
// Make newform 2001.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2001_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_2001_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_2001_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_2001_a();"
function MakeCharacter_2001_a()
    N := 2001;
    order := 1;
    char_gens := [668, 1132, 553];
    v := [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_2001_a_Hecke(Kf)
    return MakeCharacter_2001_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], [-1], [3], [0], [3], [3], [0], [8], [-1], [1], [7], [-5], [3], [2], [-6], [-2], [1], [7], [-9], [3], [-12], [-14], [14], [-10], [14], [5], [1], [8], [2], [-6], [-5], [12], [8], [18], [-3], [-8], [2], [-17], [-3], [18], [20], [10], [3], [4], [10], [21], [-3], [8], [6], [-21], [-28], [19], [12], [-9], [-6], [16], [-21], [-15], [-17], [-18], [23], [20], [29], [6], [-26], [-7], [-28], [-29], [-12], [25], [2], [-24], [-34], [6], [-14], [18], [20], [-2], [-27], [-38], [6], [9], [30], [-14], [38], [30], [7], [-16], [-3], [4], [-25], [-23], [-40], [30], [14], [-8], [38], [-2], [11], [-2], [-28], [-27], [-11], [28], [-31], [-28], [40], [-16], [-12], [6], [16], [38], [-14], [14], [23], [12], [4], [-17], [23], [28], [-44], [-6], [4], [9], [34], [-11], [-28], [-15], [-34], [-25], [-15], [-9], [-12], [-5], [50], [27], [20], [17], [-18], [-31], [12], [24], [-9], [-35], [24], [33], [40], [54], [20], [21], [-58], [-54], [-8], [-8], [10], [-9], [-55], [-38], [-2], [11], [52], [-6], [-32], [1], [-7], [-49], [20], [10], [16], [-14], [-24], [2], [-18], [26], [-24], [33], [56], [-2], [54], [10], [-3], [33], [-19], [-22], [-32], [39], [-10], [-54], [24], [24], [19], [34], [20], [42], [-17], [-26], [-57], [-8], [38], [39], [30], [-26], [8], [14], [-24], [12], [-34], [-27], [-32], [-17], [30], [36], [-12], [-12], [-44], [26], [-38], [43], [8], [39], [-18], [-41], [21], [58], [8], [-34], [-32], [50], [-28], [-33], [-68], [-4], [-58], [-18], [-19], [12], [5], [-52], [-61], [-56], [-28], [4], [34], [-34], [-18], [-40], [-40], [-46], [-45], [-57], [-34], [-11], [16], [49], [-14], [15], [-62], [-32], [-42], [47], [-8], [-60], [-7], [-19], [18], [-50], [-2], [-28], [16], [-24], [-27], [44], [6], [30], [-60], [40], [-44], [22], [35], [8], [5], [72], [-32], [8], [-56], [-30], [-70], [32], [40], [54], [18], [-18], [-68], [12], [36], [65], [-6], [-82], [-31], [12], [69], [-31], [-54], [-86], [72], [38], [24], [64], [-30], [-74], [15], [86], [-75], [-50], [82], [-62], [28], [-55], [56], [-27], [-72], [32], [84], [16], [50], [67], [17], [78], [85], [-3], [-44], [-18], [-70], [74], [-42], [36], [20], [-82], [-6], [-45], [44], [83], [-15], [-25], [54], [0], [8], [21], [-5], [-68], [50], [23], [-17], [49], [74], [-53], [33], [-18], [21], [75], [-68], [-54], [54], [-80], [-67], [-6], [58], [39], [38], [-14], [-26], [-32], [54], [75], [-59], [82], [49], [-51], [19], [-66], [98], [-39], [-81], [5], [44], [1], [72], [-82], [38], [-41], [-47], [-74], [-40], [-47], [-57], [92], [0], [-16], [-55], [-82], [-61], [3], [6], [23], [56], [94], [60], [71], [58], [-83], [29], [-30], [-80], [36], [-23], [18], [18], [98], [12], [41], [61], [-42], [-88], [-18], [-7], [0], [52], [-90], [-60], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2001_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2001_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_2001_2_a_a(:prec:=1)
    chi := MakeCharacter_2001_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_2001_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_2001_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<5,R![-3, 1]>],Snew);
    return Vf;
end function;