
// Make newform 1001.2.a.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1001_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_1001_2_a_c();".
// 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_1001_2_a_c();".
// 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_1001_a();"
function MakeCharacter_1001_a()
    N := 1001;
    order := 1;
    char_gens := [430, 365, 925];
    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_1001_a_Hecke(Kf)
    return MakeCharacter_1001_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], [-1], [-1], [-1], [-1], [2], [-1], [-5], [-5], [-9], [10], [-2], [-13], [-7], [-1], [8], [-2], [2], [6], [5], [11], [-3], [5], [5], [-12], [8], [20], [4], [1], [-8], [-12], [22], [0], [8], [-14], [2], [-16], [-17], [-14], [-9], [-20], [0], [-6], [-22], [2], [3], [-19], [24], [18], [11], [24], [-7], [24], [24], [-27], [12], [16], [-31], [-24], [0], [-1], [-3], [8], [12], [-16], [-2], [-23], [24], [1], [-6], [-2], [-36], [-34], [16], [32], [-6], [33], [24], [9], [-4], [2], [-10], [0], [6], [31], [0], [-22], [-2], [-12], [-32], [-33], [-38], [0], [-26], [-30], [39], [32], [-38], [-18], [-19], [24], [0], [7], [-5], [10], [21], [15], [-33], [-22], [20], [-44], [30], [-28], [-18], [-15], [-4], [24], [-22], [-47], [25], [49], [-12], [-28], [-15], [5], [4], [4], [-14], [41], [-8], [-24], [27], [-27], [-13], [-45], [-6], [3], [-22], [-15], [-12], [18], [40], [20], [-4], [20], [31], [22], [-50], [12], [2], [42], [16], [-12], [-5], [25], [-32], [15], [6], [33], [30], [9], [-44], [14], [-36], [-37], [8], [48], [-28], [-46], [24], [2], [-15], [42], [-20], [-28], [-4], [-18], [52], [8], [-20], [-12], [38], [43], [-20], [52], [-9], [55], [-6], [-9], [-34], [54], [-29], [-57], [9], [30], [16], [-11], [-3], [-66], [26], [30], [13], [56], [45], [60], [-61], [-13], [6], [-18], [-9], [-54], [-16], [-72], [-71], [-49], [59], [4], [-44], [-22], [17], [-56], [-48], [46], [30], [15], [-48], [-3], [20], [0], [14], [-66], [20], [-38], [50], [21], [66], [28], [-35], [14], [24], [-22], [56], [-66], [-7], [-62], [-55], [-45], [46], [17], [12], [34], [-50], [-26], [-26], [-16], [71], [28], [0], [25], [-49], [25], [41], [-56], [30], [30], [-42], [-75], [-44], [26], [8], [-58], [58], [-79], [70], [16], [66], [18], [0], [80], [74], [-10], [-56], [13], [-4], [15], [3], [-52], [-20], [-18], [43], [28], [-50], [59], [-71], [-28], [42], [56], [6], [-13], [-48], [18], [-25], [31], [-11], [48], [28], [-10], [-34], [47], [50], [-44], [-30], [-51], [62], [28], [-49], [-12], [79], [-51], [20], [8], [-36], [40], [-56], [46], [-44], [85], [-24], [-62], [80], [3], [40], [-15], [32], [11], [-46], [-58], [-50], [34], [10], [38], [-50], [-28], [38], [44], [-36], [-46], [39], [40], [-93], [-88], [-6], [-74], [-90], [-28], [-43], [69], [42], [24], [82], [-6], [3], [74], [-92], [-65], [-13], [0], [-17], [-36], [-16], [-86], [39], [-10], [-29], [-80], [88], [82], [20], [66], [39], [-54], [18], [18], [9], [1], [-78], [68], [-10], [82], [10], [21], [-12], [68], [-36], [49], [16], [-54], [-11], [-80], [23], [-21], [32], [65], [23], [-26], [12], [-70], [13], [21], [-28], [73], [-75], [60], [-45], [66], [88], [12], [2], [-46], [59], [90], [18], [-66], [-30], [-81], [51], [80], [-35], [-78]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1001_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1001_2_a_c();".
// 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_1001_2_a_c(:prec:=1)
    chi := MakeCharacter_1001_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_1001_2_a_c();".
// 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_1001_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1001_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;