
// Make newform 1183.2.a.k in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1183_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_1183_2_a_k();".
// 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_1183_2_a_k();".
// 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
        poly := [1, 3, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -1, 1, 0], [2, -6, -1, 1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1183_a();"
function MakeCharacter_1183_a()
    N := 1183;
    order := 1;
    char_gens := [339, 1016];
    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_1183_a_Hecke(Kf)
    return MakeCharacter_1183_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, -1, 0, 0], [0, 0, 0, -1], [-2, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [-1, -1, -1, -1], [0, 3, 1, 2], [0, -1, 1, 1], [0, 0, -1, -1], [-1, 0, -1, 0], [-2, -1, 0, 1], [-6, 2, 1, 0], [1, -1, 0, 0], [1, -3, 0, -1], [1, 2, 2, 4], [-1, -2, -1, 2], [2, 2, 1, 2], [-1, -6, -1, -4], [-4, 0, -2, -2], [-2, 2, 3, 2], [-6, 1, -1, 3], [-1, 4, 1, 0], [0, 1, -1, 2], [-1, 2, -1, -5], [-2, -3, 0, -2], [5, -1, -2, -3], [6, -1, -1, 0], [7, -3, 0, -3], [-5, -4, 2, 0], [5, 2, 1, 5], [-9, 0, 1, 3], [-10, -2, -3, -5], [3, 5, -2, 2], [-4, 2, 3, -1], [-3, -1, -2, -5], [3, -8, -2, -7], [10, -5, 1, 3], [-7, 4, -1, -4], [-7, 1, 0, -3], [1, 3, -4, 1], [2, -4, 1, 2], [-6, 6, 2, 3], [-8, 5, 4, 3], [-16, -5, -1, -4], [-3, 3, 2, -1], [-7, 6, 1, -4], [5, 2, -3, 3], [-9, -4, -1, 2], [2, 0, 0, -2], [-4, 11, 3, 8], [-1, 2, 1, 0], [1, -4, -2, -1], [-2, 4, -4, 6], [3, 6, 2, -1], [13, -6, -5, -2], [9, 6, 3, 1], [7, -6, -1, -8], [-1, -10, -4, -2], [-8, 1, 0, -7], [4, -11, -1, -5], [-4, 6, 3, 0], [9, 1, 2, -2], [-9, 5, 4, -4], [6, 0, 2, 2], [-7, -6, -4, -4], [0, 0, -4, -3], [-2, -6, -7, -1], [-2, -3, -3, -7], [11, -11, -4, 3], [-16, -5, -4, -6], [5, 4, 5, 1], [4, 15, -1, 5], [15, -7, -5, 0], [12, 4, 2, -3], [-1, -5, 4, -2], [-4, -3, 2, -5], [-2, 4, 4, -4], [-24, 2, 3, 1], [8, -9, 0, 4], [9, -2, 3, 4], [-12, -1, -2, 5], [-3, 2, 7, 3], [-13, -1, 3, 3], [-7, -5, 2, -2], [-4, -4, -4, 6], [-6, 14, 4, 6], [22, 1, 0, 3], [10, 9, 2, 8], [11, -2, -3, -4], [-5, 12, 5, -4], [6, 6, 8, 3], [1, 6, 3, -3], [-3, 14, 7, 3], [3, -7, 2, 4], [-4, -10, 4, 3], [-3, 13, 3, 15], [-6, 6, 5, 0], [-10, -2, 0, -4], [18, 2, -1, 3], [-7, 2, 1, -2], [11, -5, 1, -6], [20, -3, -5, -1], [-6, -10, 0, -10], [-2, -5, -7, -6], [-19, -3, 3, 5], [-5, 10, 9, 5], [7, -8, 6, 5], [3, -6, -5, -11], [-16, 7, -4, -2], [-9, 7, 0, 7], [-13, 1, -7, 6], [-7, 14, 6, 6], [-20, 7, 1, 1], [24, 3, -1, 0], [-7, 11, -3, 6], [-5, 3, 2, -1], [6, -4, 4, 10], [-1, 12, -1, -9], [-6, -4, 2, -8], [6, -2, 5, -4], [7, 12, 6, 8], [4, 5, -5, -6], [-23, 19, 6, 5], [-4, -1, -5, 6], [18, 4, 2, 2], [-6, 4, -3, -9], [13, 10, -3, 2], [-9, -9, 2, 1], [-5, 2, 2, 1], [2, -16, 2, -2], [11, 0, -1, -11], [-5, 0, -9, 6], [-16, 8, 0, 8], [7, -12, 1, -9], [11, -13, -4, 3], [-21, -3, 6, 1], [0, -5, 7, -3], [6, 4, -4, 6], [-21, -6, -8, -6], [16, -2, -2, 1], [-5, -9, 4, 1], [7, 3, 6, -7], [-15, -5, 4, -4], [-10, -11, -4, 4], [-14, -9, 3, -8], [17, 7, 7, 13], [-13, 14, 0, -1], [-8, -1, 7, 5], [0, 9, -7, 3], [-11, 12, -2, 2], [-17, -9, -5, -3], [-3, -2, 9, -6], [-14, 18, 4, 8], [-14, 1, -1, -9], [24, -5, -1, -12], [6, 4, 0, 12], [5, -17, -1, 5], [11, 9, -5, 3], [-9, -17, 4, 1], [-14, 6, -2, -1], [-3, -7, 10, -9], [29, 4, 3, 4], [-23, -1, -6, 7], [-33, -2, 4, -2], [-3, -12, -5, 1], [14, 8, 8, 9], [7, 11, 3, 3], [-15, 0, -7, -13], [7, -17, 1, 1], [-5, 19, 0, -2], [-3, 4, -4, -7], [17, 12, 7, 7], [5, 9, 3, 10], [-12, -4, -4, 9], [5, 17, 9, 11], [10, 0, 4, 14], [-15, -9, -3, -16], [-23, -6, 3, 3], [21, 2, -3, -3], [21, -8, 3, -4], [-6, -5, -7, 6], [5, -20, -2, -10], [-28, -7, -13, -10], [12, 13, 9, 8], [-9, -7, -11, -1], [-6, -4, 6, 10], [45, -10, -3, -1], [-8, 3, -2, 7], [10, 3, -11, -8], [29, -3, 5, 7], [35, -7, -6, 0], [0, -9, -5, 6], [-8, 1, 9, 8], [10, -22, -4, -5], [-15, -5, -4, 9], [-9, 13, -1, 20], [15, -21, -11, -8], [-28, -2, -1, 16], [27, -7, 6, 5], [21, -15, -5, 8], [-10, -17, 3, -2], [-1, 1, -1, 11], [30, -5, 0, -2], [15, -17, -8, -9], [-1, 9, 11, 3], [-10, -7, -11, -9], [-43, -2, -5, 4], [-3, 3, -9, -6], [-10, -18, 8, 2], [4, -10, -3, -13], [-7, -8, -1, -11], [5, -19, -4, -19], [13, -9, -2, 3], [-11, 0, -1, -19], [-13, -7, -13, -19], [25, 12, 15, 16], [7, -8, 0, -17], [3, -10, -11, -3], [-6, 9, 13, 4], [6, 4, 3, 5], [-9, 9, -2, -7], [-20, 17, 2, 19], [-3, -12, 1, -13], [0, -15, -15, -14], [18, 6, 5, -7], [21, -15, 3, -13], [3, 4, -1, 3], [13, -18, -11, 1], [11, -25, -6, -7], [2, -13, -2, 7], [-27, -9, -4, -9], [-19, 15, -6, -18], [22, 5, -10, 1], [-42, 9, -1, 3], [-5, 19, 4, -5], [-7, 12, -7, -11], [11, -2, 6, -12], [-4, 6, -4, -6], [9, -20, -13, 2], [-10, 2, -6, -13], [-5, 6, 9, -12], [-19, 12, -3, 3], [-13, 12, -2, -10], [-10, -18, -9, -8], [10, 15, 7, 19], [-8, 9, 5, 11], [-28, 13, 1, 4], [-14, -1, 13, -1], [7, 12, -3, 5], [-5, 8, 7, -11], [1, 2, -7, 5], [7, -3, 4, -9], [-25, 15, 5, 11], [-17, 5, 14, 5], [17, -17, -8, -19], [18, 1, 13, 4], [-13, -17, -10, -17], [9, -9, -6, -3], [-27, 16, 8, 11], [-39, -11, 6, -5], [25, 1, -2, -13], [-5, -19, -5, -1], [13, 5, 1, -17], [-32, -2, -1, 20], [22, -1, 1, 1], [9, -21, -8, -9], [21, -4, 4, -19], [1, -3, -8, -1], [-29, 20, 0, -4], [-1, 8, -6, 7], [-13, 8, 19, 3], [14, -18, 3, 6], [-9, 11, -4, -19], [6, -2, -18, -8], [2, 14, 24, -1], [-15, 18, 3, 10], [12, 3, -2, -5], [-22, 19, -4, 3], [-13, -15, 2, -11], [-23, 12, 13, -1], [-5, -12, -7, 3], [0, 28, -2, 15], [-20, 2, -3, 20], [-1, 5, 12, 5], [50, 6, 2, 8], [7, -3, -7, -4], [-11, -15, -5, 14], [4, 22, 14, 16], [-10, -19, -4, -14], [23, -17, -8, -11], [-5, -8, 15, 3], [6, 9, -4, 11], [-19, -13, 4, -8], [-40, 0, 0, 10], [39, 8, 16, 15], [-18, 7, -9, -4], [-11, 22, 6, 25], [-18, 7, -3, 8], [-10, 22, -4, 5], [20, 6, 7, -18], [-16, 10, 1, 5], [-2, 14, -4, 6], [-5, 5, -8, -1], [-43, 17, 4, 1], [3, -12, 8, 6], [31, 1, 0, -5], [-11, -15, -3, -3], [-1, -18, -7, -8], [22, 25, -4, 13], [-23, -5, 4, 15], [-3, 7, -4, 23], [-9, -2, 12, -8], [-10, 6, 0, -11], [19, -6, -1, -3], [11, -18, -4, 1], [43, 8, 11, 22], [-32, -7, -9, -5], [-39, -7, -8, 5], [4, -1, -6, 3], [-21, 6, 5, 8], [1, 0, -5, -9], [7, -7, -13, 5], [-10, -7, -5, 20], [-34, 18, -2, 0], [-11, -2, -6, -19], [-14, -13, -5, -4], [-18, -2, -14, -25], [18, -2, -4, 5], [32, 13, 12, 17], [-33, -4, -10, -16], [-4, -6, -15, 5], [-14, 12, -4, 22], [23, 1, -2, -7], [-5, -10, 13, -1], [-6, 0, -16, -4], [33, 12, -2, -6], [-42, -8, 0, 16], [-23, -2, -4, -3], [-3, -2, 1, -16], [25, -18, -1, 1], [-10, -17, -6, 9], [-51, 22, 8, -3], [6, -29, 1, -3], [-2, -20, -18, -22], [14, -41, -11, -25], [22, 25, -6, 2], [2, 16, 10, 20], [-39, -4, -11, -24], [-21, 7, -10, 15], [-35, -3, 6, 9], [-34, -2, -8, -8], [-31, 0, -14, -10], [-31, -7, -14, -24], [-5, 6, -6, -24], [-27, -16, 4, 1], [30, 15, 13, 26], [0, 22, 18, 20], [30, 6, -1, -1], [-12, -14, -8, -14], [15, 1, 11, 8], [12, -10, 1, -14], [-6, 22, 10, -10], [35, 8, -5, 2], [24, -6, -6, 10], [39, -5, -2, 3], [6, 26, 0, 10], [30, -24, -18, 3], [-41, 13, 15, 2], [-20, -4, -10, -30], [-17, -12, 11, 12], [18, 17, -3, 18], [-27, -17, -13, -14], [6, -11, 7, 9], [2, -4, 11, -15], [17, 7, -14, -7], [22, 15, 1, 11], [39, 16, 1, 13], [-23, -11, 3, -15], [53, -3, -1, -7], [55, 8, 11, -1], [-51, -16, 6, 0], [-3, -6, 15, 18], [7, -7, 4, 23], [-37, 5, -10, -14], [-20, 6, 12, 12], [-5, -30, -15, -18], [-19, -7, -4, -3], [-23, 14, 1, -14], [-12, 29, 9, -7], [12, 0, 11, -7], [31, 7, 9, 19], [-57, 5, 12, 4], [25, -8, -13, 4], [19, 17, 4, 16], [-20, -6, -6, 24], [-57, -1, 4, 6], [16, 2, 8, 14], [41, 3, 4, 6], [60, -20, -3, -3], [17, 3, 6, -17], [25, 8, -7, 13], [-12, 12, -8, -4], [-15, -20, -12, -13], [23, 18, -10, 7], [5, -10, -21, -6], [-10, -26, 0, 14], [14, -17, 2, 3], [-8, 11, 1, 29], [-22, -2, -12, -6], [-29, 20, -6, -5], [10, 23, 14, 13], [55, -1, 6, 3], [34, -12, 0, 6], [17, -2, 10, 2], [-62, -8, -7, -14], [0, -8, 14, 8], [-32, 12, 4, 5], [-13, 9, 10, 11], [-63, 4, -9, -18], [-43, 14, 6, 16], [13, 37, -5, 11], [-19, 19, 4, 19], [59, -42, -15, -2], [-12, -12, 10, -4], [-21, 44, 10, 27], [-30, -11, 7, 3], [-2, 20, 23, 3], [29, -9, -2, -9], [-13, -30, -7, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1183_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1183_2_a_k();".
// 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_1183_2_a_k(:prec:=4)
    chi := MakeCharacter_1183_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_1183_2_a_k();".
// 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_1183_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1183_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, -6, 1, 1]>,<11,R![-6, -16, -9, 1, 1]>],Snew);
    return Vf;
end function;