
// Make newform 1003.2.a.d in Magma, downloaded from the LMFDB on 19 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_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_1003_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_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 := [[2], [0], [-2], [-2], [-3], [4], [1], [-1], [1], [-6], [-8], [-2], [0], [4], [12], [-3], [1], [5], [10], [-4], [15], [6], [2], [-14], [-19], [-4], [-10], [-10], [3], [-2], [20], [-20], [9], [-8], [-22], [-16], [10], [12], [-22], [-5], [12], [-8], [8], [2], [-12], [-20], [-1], [21], [12], [6], [-3], [-7], [12], [-21], [-15], [23], [-17], [-25], [8], [-27], [-4], [9], [-15], [10], [-10], [-28], [-8], [13], [5], [6], [-6], [15], [-5], [-3], [28], [-3], [-6], [-9], [7], [20], [-7], [10], [-15], [23], [40], [16], [28], [-42], [13], [14], [6], [0], [-22], [17], [20], [27], [-26], [32], [9], [-42], [16], [-27], [14], [4], [-20], [43], [28], [-33], [12], [-14], [-16], [-14], [-18], [24], [13], [24], [26], [-43], [36], [10], [-34], [11], [-42], [-4], [-35], [-48], [20], [-8], [13], [37], [-26], [-12], [53], [1], [-25], [40], [4], [-14], [-44], [50], [13], [-27], [29], [-18], [5], [-31], [-46], [-47], [-50], [4], [-18], [-19], [45], [0], [50], [-22], [-48], [-33], [52], [-42], [32], [59], [12], [-15], [6], [3], [9], [46], [-14], [-27], [24], [-28], [18], [40], [16], [-10], [20], [39], [45], [-14], [3], [42], [17], [-6], [-16], [-18], [-2], [-6], [13], [3], [53], [-52], [16], [-1], [18], [-1], [-2], [-1], [57], [-32], [32], [-42], [-31], [4], [-20], [-58], [-14], [-54], [-6], [38], [14], [-21], [-24], [-57], [-20], [58], [-44], [69], [15], [-17], [-16], [9], [18], [62], [-64], [-26], [-6], [24], [2], [48], [-52], [47], [-10], [-72], [46], [-40], [-31], [-31], [-57], [31], [58], [-28], [-36], [-75], [54], [58], [17], [-17], [-44], [3], [-59], [-62], [22], [20], [-18], [-20], [-26], [-53], [10], [35], [-39], [3], [-74], [2], [-38], [36], [-17], [70], [-56], [-39], [-54], [-32], [-14], [40], [58], [-70], [-24], [-47], [22], [19], [-45], [-9], [28], [-16], [-57], [-24], [-9], [24], [32], [-17], [54], [84], [-31], [-67], [42], [42], [-39], [15], [13], [14], [39], [60], [35], [-36], [-52], [-25], [88], [-23], [-6], [-16], [16], [-40], [30], [50], [80], [46], [-26], [-32], [22], [-3], [86], [-3], [-70], [49], [0], [24], [-75], [-44], [-12], [18], [23], [51], [-24], [-51], [2], [6], [19], [-18], [25], [29], [30], [-30], [-33], [8], [6], [39], [43], [-19], [51], [75], [-30], [58], [-15], [-49], [5], [48], [-34], [-20], [-6], [39], [93], [-40], [12], [-87], [-63], [-48], [-77], [-9], [23], [65], [-53], [84], [6], [-18], [-46], [75], [80], [4], [-10], [65], [30], [-84], [-62], [-17], [-20], [57], [-49], [1], [-60], [-52], [-58], [5], [86], [-48], [0], [22], [-77], [89], [32], [-101], [76], [70], [96], [-92], [-35], [55], [2], [-51], [-42], [18], [-48], [-45], [74], [32], [-49], [42], [-59], [-77], [-18], [-105], [-14], [22], [-42], [-44], [-32], [-10], [-3], [88], [1], [-60]];
    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_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_1003_2_a_d(: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_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_1003_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 1]>,<3,R![0, 1]>],Snew);
    return Vf;
end function;