
// Make newform 9800.2.a.cl in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9800_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_9800_2_a_cl();".
// 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_9800_2_a_cl();".
// 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 := [14, 8, -7, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, -1, 1, 0], [0, -4, -1, 1]];
    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_9800_a();"
function MakeCharacter_9800_a()
    N := 9800;
    order := 1;
    char_gens := [7351, 4901, 1177, 5001];
    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_9800_a_Hecke(Kf)
    return MakeCharacter_9800_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, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, -1, 0], [-2, -1, -1, 1], [-2, 1, 0, -2], [0, 0, -1, -1], [2, -2, -2, 1], [0, -1, -3, 0], [-2, -2, -2, 1], [0, 0, -2, -1], [4, -2, -1, -2], [2, 0, 2, -2], [0, 1, 1, 1], [0, 2, -2, 1], [2, 0, 5, 0], [6, -2, 2, 2], [2, 0, 6, -1], [0, 2, 0, -4], [-2, -4, 5, -1], [6, -1, 5, 0], [-8, -2, 1, -1], [10, 0, -3, 1], [-6, -1, 2, 2], [6, -4, 0, -2], [4, -1, -5, -1], [4, 2, -6, 2], [4, -5, 1, 0], [-2, 2, 4, -5], [-2, 4, 0, -1], [2, 2, 1, 0], [0, 0, -8, 0], [6, -2, -5, 0], [10, -6, -2, 2], [6, -5, -5, 2], [-6, 0, -2, 4], [8, 4, -8, 1], [4, 5, -3, 1], [-10, 1, 1, -3], [12, -6, -2, 2], [4, -8, 0, 3], [-6, -5, -7, 6], [0, -4, 0, 4], [0, -2, 2, 3], [-10, -2, 6, 1], [6, -5, -1, -4], [16, -3, 3, 3], [0, 5, 8, -4], [0, 4, -4, 1], [8, -8, 4, 2], [6, -7, 5, 2], [4, -4, -3, 4], [2, 6, 7, -6], [-10, -2, -3, 1], [16, 2, -6, 0], [0, 6, 14, -3], [-20, 4, -4, -4], [12, 0, 4, 1], [-10, 3, -5, 2], [8, -1, 4, -4], [2, 1, -13, -1], [0, 7, -2, 2], [8, 2, -6, 4], [6, -3, -6, 2], [2, 0, 10, -2], [-2, 4, 6, 4], [0, 0, -14, -1], [0, 4, -4, -3], [-10, 8, -10, -2], [-10, -3, 12, -4], [8, 0, -20, 0], [8, -1, -7, -5], [10, -2, 8, -4], [-2, 4, -6, -2], [-8, -4, -18, 4], [8, 5, 11, -6], [10, -1, -5, -1], [-4, 3, -9, -4], [-2, -6, 11, 3], [2, 0, -5, 2], [-4, -1, 15, 2], [-2, 1, 13, -6], [2, -2, -13, 5], [-18, -4, -6, 3], [12, 2, 2, -2], [0, 3, -1, -8], [14, 0, 6, 3], [-10, -2, -12, -2], [12, 0, -8, 6], [-12, 5, 6, 4], [2, 0, 0, -1], [-6, -6, -10, 1], [10, 5, -3, 4], [18, -15, 1, 4], [4, 3, 5, -1], [0, 6, -10, 1], [-6, 10, 3, -3], [4, 2, -1, -1], [4, -7, 5, 2], [-2, 10, 12, -8], [-6, -2, -12, 0], [-20, 8, -1, -1], [14, -2, 2, 6], [26, -8, 2, 0], [2, -7, -6, 6], [-12, 2, -15, -3], [2, 5, 12, -6], [2, 3, 17, -4], [2, 10, 5, -5], [12, 1, 7, 5], [-6, 4, 16, -4], [4, 8, 10, -5], [-6, 2, 17, -8], [10, 1, -5, 4], [-14, -2, 6, -8], [-12, 5, 6, -12], [-24, -4, -10, 4], [6, 4, 0, -8], [16, -5, -7, 2], [6, 6, -2, -2], [4, 0, -12, 2], [-14, 3, -5, -1], [16, -12, -4, -2], [24, 2, -5, -5], [-8, 3, -11, 2], [0, 3, 1, -4], [14, 0, 8, -5], [4, 2, 12, -2], [10, 5, 3, -1], [8, 5, 7, -10], [6, -8, 0, 7], [2, -5, 3, -2], [26, -10, -10, -2], [8, 8, 7, 2], [18, -6, -11, -3], [-6, 19, 9, -5], [0, -1, 16, 0], [-2, 13, 9, -3], [2, -1, 11, 6], [6, -2, 19, 4], [-24, 19, -1, -4], [30, -8, -2, -1], [-2, -2, 8, -1], [-20, 20, 6, 1], [2, 4, 0, -1], [-22, 8, -10, -2], [10, -2, 1, -3], [16, 2, -9, 3], [2, 0, 14, -5], [2, -12, -16, 8], [14, 8, -17, 3], [0, 0, 24, 2], [-32, 2, 8, -2], [22, -14, 0, -1], [-8, -16, -12, 4], [-18, 9, 11, 8], [4, 14, 1, -2], [6, 13, 20, -4], [2, -6, 10, 4], [24, 2, -10, 1], [-16, -6, 12, -7], [8, -14, 0, 0], [34, -4, -7, 0], [-8, -10, 2, -2], [-24, -1, 3, -1], [16, 6, 0, -10], [-30, 7, -11, -7], [2, -1, 19, -10], [-14, -2, -8, 8], [8, -2, -2, -8], [22, 4, 6, 4], [6, 5, -5, -6], [-18, 6, 16, 3], [8, -2, -8, 6], [10, 0, 11, 1], [14, 5, -15, 2], [-16, 11, -3, 4], [16, -2, 14, 4], [2, 6, 10, -10], [-24, 16, -6, -2], [40, -6, 7, 3], [-14, -8, -20, 6], [-10, 14, -9, -7], [4, -8, -14, -2], [4, 14, -14, -1], [16, -18, 2, -3], [4, -13, -24, 6], [-14, -11, -3, 0], [12, -8, -2, 0], [-22, 2, 9, -5], [4, -18, 2, 6], [6, -1, 11, -12], [12, -4, 8, -1], [-8, 0, -4, 2], [10, -13, 2, 12], [20, -3, -19, -4], [-12, -14, -4, -1], [-18, 11, -4, -6], [8, -5, 9, -5], [-10, 2, 2, 6], [-24, -10, 24, -4], [10, 0, -12, 8], [30, -2, 1, -11], [-4, -4, -11, 11], [14, -1, 5, 5], [2, 12, 16, -3], [-30, 18, 0, 0], [-8, 4, -16, -8], [20, -18, -7, -1], [-8, 14, -4, 5], [-4, -6, -16, 15], [14, 4, 0, -1], [16, -7, 18, -6], [12, -6, -30, 0], [-22, 12, -11, -1], [2, 18, 12, -9], [-12, 2, 5, -6], [-16, -6, 22, -2], [28, -4, 22, 3], [6, -4, -22, 14], [-12, 10, 18, 0], [-32, -1, -13, 0], [4, 10, -8, 2], [8, 9, -18, -2], [-2, 0, 22, 2], [22, 1, 0, 0], [2, 1, 13, -8], [20, 7, -3, -5], [-2, 4, -2, -2], [-2, -10, 6, 10], [-32, -8, 11, -1], [44, 2, 16, 0], [-16, -13, 27, -6], [-12, -11, 22, -8], [-28, -1, -19, 7], [12, -6, 21, 6], [-14, 6, 12, -4], [-2, -3, 1, -6], [6, -8, -14, -3], [2, 2, 8, 6], [-12, -18, -9, 9], [8, -13, -11, 7], [-2, 18, -2, -6], [-2, -13, -2, 6], [0, 2, 28, -2], [28, 0, 20, -8], [4, 8, 23, -7], [-12, 19, 9, 2], [-32, 14, -14, -14], [18, 4, -14, 6], [22, 6, 7, -7], [-30, -12, 6, 3], [18, -8, 1, 1], [22, -8, -12, 6], [14, -9, -9, 16], [38, -10, 12, 12], [-4, 8, 11, -13], [-26, 9, -5, 1], [-22, 13, 14, -4], [16, -14, 20, 4], [44, -16, -8, 3], [52, 4, -6, 1], [42, -7, -5, 5], [-26, 18, 13, -5], [-38, 2, 5, -12], [-12, -17, -25, 4], [-38, 10, -1, -1], [10, 8, -2, -9], [-18, -20, -4, 4], [28, -12, 0, -5], [40, 10, -18, 5], [18, -9, -6, -12], [-44, 18, 2, 0], [-18, 18, 15, -3], [44, -7, -5, 7], [26, -14, 12, 4], [-34, -6, 14, -8], [6, 11, -9, 6], [-36, 24, -7, -13], [4, -13, -3, -5], [30, 19, -19, 6], [4, -12, -2, 14], [-16, -2, 24, 1], [4, -17, -16, 14], [34, 1, 9, 2], [-2, 16, 0, 6], [-14, 0, -2, 2], [-6, 6, -34, -5], [-16, -6, 19, -18], [-20, 9, 25, -4], [-4, 27, 14, -12], [-44, -6, 0, 1], [-30, -4, -13, 10], [-10, -12, 6, 6], [-28, 8, -8, 9], [6, -12, -2, 11], [-10, -3, -3, 1], [-32, 22, 11, 7], [28, 9, 24, -10], [-10, 3, -2, 4], [-32, 0, -2, -3], [-50, -7, -17, 4], [-18, -14, -24, 6], [28, -1, -11, 2], [18, 2, 4, -1], [6, -20, 14, -8], [6, 14, -14, 0], [-24, 6, 2, 16], [-20, 6, 22, -1], [-20, 4, 0, -2], [20, 11, 5, -6], [4, -12, -16, 8], [-26, -12, -18, 4], [-32, 14, -14, -8], [-18, -8, -21, 15], [32, -8, 7, 3], [-6, -25, -11, 8], [-6, -13, 14, 4], [-26, 9, 13, -16], [-16, 0, -1, 7], [-14, -4, 10, 5], [12, -6, -8, 3], [30, -16, 22, 5], [-8, 6, -10, 6], [-8, 14, -1, -4], [-12, 5, -13, -2], [-16, -14, 7, -3], [-6, -8, -24, 9], [-20, 6, -12, -3], [34, -10, -2, -6], [-24, 10, -24, 3], [-10, -16, 0, 9], [20, -3, -24, 10], [12, 22, 30, -18], [4, 5, -36, 10], [-8, -1, -35, 2], [-30, -4, 27, 1], [10, 2, 33, -9], [32, -15, -23, -7], [-20, -4, -4, -1], [-44, 6, 12, 5], [4, 6, 18, 5], [-6, 3, -3, -14], [0, 4, -18, -7], [2, 10, 8, 6], [0, -2, 10, -17], [10, -6, 8, 14], [-6, 0, -34, -3], [-18, -17, -3, 3], [14, 24, -23, 10], [42, 2, -14, 2], [16, -7, -15, 14], [44, -13, 3, 3], [-4, -13, -5, 12], [-22, 20, 11, -11], [-24, 10, 24, 6], [-18, 14, 21, -14], [-2, -8, -20, 0], [10, 10, 18, 1], [18, 10, -12, -4], [16, -22, -23, 2], [-14, -4, 12, -3], [38, -5, 15, 0], [-24, -14, -7, 7], [-2, 0, 4, -2], [42, -11, 1, -11], [-8, 14, -4, -12], [-6, -1, -45, 2], [2, 1, -27, 0], [32, -14, -19, -11], [-2, 0, -14, 3], [4, 17, 29, -6], [8, 14, -14, 8], [38, 12, -18, 0], [-24, 18, 9, -15], [-46, 19, -5, 2], [10, 19, -14, -4], [-32, 8, -7, -6], [-26, 5, 0, 10], [-34, 16, -10, -18], [-42, 14, 24, 13], [10, 8, 34, -19], [-32, 12, 2, 5], [18, 4, -13, -2], [-24, 23, 21, -3], [50, -5, 1, -6], [-58, 5, -25, -5], [-56, 8, -4, -2], [-48, -1, 15, 3], [4, -3, 5, 4], [30, 7, 13, -3], [22, -17, -5, 12], [-20, 10, -19, -11], [6, -5, 27, -14], [0, -12, -4, 4], [-36, -8, -26, 8], [-4, -6, 13, 10], [22, 1, -15, 14], [14, -6, -24, -10], [26, 2, 6, 8], [12, -14, -24, -5], [-2, -10, -12, 17], [30, -19, -12, 2], [-22, 14, -4, -18], [0, 14, 4, -4], [12, -28, -8, 5], [-14, -7, 9, 8], [-32, 26, 3, -3], [12, -22, 29, 7], [-30, 15, -4, 12], [-38, 20, 4, -10], [-12, 16, -8, 5], [20, 6, -18, -2], [50, -2, -4, -15], [-22, 22, 24, -12], [-14, -13, 33, -10], [-8, 28, -18, 0], [-22, 10, 1, 5], [-16, 7, -11, -11], [-12, 15, -7, 4], [-14, 4, 44, 0], [20, -14, 10, 0], [-16, 6, 3, -17], [6, -3, 16, -6], [2, -31, 3, 7], [-28, -14, -14, -5], [12, -28, -8, 18], [-26, 8, 11, 12], [28, 8, -42, 14], [-4, -8, -33, 4], [22, -8, 30, -2], [14, 6, -44, 2], [-8, 25, -29, -3], [2, -29, 17, -1], [26, -12, 3, 11], [28, -23, 1, 16], [8, 13, 31, -28], [32, 22, 10, -7], [32, -32, -10, -8], [-4, -9, 14, -16], [-32, 5, 5, 0], [-22, 6, 28, -14], [-22, 17, 3, -18], [26, 0, 43, -7], [-64, 10, -14, -4], [0, 25, -16, 4], [44, -12, -30, 4], [14, -28, 3, 1], [-32, -6, -40, 9], [30, -18, 12, 6], [-8, 0, -48, 8], [-14, 2, 16, -10], [-52, 6, 1, 10], [36, -6, 38, 2], [-16, 9, -7, 8], [16, -17, 21, -2], [12, 18, -7, -9], [30, -5, -31, -11], [-4, 14, -38, -2], [12, -3, 55, 8], [10, -1, 3, -4], [-4, 11, -3, -8], [24, 17, -19, -4], [-48, 1, 10, 10], [42, 8, 2, -1], [-44, 2, -2, 6], [8, -3, 30, -2], [-6, -2, -10, 8], [-48, 18, 5, 1], [4, 12, 14, -12], [34, -10, 8, -15], [18, 2, -8, -19], [34, 23, 23, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9800_2_a_cl();".
// 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_9800_2_a_cl(:prec:=4)
    chi := MakeCharacter_9800_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(3361) - 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_9800_2_a_cl();".
// 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_9800_2_a_cl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![14, -8, -7, 2, 1]>,<11,R![-4, 20, -11, -2, 1]>,<13,R![-124, -52, 19, 10, 1]>,<19,R![8, 24, -26, 0, 1]>,<23,R![16, -48, -38, -4, 1]>],Snew);
    return Vf;
end function;