
// Make newform 1134.2.a.k in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1134_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_1134_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_1134_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 := [-8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_1134_a();"
function MakeCharacter_1134_a()
    N := 1134;
    order := 1;
    char_gens := [407, 325];
    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_1134_a_Hecke(Kf)
    return MakeCharacter_1134_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, 0], [0, 0], [1, 1], [1, 0], [2, -1], [2, 0], [-2, 1], [5, 0], [-5, 1], [-2, -2], [2, 0], [2, 0], [8, -1], [2, -3], [0, 0], [-2, -2], [0, -3], [-7, 3], [8, -3], [3, -3], [2, 3], [5, -3], [4, 4], [8, 2], [2, -3], [5, -1], [-10, 0], [-4, -1], [14, 0], [-1, -1], [-7, 3], [-5, 1], [14, -1], [-7, 6], [10, -2], [-1, 3], [-1, 3], [2, -6], [8, -4], [-6, 0], [10, -2], [11, -3], [-9, -3], [-7, 0], [-6, 0], [-10, 0], [-16, 0], [-4, 0], [-19, 2], [-13, 3], [-7, 2], [-7, 5], [8, 3], [-9, 0], [10, -5], [11, -1], [-5, 1], [-4, 6], [8, -6], [13, 1], [17, 3], [-7, -1], [-13, 0], [-12, 6], [-10, -3], [-6, 0], [2, 6], [2, -3], [0, -3], [-22, 0], [-10, -1], [5, 5], [8, -6], [-10, 0], [2, -3], [-26, -2], [-16, 8], [-22, 0], [-5, -2], [8, -9], [11, -7], [14, 6], [-20, 4], [-10, -3], [8, 0], [-30, -3], [33, 0], [5, -12], [19, -5], [-13, -3], [-4, 11], [26, 2], [5, 9], [-22, -1], [8, 3], [-12, -6], [-2, -8], [-8, -5], [-25, -3], [14, -6], [8, 3], [16, 4], [3, 0], [6, 3], [8, -9], [20, -3], [1, 4], [8, 2], [24, 0], [14, 3], [2, 6], [-16, -6], [2, 5], [5, 12], [-7, -9], [-13, 14], [-4, 9], [-8, 4], [-20, -2], [-14, -2], [17, 3], [-13, 3], [-6, 12], [-14, 13], [-13, -3], [-18, -6], [44, 0], [-4, 2], [-16, -6], [-31, -3], [2, -3], [20, -10], [-7, -3], [26, 6], [22, 4], [-10, 0], [-23, -5], [-4, 0], [-25, 5], [10, -5], [32, 3], [-24, 6], [8, -6], [12, 0], [-34, 6], [-8, 4], [-25, -3], [14, -16], [44, -3], [-7, 5], [2, -18], [0, -6], [-10, -9], [-8, 10], [-16, -3], [-41, 1], [-37, -3], [24, 6], [-10, 12], [9, 3], [24, 3], [18, 3], [41, -3], [5, -1], [30, 3], [-46, 2], [8, 12], [5, -3], [26, 0], [-5, 13], [50, -1], [11, -9], [13, 13], [-19, 18], [-40, 0], [20, 2], [-10, -3], [-20, -2], [-22, -6], [41, -3], [11, 9], [16, -11], [-58, 0], [38, -1], [15, 9], [2, 8], [2, -6], [-49, -3], [-37, 6], [-20, 4], [20, -3], [22, 7], [20, 3], [39, 3], [-2, -5], [-12, 9], [17, 6], [-40, 6], [22, 7], [8, 2], [-6, 12], [38, -12], [17, 15], [-4, 9], [27, 12], [1, 1], [-10, 18], [4, 1], [-26, -11], [5, 18], [35, 6], [14, -22], [44, -6], [3, -3], [36, -6], [-22, 9], [29, -3], [-28, 11], [8, -22], [-28, 2], [-4, -6], [-10, 12], [-47, -8], [-16, -6], [-16, 5], [-16, 6], [24, -18], [37, -5], [56, 0], [12, 0], [-16, -6], [5, 15], [29, 9], [-30, -3], [11, -15], [50, -4], [20, -6], [-10, 20], [-16, -1], [6, 6], [56, -1], [44, 0], [-28, 12], [26, -18], [50, -1], [0, 18], [2, 0], [-21, -12], [-16, -12], [15, 3], [20, -12], [8, 5], [-27, 15], [32, -6], [-10, 8], [20, 5], [2, -12], [11, 9], [9, -3], [44, -9], [-16, 18], [-44, 1], [-10, -12], [-7, -3], [60, -6], [-4, -9], [-42, 12], [-8, -11], [20, 9], [0, 6], [14, 0], [-4, -12], [-16, 9], [-16, 12], [-22, 3], [8, 0], [4, 16], [-58, -6], [-58, -3], [-15, -6], [-48, 0], [-16, 18], [34, -8], [53, -3], [23, -6], [-19, -1], [5, -3], [-52, 2], [2, 18], [42, 3], [-4, 2], [-29, 10], [13, 10], [-52, 5], [-10, 0], [-57, 9], [50, 6], [-5, -17], [-28, -4], [-4, 0], [56, -9], [-16, -16], [-13, 15], [-4, -4], [38, 6], [-25, 3], [32, -16], [-7, -9], [52, -8], [38, 6], [-6, 24], [-2, -14], [-18, 12], [2, 6], [-27, -15], [-16, 21], [-5, -8], [51, -15], [-46, 3], [-22, 8], [-4, 9], [35, -18], [-42, 18], [-25, 9], [-19, 5], [-4, 3], [-16, -9], [8, -3], [-3, 21], [30, -12], [-22, 0], [12, -12], [-4, 24], [-19, 14], [-22, 21], [21, -9], [38, 0], [22, -17], [26, 15], [44, -12], [38, -12], [31, -5], [2, 14], [-67, -3], [12, -18], [-42, -15], [-49, 3], [-16, 21], [-38, 10], [86, -4], [-13, -15], [-43, 12], [-40, -10], [-4, 0], [-4, 24], [54, -12], [46, 4], [73, -5], [-59, 13], [-32, -8], [-40, 0], [-54, 3], [-20, 16], [-12, -3], [-16, 3], [53, -18], [-38, 10], [8, 0], [17, -12], [-52, 8], [8, -3], [-23, -11], [-8, -8], [14, -24], [14, 6], [-21, 6], [-39, -9], [-40, 6], [-34, -7], [-16, 6], [-1, 5], [17, 20], [47, 9], [23, -4], [41, -6], [-56, -2], [56, 12], [5, 27], [2, -30], [-16, -10], [26, -15], [9, 9], [34, -17], [-13, -6], [12, -30], [47, -21], [56, -6], [-2, -23], [2, 3], [-42, 0], [41, 3], [-7, -16], [-16, 0], [88, -2], [19, -29], [14, -30], [26, -6], [-11, 16], [29, 0], [-12, 12], [-4, -3], [-18, 12], [-36, -9], [-10, -9], [-7, -3], [35, -7], [8, 20], [38, -18], [78, -3], [-32, 4], [-58, 14], [17, -27], [-47, 1], [11, -22], [-16, -18], [-18, 18], [72, -3], [-3, 9], [-25, 21], [-46, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1134_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1134_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_1134_2_a_k(:prec:=2)
    chi := MakeCharacter_1134_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_1134_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_1134_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1134_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-6, -3, 1]>,<11,R![-6, -3, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;