
// Make newform 1003.2.a.b in Magma, downloaded from the LMFDB on 25 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_b();".
// 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_b();".
// 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 := [[0], [2], [-2], [2], [-5], [2], [-1], [-5], [-5], [-6], [8], [2], [-6], [-2], [6], [-11], [-1], [-1], [-8], [-6], [13], [4], [6], [10], [3], [-2], [-6], [6], [-7], [6], [16], [-4], [9], [-4], [-4], [-8], [10], [0], [12], [1], [-18], [22], [0], [-8], [26], [-24], [-23], [9], [-8], [-20], [-25], [-15], [2], [27], [17], [27], [5], [-9], [-26], [-3], [16], [-11], [25], [8], [6], [-32], [-8], [27], [-17], [-26], [-4], [-25], [13], [-7], [34], [1], [6], [-7], [1], [-22], [-17], [10], [-17], [11], [18], [-8], [-30], [-14], [-15], [-26], [20], [-12], [-24], [-15], [-20], [1], [-36], [-42], [33], [-14], [28], [-39], [-28], [10], [-28], [-25], [-28], [39], [-24], [38], [-16], [-16], [20], [-2], [-15], [24], [6], [17], [48], [-6], [22], [37], [-16], [36], [39], [20], [-20], [0], [33], [9], [-50], [-24], [-45], [1], [-45], [14], [-46], [18], [-12], [-50], [-53], [-13], [31], [-4], [-55], [35], [34], [3], [58], [-32], [-36], [3], [17], [48], [-28], [26], [-10], [29], [-8], [-6], [42], [3], [-50], [29], [4], [53], [-37], [-8], [62], [27], [-28], [0], [0], [-34], [-4], [6], [2], [21], [21], [14], [59], [58], [-49], [-58], [-24], [-38], [10], [46], [-29], [29], [-17], [16], [-12], [51], [52], [-51], [50], [37], [27], [-8], [46], [0], [-27], [48], [44], [2], [16], [36], [50], [48], [-2], [-33], [-4], [3], [6], [58], [32], [73], [-47], [-9], [24], [-49], [-50], [-38], [-12], [-14], [20], [32], [-20], [-32], [-64], [-3], [-34], [-12], [6], [48], [15], [39], [-3], [-33], [-42], [-40], [-24], [29], [-46], [-26], [11], [-35], [-8], [51], [-19], [-50], [-8], [-58], [-46], [60], [-12], [13], [26], [-53], [71], [47], [46], [-46], [-44], [-16], [-25], [-72], [-8], [-15], [-50], [8], [-80], [-16], [-78], [64], [-52], [25], [-80], [39], [63], [65], [32], [76], [-23], [-26], [13], [-58], [-38], [-37], [-62], [-4], [53], [71], [-56], [-62], [-83], [-21], [15], [-12], [11], [74], [69], [84], [-32], [17], [0], [27], [-48], [70], [0], [-30], [-18], [-80], [76], [12], [26], [88], [-22], [-55], [80], [55], [58], [-67], [-74], [-38], [71], [20], [0], [-38], [65], [33], [-68], [-55], [-22], [4], [-85], [34], [-1], [77], [-18], [-42], [-31], [46], [26], [69], [13], [-43], [5], [-95], [-6], [-10], [-27], [-39], [-11], [-24], [60], [54], [70], [7], [-33], [-12], [48], [-9], [41], [-98], [-91], [-9], [-99], [17], [-71], [48], [38], [-14], [-12], [-19], [32], [56], [-60], [57], [-20], [96], [32], [-11], [-70], [27], [71], [25], [-82], [34], [82], [51], [30], [48], [-84], [34], [51], [-51], [-22], [-53], [-22], [48], [-8], [-8], [-15], [-75], [92], [9], [-66], [-6], [16], [19], [-60], [-62], [-61], [16], [-69], [21], [42], [7], [-58], [40], [28], [-52], [-14], [-2], [-53], [24], [-47], [-36]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<3,R![-2, 1]>],Snew);
    return Vf;
end function;