
// Make newform 4851.2.a.bn in Magma, downloaded from the LMFDB on 10 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4851_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_4851_2_a_bn();".
// 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_4851_2_a_bn();".
// 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, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 0, 1]];
    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_4851_a();"
function MakeCharacter_4851_a()
    N := 4851;
    order := 1;
    char_gens := [4313, 199, 442];
    v := [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_4851_a_Hecke( : Kf := Kf)
    return MakeCharacter_4851_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]];
    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, -1], [0, 0, 0], [-1, 1, -1], [0, 0, 0], [1, 0, 0], [4, -1, 0], [-1, 0, -1], [3, 2, -1], [-4, 0, 1], [2, 3, 1], [2, -3, -1], [-2, 2, 2], [-2, 1, 4], [1, -1, -2], [-1, 0, -1], [-6, 1, -2], [3, -1, 6], [8, 0, -2], [-6, 2, 0], [-1, -4, 3], [5, 5, 0], [0, 3, -1], [-3, -2, -1], [0, 1, -1], [-4, 3, 0], [0, -2, -3], [0, -2, -1], [1, -3, -3], [-11, 1, -3], [1, 6, -3], [-1, 5, 6], [-6, -1, -3], [3, -3, -2], [7, -2, 0], [1, 0, 0], [12, -3, -2], [5, -2, 4], [-5, 0, -6], [-8, 5, -1], [10, -5, 0], [5, 5, 2], [7, 1, 8], [-5, 4, 4], [8, -7, 6], [6, -8, 5], [2, 5, 3], [-2, -6, 4], [-13, 0, -4], [1, 7, -5], [7, -2, -8], [9, -4, 4], [8, -5, 7], [9, -5, 2], [15, -2, 5], [6, -9, 9], [6, -6, -3], [-6, -3, 3], [5, -4, 12], [-3, 11, -4], [-17, -1, -1], [0, -6, 7], [9, -4, -2], [1, -5, 9], [23, -1, -5], [-7, 8, 9], [-9, -1, 4], [1, -6, 10], [15, -8, -5], [10, -5, 1], [7, -4, -4], [0, 7, 4], [-2, 12, 1], [2, 5, -5], [-1, 14, -8], [-11, 4, -5], [14, -9, 8], [-12, 7, 9], [18, 6, 2], [15, -7, -6], [14, 3, -9], [13, 4, 6], [-14, 10, -9], [5, 2, 4], [-2, 6, 8], [-4, 2, 5], [-15, -1, -1], [-15, 0, -10], [5, 4, -1], [-15, 10, 3], [-2, -1, 9], [6, 8, 5], [8, -1, 11], [5, 1, 4], [-9, 12, 12], [2, -7, -7], [12, -7, 4], [-4, 2, 9], [10, 2, -7], [7, 0, 3], [8, -7, -5], [-3, -10, 10], [-23, 7, -3], [-19, 15, 9], [24, 6, -1], [-5, -14, -3], [-23, 4, 3], [-15, -7, 4], [18, 3, 3], [3, -13, -2], [25, 5, -5], [-8, -1, -12], [16, 8, 6], [24, -4, -8], [-8, -2, -16], [-9, 9, -6], [24, 1, -9], [-4, -4, -2], [6, -4, 2], [26, 3, 4], [8, 2, -3], [25, 11, -5], [16, 12, -6], [-1, -22, 7], [-6, 3, 8], [12, 12, 0], [3, -2, -9], [12, -9, 23], [19, 13, 1], [-26, 15, 7], [31, -6, 7], [9, -13, 20], [0, -1, 5], [31, -3, -5], [-1, -17, 13], [6, 4, -9], [-25, 18, 12], [3, 10, 1], [-2, 7, -10], [14, -16, -3], [-3, 6, 16], [25, -7, 13], [-3, -27, 3], [2, -5, -17], [16, -3, -2], [29, 3, 0], [-15, -3, -11], [-8, -14, -6], [20, 10, -9], [8, -21, 12], [29, -7, -9], [7, 3, -12], [-12, 11, 16], [-44, 1, -8], [1, -2, -17], [-7, 2, 6], [9, -13, 11], [-9, 0, -4], [21, -3, -3], [-6, -7, -13], [7, -3, -4], [-5, -13, 7], [-14, -12, 11], [-19, -5, 12], [2, 2, -11], [-8, -6, 5], [3, -15, -8], [-5, -4, -5], [6, 5, -10], [22, -17, -3], [14, -18, 17], [30, 1, -8], [22, 0, 13], [17, -19, 3], [-17, 2, 5], [16, -2, 3], [25, 7, -4], [-4, -12, 5], [19, -17, -4], [-18, 1, -10], [17, -10, -6], [33, -8, 12], [11, -3, 1], [-26, -5, 4], [1, 5, 1], [-1, 24, -14], [6, -20, 0], [-11, -19, 15], [-18, -9, -2], [-4, 7, -1], [-23, 16, -5], [-22, 0, 12], [-19, -11, 8], [5, 2, 3], [22, -20, -4], [8, 3, 16], [12, -16, -8], [-10, 23, 5], [26, 2, -3], [-18, 1, -9], [-42, -3, -7], [11, 1, 2], [2, -24, -1], [29, -10, -11], [-17, -10, 6], [-42, 8, -4], [3, 20, 3], [4, -13, -16], [9, 5, -20], [27, -17, 1], [19, -4, -21], [5, 10, 1], [10, 5, -20], [-10, -19, 10], [-43, 7, -23], [-39, 10, 5], [32, -1, 9], [-7, 9, -19], [0, 16, 10], [18, 5, -13], [31, -4, -3], [-21, 3, 16], [2, 25, -6], [12, 6, 5], [-14, 7, 21], [-15, 4, 6], [-11, 4, 0], [1, -10, -28], [21, 8, -9], [5, -23, -3], [-2, 6, -10], [14, -1, -17], [4, -5, -4], [-26, 12, -2], [-45, 4, -4], [9, 17, 5], [-12, 2, -6], [39, -10, -13], [-18, 10, -13], [3, -26, 3], [11, -17, 19], [-30, 5, 6], [-28, 20, -9], [-30, -8, -3], [14, 6, 14], [6, -6, 10], [-24, 15, -22], [11, 4, 5], [-17, 25, -8], [12, 9, 10], [27, 16, 1], [-24, 3, 5], [35, 20, -5], [-7, -7, -11], [-7, -5, -25], [-28, -6, -1], [42, -13, -10], [-1, -12, 1], [-13, -6, -18], [-1, 35, -7], [4, -11, -8], [16, -24, 8], [14, -20, -7], [33, 3, 22], [34, -8, -16], [15, 11, -8], [3, 5, -1], [-6, 18, -16], [52, 10, -7], [19, 14, -7], [43, -2, 7], [23, 22, 0], [34, 10, 9], [21, -22, -2], [2, -16, 6], [14, -4, -17], [-23, 6, 20], [-6, 3, 4], [38, -3, -2], [-11, -7, -14], [9, -2, 29], [18, -11, 30], [8, 12, -17], [27, -6, -22], [-19, 30, -8], [-1, 12, -3], [9, 31, -2], [13, -7, -3], [38, 0, -13], [9, 19, -3], [9, -29, -6], [-39, -3, 3], [50, -18, -17], [6, -8, -16], [-10, -9, -15], [4, 9, 13], [20, -11, -7], [-6, -13, 27], [10, 25, -9], [-40, 5, -11], [-33, 8, -7], [-23, 5, -26], [2, -26, 3], [38, -6, 22], [-51, -5, -2], [-37, 17, 15], [-7, 2, -27], [17, -15, 33], [20, 8, 4], [72, 2, 0], [-3, 24, -1], [-3, -16, 16], [-26, -1, -4], [-29, 19, -6], [21, 20, 5], [15, 32, -8], [6, 3, 9], [19, 11, -12], [9, 28, -12], [18, -7, 13], [-15, 5, -29], [55, -12, 4], [26, -9, -5], [13, -25, 4], [28, -29, -4], [33, 2, 23], [-33, 4, 15], [26, -10, -5], [12, -14, 18], [-18, -5, 13], [-15, -19, 16], [15, -30, -19], [-32, 25, 8], [-26, -8, 6], [-10, -1, 1], [1, 4, 12], [-51, -3, 15], [34, 4, 17], [-24, 4, -13], [18, -22, -14], [-3, 27, -8], [8, -19, 14], [27, -15, 14], [-10, 11, 2], [1, -8, 2], [-24, 10, -25], [-33, -3, -25], [19, -18, 19], [4, 33, 2], [-36, 16, 10], [-54, 2, 3], [39, -15, -15], [14, -20, 17], [44, -5, 22], [-2, 6, 5], [0, -7, -26], [-3, -4, 7], [58, -9, -15], [-2, -24, 6], [14, 12, -28], [9, 33, -16], [32, -7, 11], [-11, 9, 17], [-19, 10, 29], [-9, 6, -25], [-8, 3, -4], [6, 22, -16], [43, -29, -9], [43, 3, -11], [-4, 2, -4], [18, 20, 3], [-17, -17, -4], [8, -9, 28], [-8, 3, 19], [-23, 13, 15], [-36, -19, -4], [36, -12, -8], [-14, 26, 1], [-56, 12, 18], [14, -21, -23], [18, -6, 38], [-61, 21, 20], [-76, 3, 8], [-37, 14, -27], [73, 8, -1], [-28, -13, 21], [-17, -6, 16], [-10, -21, 9], [-3, 9, 17], [-8, -26, -20], [8, -11, 13], [13, 5, -9], [-2, -7, -25], [8, -5, 6], [-59, -13, 11], [-69, 9, -16], [-27, 0, 1], [7, -35, -7], [-10, -28, 1], [20, -7, -33], [-17, -1, 11], [66, -14, 3], [-27, 36, 8], [-6, -15, -25], [-6, -8, 7], [21, 15, -5], [-64, 9, 19], [-26, -23, 23], [57, 5, -4], [16, 42, -15], [4, 24, 8], [6, -4, 10], [-3, 23, 6], [-18, 34, -28], [-23, -6, 24], [25, -14, 5], [-72, 13, -1], [-62, 2, 2], [13, 15, -32], [-29, -2, -28], [-55, -9, 9], [38, -9, 11], [-24, 45, 4], [-73, -1, -1], [-11, -9, -22], [37, -26, -21], [-15, 9, -9], [-9, 7, -23], [-42, -13, 21], [-50, 1, 21], [34, -29, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4851_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4851_2_a_bn();".
// 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_4851_2_a_bn(:prec:=3)
    chi := MakeCharacter_4851_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_4851_2_a_bn();".
// 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_4851_2_a_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4851_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, -5, 0, 1]>,<5,R![-5, -7, 2, 1]>,<13,R![-35, 36, -11, 1]>,<17,R![-7, -2, 3, 1]>],Snew);
    return Vf;
end function;