
// Make newform 4851.2.a.ba in Magma, downloaded from the LMFDB on 23 April 2024.
// 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_ba();".
// 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_ba();".
// 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 := [-5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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)
    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+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, 1], [0, 0], [3, 0], [0, 0], [1, 0], [-1, 0], [4, -2], [3, -2], [2, -2], [1, -4], [0, 2], [1, 0], [-4, 4], [-2, -2], [5, 2], [6, -2], [1, -2], [-10, 0], [5, -2], [-4, 4], [-7, 0], [0, -4], [-8, 2], [2, -4], [6, 2], [8, -2], [-6, -4], [9, -2], [0, -2], [-2, 4], [-8, 2], [-16, 0], [8, -2], [-4, 4], [1, 4], [0, -2], [-12, 4], [5, -2], [6, 6], [0, 4], [0, 8], [2, -2], [-8, 2], [-6, -2], [2, 4], [6, -2], [-6, 4], [-6, 0], [-22, 0], [10, 6], [14, 0], [-13, 2], [3, -4], [11, 2], [19, 0], [5, -10], [10, 0], [-9, -2], [-12, 4], [13, -8], [-17, 6], [0, 0], [0, 0], [0, -8], [-16, 2], [26, 2], [4, 4], [14, -2], [-12, 8], [15, 0], [17, -4], [16, 4], [-22, 0], [-24, 6], [-9, 2], [-4, -8], [12, -8], [28, -2], [-14, 10], [6, 8], [-1, -2], [-5, 16], [-7, 6], [4, -4], [-21, -2], [16, -4], [4, -8], [-2, 10], [4, -8], [5, -2], [-35, 2], [-12, 2], [-4, -8], [5, -10], [31, -6], [10, 6], [6, 0], [23, 4], [-19, -2], [4, -8], [8, 0], [5, 8], [-34, 2], [-2, -16], [36, 4], [4, -10], [27, -6], [16, 0], [0, 4], [-3, 12], [11, -6], [-16, 6], [-10, -4], [4, -14], [16, -4], [-40, 2], [-6, 18], [7, 10], [16, -8], [25, -2], [14, -14], [22, 6], [-38, 4], [8, -14], [-10, 0], [-10, 0], [-1, -16], [11, 14], [-2, -10], [34, 0], [-14, 10], [21, -18], [9, 10], [13, -8], [28, -8], [15, 12], [-1, -4], [-35, 2], [-31, 12], [5, 8], [13, 6], [47, 0], [-27, 2], [9, 10], [40, 0], [25, 10], [-10, 4], [-22, -8], [-12, 0], [-18, -2], [-12, -12], [-9, 16], [-45, 6], [28, -6], [28, -8], [-18, -12], [-26, -10], [-7, 8], [-10, 0], [28, -4], [-38, 4], [21, -4], [-6, 4], [-9, 2], [-34, -6], [-12, -4], [-31, -6], [18, 16], [30, 0], [-5, 0], [-19, 2], [30, 4], [0, 0], [-12, 18], [-22, -8], [26, -16], [-34, 2], [-2, 20], [17, -10], [-35, 12], [-23, 2], [-18, -2], [10, -12], [-8, -20], [36, 0], [-1, 0], [24, 4], [16, -10], [12, 2], [-12, -10], [10, 16], [-1, 18], [17, -10], [-7, 8], [-23, 10], [-11, -4], [32, -18], [34, 12], [-22, 6], [-10, -10], [2, -16], [-43, -6], [6, -14], [-15, -12], [-16, 24], [-22, 16], [4, 20], [-9, -6], [-10, 12], [-46, -2], [-2, 8], [1, 24], [19, -6], [36, -4], [2, 4], [2, -12], [-4, -8], [16, -6], [12, -10], [-40, -6], [32, 10], [1, 18], [14, 2], [-21, 14], [-8, 4], [-34, 8], [-31, 8], [42, 4], [-43, 2], [-17, -10], [18, -24], [17, -6], [-14, 24], [21, 0], [-14, 10], [-30, 20], [4, -10], [-31, 8], [12, 6], [-7, -10], [-10, 14], [-12, 16], [-16, 26], [2, 12], [44, 8], [36, 12], [16, -28], [21, 6], [28, -12], [13, 18], [42, 8], [-38, -2], [-22, 24], [30, -4], [38, -4], [-36, 0], [41, -4], [-11, 14], [-15, -8], [5, 12], [-66, 4], [-12, 8], [6, -8], [21, 8], [-3, 20], [16, 10], [-38, 4], [-21, 0], [12, 12], [-34, 16], [8, 24], [17, -10], [42, 6], [20, -22], [5, -20], [-8, -18], [12, 16], [-34, 4], [20, -14], [14, 6], [-42, -6], [8, -16], [48, -6], [-39, -12], [-29, -6], [14, -14], [55, -8], [-2, 12], [24, -2], [52, -14], [24, 6], [-36, -16], [-67, 4], [50, 2], [-16, 12], [66, 2], [46, 12], [2, 8], [6, -2], [-29, 2], [-25, 0], [-17, -20], [-22, 22], [42, -10], [-7, 22], [-5, 12], [50, 10], [34, -14], [34, -14], [14, 6], [2, -4], [-24, -12], [-63, -4], [12, 24], [-56, 8], [6, 16], [-40, 20], [56, -16], [18, 10], [-34, 4], [55, -10], [-5, 4], [-18, 12], [-52, 4], [3, -12], [0, 12], [12, -32], [34, -10], [19, -2], [33, 4], [-28, 0], [-12, 4], [27, 2], [-16, 6], [38, -14], [-47, 22], [-13, -4], [-34, 10], [-28, 22], [-28, 16], [-6, 20], [8, 10], [24, -6], [63, -10], [36, -22], [-19, -2], [-70, -4], [28, 4], [32, -12], [-26, -16], [24, -4], [-15, -4], [-72, -4], [-29, 10], [-32, 6], [-28, -10], [-11, 22], [40, -8], [1, -8], [-42, 22], [-72, 8], [36, -12], [-12, 4], [17, -2], [-20, 0], [-49, 4], [52, 2], [14, 4], [26, 16], [-64, 14], [-8, -10], [31, -30], [-57, 0], [-4, -24], [-69, -4], [11, -6], [-1, -6], [-51, 4], [58, -8], [30, 24], [45, 12], [2, 2], [-50, -4], [49, 4], [36, -16], [49, 2], [-39, -10], [42, -20], [-36, -10], [-29, -2], [-15, -4], [2, -20], [-46, -14], [-14, 2], [-18, 4], [16, 30], [-37, 26], [29, -16], [68, -20], [81, -8], [23, -8], [-32, 10], [-60, 2], [26, -24], [-54, 18], [-35, -14], [19, 8], [58, 8], [-46, -4], [28, 4], [-34, -12], [70, -12], [-16, 36], [26, -22], [-22, 28], [7, -36], [1, 18], [-28, -18], [10, -12], [8, -22], [36, 22], [31, -4], [-72, -6], [-40, 14], [-38, -16], [-1, -4], [40, -22], [25, 4], [-2, 0], [0, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4851_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4851_2_a_ba();".
// 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_ba(:prec:=2)
    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_ba();".
// 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_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4851_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, -1, 1]>,<5,R![-3, 1]>,<13,R![1, 1]>,<17,R![-12, -6, 1]>],Snew);
    return Vf;
end function;