
// Make newform 1003.2.a.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1003_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_1003_2_a_a();".
// 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_1003_2_a_a();".
// 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_1003_a();"
function MakeCharacter_1003_a()
    N := 1003;
    order := 1;
    char_gens := [768, 120];
    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_1003_a_Hecke(Kf)
    return MakeCharacter_1003_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 := [[-1], [3], [1], [1], [0], [-2], [1], [5], [4], [-9], [10], [-8], [-3], [10], [12], [3], [-1], [8], [-8], [8], [-6], [-9], [-16], [-14], [2], [8], [14], [11], [6], [16], [-7], [10], [-9], [-8], [-4], [-4], [-20], [-12], [-7], [-14], [24], [7], [8], [-19], [18], [-11], [-22], [-12], [-18], [18], [12], [-1], [3], [15], [-27], [-1], [-14], [-7], [-7], [-33], [-16], [21], [3], [-11], [-28], [2], [13], [-14], [-28], [-6], [24], [-15], [-32], [6], [25], [12], [18], [12], [-14], [14], [14], [-26], [36], [-25], [-8], [22], [-5], [24], [-14], [-4], [18], [24], [17], [-13], [35], [12], [-8], [2], [-21], [12], [28], [27], [14], [10], [4], [37], [22], [-27], [21], [-8], [-19], [34], [27], [-3], [28], [30], [35], [23], [33], [28], [35], [-34], [-6], [14], [4], [-42], [-13], [-14], [-8], [-14], [-38], [24], [-46], [-47], [-7], [4], [46], [52], [-2], [26], [4], [-18], [44], [-36], [-7], [26], [-1], [-20], [-8], [46], [-33], [-28], [27], [-42], [23], [-7], [0], [-12], [22], [18], [-37], [50], [18], [-33], [-24], [-6], [0], [7], [-44], [0], [-24], [-4], [-3], [-2], [25], [-1], [41], [-6], [-24], [-5], [-51], [-45], [-4], [3], [20], [30], [58], [-6], [22], [30], [-16], [-22], [-56], [-22], [27], [14], [10], [8], [-36], [-14], [-34], [-57], [38], [-38], [7], [8], [58], [-18], [-6], [32], [-28], [-69], [27], [-33], [25], [40], [16], [-18], [6], [-50], [14], [36], [-72], [5], [-16], [4], [21], [36], [14], [15], [8], [-16], [32], [12], [58], [-13], [-22], [74], [0], [40], [-5], [56], [75], [-75], [39], [-56], [44], [40], [-53], [-3], [-23], [28], [34], [-25], [45], [40], [79], [46], [-8], [38], [66], [69], [-35], [26], [31], [-24], [-11], [-32], [-65], [-75], [-48], [4], [-68], [19], [-5], [26], [-72], [-65], [-20], [79], [-72], [78], [-2], [-40], [-72], [72], [6], [-27], [-34], [-65], [54], [12], [26], [-16], [-33], [-66], [-84], [69], [-20], [2], [-39], [24], [-10], [48], [-85], [74], [67], [58], [0], [-10], [-41], [-55], [-45], [-22], [47], [-56], [-38], [61], [-14], [-54], [-52], [66], [-13], [1], [27], [42], [60], [64], [-24], [6], [56], [-18], [0], [51], [-40], [69], [1], [30], [-50], [-79], [-30], [30], [78], [8], [54], [-36], [-2], [11], [-12], [12], [-18], [43], [-3], [-94], [-34], [6], [-4], [-29], [-93], [-33], [42], [65], [69], [6], [48], [-78], [58], [75], [-52], [-28], [82], [6], [54], [12], [17], [66], [44], [-38], [74], [65], [51], [-54], [-68], [76], [25], [-78], [-13], [-26], [36], [-46], [-22], [-94], [32], [-102], [-36], [34], [-5], [-73], [41], [61], [-86], [-14], [-15], [-26], [97], [28], [-76], [3], [-6], [-24], [-39], [90], [-4], [-19], [-1], [39], [-20], [4], [-36], [57], [-2], [58], [-72], [1], [-23], [53], [60], [22], [-101], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1003_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1003_2_a_a();".
// 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_1003_2_a_a(:prec:=1)
    chi := MakeCharacter_1003_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_1003_2_a_a();".
// 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_1003_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<3,R![-3, 1]>],Snew);
    return Vf;
end function;