], [15, -7, 1, -8], [13, 3, 6, 9], [19, -2, 6, -13], [-7, 6, -5, -4], [-1, -9, -1, -8], [-4, -6, -2, 2], [24, -2, -4, -4], [18, 4, 4, 4], [2, 9, -1, 9], [14, -2, 0, 4], [11, 0, 3, -4], [24, -4, 8, 0], [11, 2, -4, 7], [-11, -15, 2, 3], [-1, -4, 6, 7], [14, -10, 1, -1], [-18, -4, -6, 6], [16, 4, -10, 2], [30, -6, -4, -4], [4, 8, 0, -2], [-34, -4, -2, 4], [14, 2, 2, -2], [-6, 12, -2, -4], [28, -6, 2, -6], [-12, 6, -2, 12], [37, 4, -2, -3], [20, 0, 5, 5], [46, -2, 0, -6], [12, 2, -4, -4], [6, 4, -1, -5], [-21, 18, 0, 11], [18, 19, -7, 7], [-26, -10, 0, 0], [-29, 6, -5, 2], [-2, 8, -2, 4], [-21, 6, 0, -3], [22, 5, 3, 7], [24, -2, 4, 0], [-7, 5, 0, 5], [4, 6, 2, 2], [22, 9, -7, 7], [-11, 8, -5, -4], [-26, 2, -8, 8], [2, 15, 1, 5], [9, -5, 2, -1], [1, 8, -5, -8], [23, -10, 5, -12], [0, 1, -3, -3], [4, 0, 2, 2], [34, -1, -1, -5], [-9, -12, 9, -10], [12, 6, -10, 2], [-41, -8, 6, -1], [-19, 11, -3, 12], [-12, 10, 2, 10], [-16, -8, 6, -4], [-23, -6, -1, 12], [35, 5, 3, 0], [28, -13, -5, -3], [32, -4, -4, -4], [-8, -12, -2, -6], [19, -6, 8, -7], [-38, -10, 4, 0], [-10, 7, 1, 9], [7, -2, 0, 7], [42, 0, -4, 0], [15, -12, 5, -14], [26, -12, 1, -5], [10, 8, -6, -2], [-3, 12, -2, 1], [-4, -12, 3, 7], [-33, -10, 3, -4], [-30, 7, 3, -3], [-14, -1, -5, -9], [21, -10, 5, -10], [34, -2, -2, 8], [26, 2, 4, 8], [6, 14, -8, 12], [-11, -4, 5, -6], [19, 2, 2, 1], [14, -5, -1, -3], [12, 12, -6, 6], [0, 6, 4, 6], [42, 2, 4, 2], [32, 4, -1, 11], [-4, 15, 0, 6], [6, 11, -9, 11], [-11, 5, 1, 6], [-12, 0, 4, -2], [-30, -4, 6, 2], [19, -6, -6, -3], [22, -8, 2, -2], [-1, 10, -5, 10], [20, 2, -3, 7], [30, -20, 0, -16], [46, 4, -2, -4], [3, -9, -1, -4], [6, 19, -3, 11], [-20, -6, 6, -2], [-13, -4, -1, 4], [20, 11, -9, 5], [20, 11, -1, 7], [16, 10, 0, 8], [-8, -15, 1, -7], [10, -10, 4, -4], [-12, 4, 4, 12], [-22, 2, 4, 0], [12, -9, 5, 1], [0, 12, 3, 17], [29, 12, -2, 7], [-38, 2, -8, -8], [-20, 8, 1, -1], [22, 4, 4, -10], [6, -6, 10, 4], [-15, 14, -6, 5], [-14, -11, 3, -7], [-16, 10, 3, -1], [-33, 2, -9, 8], [26, -14, -2, -10], [61, -2, 2, -1], [50, -9, -4, 0], [3, 12, 1, 10], [26, -2, -7, -5], [-20, -3, -5, 17], [-17, 10, 2, 11], [-16, -13, 3, -13], [42, 6, -4, 6], [42, 2, 4, 6], [24, 5, -1, 1], [58, -6, -2, 2], [-22, -6, -4, -16], [-9, 1, 5, 4], [-32, 10, 6, 2], [-16, -5, 3, -5], [64, -2, 4, -8], [-10, 14, 0, 0], [1, 12, -4, 15], [8, -10, 6, -4], [24, 14, -10, 2], [4, -4, 7, -7], [-14, -9, -7, 5], [-7, -2, -3, -4], [34, -13, 3, -11], [28, -15, 5, -13], [12, 0, 2, 2], [-21, 2, -4, -11], [16, -2, 12, -4], [-4, 4, 0, 8], [-18, 8, 4, 4], [30, -3, 1, 7], [6, 2, -2, -6], [18, -16, 5, -7], [-4, -2, 6, -10], [-2, -12, -2, -10], [26, -5, 1, 5], [64, 8, -4, 0], [-3, 12, -1, 20], [8, 8, 6, 8], [36, 8, 4, 0], [42, -11, 9, -11], [-20, 0, 0, -14], [-28, -12, 4, 8], [-10, 15, -12, 10], [-32, 4, 0, 4], [26, 10, -4, 0], [39, 9, -1, 0], [8, -6, -10, 6], [22, -2, -4, 8], [-34, -4, -8, 6], [4, -7, 5, 1], [32, -4, 8, 0], [-10, 2, -2, -10], [36, 0, 2, -2], [48, -12, -5, 1], [66, -7, 3, -7], [0, -3, -5, -5], [-23, -4, 7, -6], [-33, 16, -3, 18], [42, -1, -1, -5], [0, -10, 6, 12], [12, 13, 1, 7], [-12, 2, -5, 7], [10, -16, 2, -22], [-9, 0, -2, -3], [-21, -8, -2, -11], [-2, 12, 2, 8], [-3, -3, -10, -7], [10, -6, 11, 5], [32, -12, -2, -10], [32, -12, -2, 4], [-1, -13, 5, -10], [12, 4, 0, 0], [14, 12, -10, 12], [-6, -1, 15, -3], [-6, -19, 6, 2], [-14, -14, -4, 14], [-29, 6, -3, -2], [-30, -1, -7, 19], [-44, -2, 7, -3], [9, 0, 6, 5], [32, -16, 6, -16], [-2, 10, 4, -2], [-31, 2, 0, 3], [29, -22, 5, -24], [13, 5, 9, 6], [54, -7, 1, -11], [-42, -12, 4, -12], [7, -6, -9, -6], [62, -9, 1, -9], [52, -3, 3, -7], [22, -6, 10, -6], [20, -4, 8, -16], [-2, 9, -7, -1], [2, 11, -4, 0], [-45, -2, -5, 4], [18, -8, -1, -11], [3, 12, -5, -6], [25, -2, 0, -5], [-14, 0, 0, 10], [-70, 2, 2, -2], [40, 8, 4, -2], [-64, 8, 4, 4], [16, 1, 1, 5], [-36, 15, -1, 9], [37, 1, 9, 16], [30, 14, 0, 2], [6, 14, 6, -6], [62, 1, 7, -3], [5, -1, -10, -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_ce();".
// 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_ce(:prec:=4)
    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_ce();".
// 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_ce( : 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, -8, -20, 0, 1]>,<13,R![128, 24, -24, -2, 1]>,<17,R![16, 0, -20, 2, 1]>,<19,R![-64, 72, -8, -6, 1]>,<29,R![-944, 320, 20, -14, 1]>],Snew);
    return Vf;
end function;