
// Make newform 9800.2.a.q in Magma, downloaded from the LMFDB on 25 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_q();".
// 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_q();".
// 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], [-1], [0], [0], [-1], [6], [-7], [-1], [-8], [-6], [-4], [-8], [5], [0], [6], [-4], [4], [-6], [5], [14], [15], [14], [1], [3], [-6], [0], [4], [-9], [-10], [-1], [-22], [-16], [-3], [-1], [4], [8], [4], [25], [-20], [-10], [15], [22], [18], [-11], [-6], [10], [-19], [10], [-28], [22], [-10], [12], [13], [25], [-18], [14], [6], [14], [2], [30], [1], [6], [15], [10], [-22], [-28], [3], [19], [23], [-8], [-14], [-6], [8], [20], [-3], [6], [-16], [30], [13], [-27], [35], [36], [-4], [9], [-20], [25], [-23], [-7], [-2], [24], [28], [-2], [22], [-20], [28], [14], [30], [-27], [23], [14], [-41], [-36], [36], [-11], [-44], [19], [-15], [-9], [-8], [35], [36], [-14], [-18], [28], [-8], [2], [40], [-8], [36], [29], [18], [-30], [-48], [-9], [43], [24], [-12], [0], [-14], [-16], [-20], [-48], [42], [-42], [-27], [53], [-12], [36], [22], [-26], [-28], [0], [10], [13], [-20], [-34], [-14], [45], [23], [-4], [-48], [-26], [11], [18], [4], [-12], [-42], [-14], [-1], [48], [20], [25], [-2], [23], [45], [36], [20], [2], [-23], [14], [-15], [-34], [14], [-1], [-36], [-21], [21], [-36], [-58], [54], [-44], [31], [54], [-21], [66], [56], [10], [-49], [25], [-30], [-1], [33], [-27], [16], [-9], [-42], [3], [-26], [43], [34], [-54], [-2], [-58], [23], [-28], [12], [-30], [56], [30], [8], [42], [0], [-64], [-37], [14], [61], [-8], [61], [8], [-36], [20], [22], [9], [-20], [55], [-32], [-57], [26], [32], [-39], [38], [-75], [-42], [35], [20], [12], [45], [-42], [29], [-60], [39], [65], [2], [58], [-49], [30], [34], [-60], [-7], [-34], [22], [-57], [42], [51], [-4], [-39], [20], [13], [-12], [-43], [46], [-43], [-22], [44], [57], [35], [-6], [-2], [27], [8], [28], [41], [-17], [-42], [38], [-48], [-33], [30], [22], [7], [-66], [32], [-16], [-30], [-35], [36], [30], [-6], [20], [50], [26], [-28], [-9], [-51], [-22], [-36], [-50], [-48], [-68], [73], [37], [-18], [22], [-33], [-59], [-7], [-4], [-56], [36], [-6], [62], [52], [-67], [-32], [66], [42], [7], [-30], [-41], [-61], [-80], [41], [0], [-64], [18], [34], [29], [-76], [36], [-76], [6], [26], [-44], [39], [51], [-20], [-74], [6], [-57], [58], [-76], [9], [-48], [8], [22], [28], [-88], [77], [6], [34], [-15], [50], [78], [26], [4], [27], [-14], [15], [-78], [-34], [24], [63], [-82], [-40], [-23], [63], [-60], [16], [-79], [-20], [71], [-82], [54], [78], [76], [-9], [78], [-49], [-79], [22], [88], [-15], [4], [-85], [76], [-18], [68], [-4], [12], [-30], [23], [-48], [25], [-51], [-18], [5], [-44], [-31], [-97], [0], [50], [41], [74], [33], [-48], [50], [60], [-2], [-12], [-75], [-29], [84], [25], [32], [-85], [90], [18], [-14], [-27], [-6], [40], [42], [-36], [69], [-79], [-28], [52], [-17], [67], [-96], [-35], [-87], [49], [-58], [-68], [-55], [-21], [14], [-52], [22], [72], [63], [54], [-16], [87], [7], [-17], [26], [78], [-14], [-12], [102], [-111], [70], [22], [-94], [80], [-15], [8], [91], [44], [-40], [-108], [-16], [-81], [22], [-82], [-5], [-63], [13], [0], [53], [-50], [27]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<11,R![1, 1]>,<13,R![-6, 1]>,<19,R![1, 1]>,<23,R![8, 1]>],Snew);
    return Vf;
end function;