
// Make newform 9800.2.a.bg in Magma, downloaded from the LMFDB on 19 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_bg();".
// 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_bg();".
// 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_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], [2], [0], [0], [-3], [-2], [4], [0], [-3], [1], [-2], [7], [2], [-1], [-12], [6], [6], [6], [-7], [-3], [-2], [-5], [-6], [-18], [-18], [-6], [8], [-8], [-7], [13], [-15], [16], [-10], [-22], [15], [-15], [10], [-4], [-14], [-12], [4], [22], [8], [9], [-3], [14], [-20], [-10], [-8], [-20], [15], [-24], [-4], [-12], [-10], [31], [-26], [14], [-22], [-27], [16], [-14], [4], [20], [4], [5], [19], [22], [-9], [28], [-6], [21], [2], [-27], [-17], [14], [-15], [32], [25], [-4], [-2], [-29], [-24], [28], [14], [-12], [15], [33], [18], [-4], [12], [-2], [19], [9], [-32], [12], [2], [32], [36], [-9], [-23], [-27], [16], [17], [-33], [0], [22], [36], [-5], [-40], [22], [23], [3], [-46], [-1], [-7], [12], [-20], [26], [0], [40], [22], [36], [-21], [-6], [-30], [6], [6], [8], [0], [33], [48], [-4], [27], [-48], [-14], [22], [-6], [36], [3], [-4], [-22], [25], [-49], [4], [22], [-52], [22], [8], [-21], [18], [-54], [-51], [50], [-44], [19], [-49], [6], [-2], [-30], [-52], [-11], [8], [20], [3], [34], [39], [-12], [-33], [36], [23], [40], [7], [30], [-50], [0], [-13], [-2], [14], [10], [11], [-16], [19], [-36], [-7], [28], [-43], [-4], [21], [-42], [48], [-39], [21], [48], [43], [6], [29], [-17], [-20], [-14], [-6], [-40], [-34], [50], [-30], [10], [-60], [12], [71], [-46], [-1], [68], [-41], [32], [-18], [-8], [-1], [56], [-31], [23], [-27], [8], [-45], [-39], [-62], [-34], [66], [-49], [16], [-31], [-45], [6], [44], [39], [-70], [22], [34], [29], [21], [56], [-71], [22], [66], [-65], [-30], [-52], [-52], [24], [44], [48], [78], [22], [40], [-20], [4], [-20], [65], [12], [-32], [-72], [64], [-29], [-46], [-54], [-2], [-80], [15], [6], [-69], [-33], [40], [52], [12], [-13], [-28], [-60], [27], [50], [-73], [-24], [-10], [68], [-66], [52], [-34], [-71], [-59], [-51], [64], [-60], [35], [-50], [39], [-42], [-19], [16], [42], [-30], [-30], [-32], [46], [-6], [31], [-9], [-37], [38], [17], [30], [31], [-2], [54], [-18], [-7], [4], [49], [-28], [-4], [88], [-38], [29], [-44], [74], [12], [1], [9], [-14], [-41], [32], [-51], [-25], [26], [67], [-64], [56], [44], [36], [-11], [-6], [14], [42], [6], [-29], [58], [-21], [67], [45], [-62], [55], [-28], [-18], [-50], [41], [50], [-44], [-49], [48], [70], [2], [-62], [60], [1], [10], [-4], [29], [-56], [-54], [52], [67], [71], [-67], [66], [-79], [66], [94], [-63], [88], [53], [-56], [-36], [46], [98], [-27], [85], [-15], [58], [40], [29], [-2], [31], [82], [-71], [-46], [77], [76], [-31], [-9], [-56], [78], [-20], [-3], [-18], [21], [-23], [30], [-96], [-18], [-63], [46], [70], [34], [50], [63], [-12], [-13], [-13], [-68], [20], [-102], [72], [54], [-34], [82], [36], [-92], [-70], [58], [67], [-22], [4], [76], [-26], [-67], [-96], [-24], [4], [12], [17], [-75], [60], [68], [-26], [-66], [-59], [5], [102], [-78], [68], [-32], [20], [-38], [92], [-84], [-69], [0], [-35], [-1], [-87], [-6], [-74], [-59], [25], [-89], [81], [-90], [-32], [-5], [-71], [12], [-54], [-10], [23], [-68], [-56], [11]];
    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_bg();".
// 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_bg(:prec:=1)
    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_bg();".
// 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_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<11,R![3, 1]>,<13,R![2, 1]>,<19,R![0, 1]>,<23,R![3, 1]>],Snew);
    return Vf;
end function;