
// Make newform 9802.2.a.d in Magma, downloaded from the LMFDB on 30 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_9802_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_9802_2_a_d();".
// 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_9802_2_a_d();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_9802_a();"
function MakeCharacter_9802_a()
    N := 9802;
    order := 1;
    char_gens := [5917, 8789];
    v := [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_9802_a_Hecke( : Kf := Kf)
    return MakeCharacter_9802_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 := [[1], [-3], [3], [2], [1], [0], [-4], [8], [0], [-1], [-3], [8], [2], [7], [-11], [1], [4], [4], [4], [2], [12], [-7], [0], [6], [6], [8], [-6], [-2], [-1], [18], [-8], [12], [20], [0], [-3], [-10], [22], [-19], [22], [-14], [-14], [-13], [0], [-10], [-2], [-2], [-25], [-26], [18], [-14], [-25], [-20], [-17], [-7], [21], [-17], [20], [-13], [14], [13], [0], [34], [29], [0], [25], [0], [-3], [4], [18], [19], [2], [-9], [-16], [-1], [28], [34], [16], [-19], [5], [-22], [-14], [28], [20], [-16], [-8], [12], [-22], [10], [30], [16], [23], [11], [-2], [5], [8], [-19], [-21], [27], [16], [8], [-2], [18], [-9], [-30], [-36], [44], [0], [17], [13], [-10], [29], [27], [-12], [-31], [-22], [36], [-2], [-20], [-14], [21], [-46], [1], [-18], [16], [-4], [31], [-35], [-6], [32], [48], [-1], [28], [32], [-28], [-34], [-48], [6], [-46], [12], [24], [-6], [37], [-16], [-23], [-28], [45], [-30], [-11], [-51], [-6], [23], [42], [-42], [-1], [-28], [-3], [-6], [-18], [14], [15], [-51], [15], [-59], [-52], [-37], [-39], [26], [16], [15], [-20], [10], [47], [30], [26], [4], [6], [16], [-24], [9], [18], [3], [15], [34], [-33], [0], [39], [32], [52], [-20], [50], [13], [-9], [31], [-32], [-12], [-62], [-8], [-14], [-9], [-52], [66], [8], [14], [2], [-20], [39], [11], [-18], [-3], [-67], [-22], [-22], [17], [1], [24], [-53], [40], [-60], [32], [-58], [-52], [58], [66], [27], [-56], [-6], [48], [56], [-37], [-30], [-16], [-56], [-28], [-48], [58], [48], [28], [-6], [-58], [-45], [-17], [-34], [74], [4], [30], [-6], [8], [-40], [-46], [33], [8], [-15], [-9], [24], [-14], [12], [-54], [3], [14], [39], [-80], [-1], [14], [-24], [12], [-45], [-24], [56], [-4], [-39], [21], [-54], [-5], [40], [8], [-23], [12], [-35], [20], [40], [-30], [-30], [16], [38], [-52], [36], [32], [-58], [46], [9], [-73], [-24], [-77], [47], [-22], [-13], [63], [81], [34], [-20], [-8], [-38], [-40], [-1], [-28], [-33], [63], [31], [-34], [-41], [44], [30], [-78], [8], [-90], [11], [-52], [54], [-25], [-12], [-11], [-9], [62], [43], [-3], [-38], [74], [4], [13], [9], [-84], [43], [36], [-44], [43], [16], [-62], [38], [-18], [-44], [-64], [-14], [20], [32], [-18], [19], [-60], [-52], [-77], [-86], [52], [-46], [90], [64], [-28], [-68], [9], [-66], [-44], [-34], [61], [-63], [-27], [42], [-77], [10], [72], [-14], [26], [-93], [-6], [-78], [30], [-68], [-71], [76], [21], [40], [-41], [-3], [-40], [-9], [-71], [60], [38], [42], [-56], [71], [-55], [-92], [-70], [-1], [-37], [-3], [17], [69], [36], [-12], [-58], [-30], [-14], [18], [-68], [27], [-75], [-20], [-73], [0], [81], [-60], [-98], [81], [-14], [40], [18], [-90], [-13], [98], [52], [41], [-94], [22], [-73], [57], [65], [-45], [4], [32], [108], [-79]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9802_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9802_2_a_d();".
// 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_9802_2_a_d(:prec:=1)
    chi := MakeCharacter_9802_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_9802_2_a_d();".
// 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_9802_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9802_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<5,R![-3, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;