
// Make newform 1143.2.a.b in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1143_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_1143_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_1143_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_1143_a();"
function MakeCharacter_1143_a()
    N := 1143;
    order := 1;
    char_gens := [128, 892];
    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_1143_a_Hecke(Kf)
    return MakeCharacter_1143_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], [0], [-1], [0], [4], [1], [-4], [-4], [-3], [9], [-1], [-11], [6], [-6], [-8], [-9], [-7], [-5], [4], [12], [-5], [-8], [-15], [15], [-8], [7], [0], [6], [-14], [8], [-1], [-20], [1], [22], [10], [8], [10], [11], [-16], [-18], [-14], [20], [18], [20], [-22], [-17], [-5], [14], [-8], [-14], [-1], [5], [-2], [6], [5], [30], [-18], [-19], [2], [-10], [18], [-26], [-2], [1], [-4], [2], [10], [-6], [33], [12], [4], [24], [-13], [-26], [-10], [20], [30], [29], [-37], [4], [-14], [24], [0], [-30], [-14], [24], [6], [-25], [14], [41], [-17], [-6], [-4], [16], [-30], [-34], [0], [-22], [11], [-14], [-26], [-16], [-4], [6], [16], [-3], [42], [-15], [35], [-4], [27], [10], [-15], [-18], [-2], [-46], [36], [21], [34], [-9], [-25], [-33], [42], [-20], [32], [33], [-26], [2], [4], [13], [-15], [11], [-26], [-3], [-27], [-10], [-48], [17], [42], [24], [5], [-13], [32], [-29], [32], [21], [56], [-39], [-14], [-21], [6], [17], [-23], [-19], [23], [18], [5], [58], [-52], [20], [12], [18], [2], [6], [-16], [-14], [38], [8], [-29], [-24], [35], [40], [24], [-41], [24], [-57], [45], [-26], [-16], [28], [-29], [-29], [10], [6], [48], [-45], [-58], [-29], [-14], [-30], [-54], [28], [46], [18], [54], [30], [20], [19], [-8], [-49], [-11], [49], [-61], [-8], [21], [6], [-33], [14], [4], [-44], [46], [42], [44], [-30], [34], [-20], [66], [-39], [-36], [36], [50], [45], [15], [-32], [56], [-53], [66], [-64], [8], [15], [-40], [15], [-72], [-22], [-34], [-21], [-16], [-74], [-12], [23], [16], [-74], [-51], [7], [-1], [-12], [-34], [42], [68], [40], [-23], [18], [-68], [28], [-59], [48], [-48], [-67], [24], [-68], [34], [-34], [-66], [35], [-37], [6], [3], [-6], [-8], [18], [-36], [24], [-50], [-42], [-50], [-14], [-48], [74], [20], [57], [-9], [-4], [-38], [-30], [-62], [8], [32], [6], [-42], [-74], [-22], [38], [-26], [-44], [0], [-42], [62], [-60], [-24], [33], [-63], [15], [-29], [-42], [-78], [44], [22], [34], [-5], [-6], [-54], [-6], [-12], [-84], [17], [-50], [-89], [-10], [77], [34], [59], [12], [55], [-4], [-46], [73], [2], [-64], [47], [39], [35], [35], [50], [54], [-48], [-32], [16], [6], [-73], [27], [34], [34], [-51], [-45], [-62], [-49], [48], [44], [-7], [42], [20], [42], [3], [-20], [-9], [-48], [32], [-95], [48], [72], [22], [27], [-82], [55], [10], [-37], [80], [68], [-18], [84], [-78], [-59], [4], [-15], [1], [99], [37], [-14], [44], [28], [-54], [-42], [44], [96], [-44], [-87], [55], [-54], [33], [41], [-41], [98], [38], [64], [-60], [-50], [-77], [-12], [-61], [72], [25], [46], [27], [-69], [-6], [-31], [-18], [-78], [-101], [50], [-88], [-22], [-20], [-92], [-50], [3], [7], [-34], [-74], [-69], [0], [36], [27], [-26], [-26], [98], [20], [-15], [-100], [-53]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1143_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1143_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_1143_2_a_b(:prec:=1)
    chi := MakeCharacter_1143_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_1143_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_1143_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1143_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<5,R![1, 1]>,<7,R![0, 1]>],Snew);
    return Vf;
end function;