
// Make newform 1856.2.a.x in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1856_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_1856_2_a_x();".
// 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_1856_2_a_x();".
// 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 := [-2, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, -2, 1]];
    Rf_basisdens := [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_1856_a();"
function MakeCharacter_1856_a()
    N := 1856;
    order := 1;
    char_gens := [639, 581, 321];
    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_1856_a_Hecke(Kf)
    return MakeCharacter_1856_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, 0], [-1, 1, 0], [-1, 0, 1], [0, 0, 0], [-1, -1, -2], [-1, -2, -1], [2, 0, 0], [2, 0, 2], [-2, 2, 0], [-1, 0, 0], [-5, 1, 0], [0, 0, -2], [4, -4, -2], [3, -1, 2], [1, -3, 2], [1, -2, 1], [-2, -2, 0], [-2, 4, 0], [-8, 4, 0], [2, 2, -4], [0, 4, 2], [-9, -1, 2], [-10, -2, 0], [4, 4, 6], [-4, 0, 2], [-4, 4, -2], [-2, 2, 0], [-4, 4, 4], [-1, 4, -3], [-2, 0, -4], [-2, -4, -2], [-10, 0, -2], [0, 0, -2], [2, -6, 0], [3, 6, 3], [-8, -4, -4], [-8, 0, -2], [5, 7, 4], [2, 2, -4], [-2, 0, -4], [6, 2, -4], [-3, -4, 3], [-2, 4, 6], [0, -4, 2], [6, 0, -4], [-16, -4, -4], [-7, 3, 4], [2, -2, 8], [2, 10, 8], [-4, 4, -2], [13, -2, -3], [10, -2, 0], [11, 6, 3], [11, 1, 4], [17, -4, -1], [7, -3, 0], [-2, -8, 4], [-15, -1, -4], [-2, -8, -4], [-19, 2, 1], [-14, -2, -4], [6, 0, -4], [-1, -9, -2], [-18, 4, -2], [5, 0, -1], [-6, -4, 4], [-7, -5, 4], [14, 4, 0], [6, 6, 8], [-15, 6, 1], [-6, -8, 0], [3, -1, 6], [-14, 0, 6], [-15, 0, -5], [-2, 4, -6], [20, 0, 4], [24, -8, -2], [17, 4, 7], [1, 6, 1], [-18, 12, 0], [2, 10, 0], [-2, 0, 4], [-10, 10, 8], [-4, -4, -2], [-6, 6, 8], [6, 0, 6], [-10, 8, -4], [-14, 8, 0], [2, 8, 8], [14, -6, -8], [11, -9, -6], [-11, 3, 4], [4, -4, 0], [9, -3, -2], [22, -10, 0], [-11, 3, -4], [-3, 2, 9], [3, 4, 9], [-8, 4, 0], [-12, -4, -10], [0, 0, -4], [-10, 16, 4], [-19, -9, -4], [6, -12, 0], [-8, -4, -8], [-14, 4, 4], [6, -6, -12], [15, -16, -7], [-17, -5, -2], [20, 0, 2], [13, 3, -4], [23, -4, 5], [16, 0, -2], [-11, 11, 0], [-10, 2, 0], [-6, 12, 4], [-32, -4, 0], [10, -6, -4], [-16, 0, -10], [3, -11, 8], [-14, -8, 0], [-13, 2, 7], [-10, -12, 0], [12, -4, -4], [0, -8, 4], [-1, -6, -5], [-1, 2, 11], [-6, 10, 12], [-10, -4, 14], [-16, -4, -14], [3, -3, 0], [-14, -4, 2], [-2, -16, -6], [18, 8, 16], [-2, -8, -4], [2, -12, -12], [-12, -4, -2], [-2, 14, 0], [-18, 4, 0], [20, 8, 10], [-16, 0, -4], [7, 6, -9], [14, 12, 6], [1, -9, 8], [38, 4, 8], [25, -7, -2], [30, 0, 4], [13, 0, 7], [-15, 1, -6], [-22, -10, -16], [41, -2, 1], [-10, -4, 0], [4, -12, -4], [-21, 13, 4], [-2, 0, -2], [1, -9, -12], [-14, 10, -4], [-10, 0, -12], [-26, -8, -12], [7, 4, 5], [-29, -9, -14], [9, 2, 13], [1, 13, -6], [18, 8, 2], [13, -8, 7], [21, 5, 6], [6, -10, -4], [-20, 8, 2], [1, 16, 3], [24, -24, -2], [-36, 0, -8], [7, 0, 1], [-10, 14, 4], [10, -16, 0], [-34, 6, -4], [-6, -8, 8], [-8, -12, 16], [-14, -4, 4], [-25, -1, -6], [18, -8, -8], [35, -5, -6], [-7, 11, 12], [-6, 0, 0], [17, 4, -1], [-12, 8, 4], [-3, -12, -5], [42, -4, 4], [-10, -12, -6], [-2, 4, -8], [28, -8, -4], [7, -2, -9], [-45, 1, -4], [-29, -5, -10], [8, -8, 6], [26, 4, 6], [2, 8, 0], [16, -12, 10], [10, 8, 16], [23, 10, 3], [-16, -8, 16], [4, -12, -2], [18, 10, 12], [0, 0, -2], [-4, 20, -10], [-14, -12, -10], [-19, 12, 11], [-41, -9, -6], [14, 14, 8], [3, 4, 9], [1, -3, -10], [24, -8, -10], [26, 8, -8], [-23, -9, -12], [-9, 3, -6], [22, 4, -10], [-7, -12, -9], [24, 12, 20], [-16, -4, 2], [32, -16, -8], [-26, 0, 4], [-10, -4, 8], [-4, -12, 0], [14, -4, -8], [-29, -9, 6], [-4, -4, -4], [-10, -12, -16], [-20, -16, -22], [2, -12, -14], [17, -3, -6], [24, -8, 4], [16, 4, -6], [16, 16, 4], [14, 4, -10], [12, -16, -6], [22, 10, 12], [26, 4, -6], [18, -4, -4], [-20, 4, -2], [-20, -4, -4], [-9, 5, -8], [-15, 25, 10], [-42, -14, -20], [-26, 10, 8], [16, -4, 18], [14, 0, 12], [-26, -14, 0], [-10, -10, -12], [-14, -6, 0], [2, 14, 12], [-29, 3, -14], [42, 0, 8], [-5, -4, 1], [29, -25, -8], [20, 12, 14], [-16, -16, -26], [-40, 12, 0], [-8, 4, -6], [19, 5, 16], [-10, -8, 4], [-29, -2, 11], [26, -12, 10], [31, -13, -6], [-30, 0, 8], [-46, 8, 8], [-2, 28, 8], [11, 7, -6], [20, 4, -18], [-2, 4, 24], [38, -24, -2], [-17, 4, 5], [1, 2, 13], [10, 10, -16], [23, -26, -9], [22, 4, -2], [8, 12, -6], [23, 17, -12], [18, -12, -10], [-47, -8, -5], [-16, 4, -14], [22, 14, 8], [10, 14, 0], [10, 22, 8], [44, -4, -8], [-34, 0, -12], [-14, 8, -14], [-6, -12, -22], [14, -4, 8], [8, 20, 10], [-16, 4, 4], [-5, 2, 15], [53, 4, -5], [42, -22, -8], [-9, -14, -13], [19, -23, -12], [-14, 16, 16], [15, -22, -5], [-11, 1, 10], [45, 10, -7], [8, 4, -16], [-2, -8, 14], [26, 8, -8], [-6, -8, 0], [-22, 28, 2], [7, -1, -2], [-10, -8, -2], [-19, 12, -5], [1, 23, -8], [3, 18, -9], [-12, -20, -8], [-67, 4, -5], [-34, 14, 12], [0, -16, -2], [6, -8, 12], [18, 6, -12], [-24, 20, 4], [21, 2, -7], [18, 16, 2], [-14, -14, -4], [-15, -16, -29], [-42, 36, 8], [-1, -3, 28], [-3, 12, -13], [18, 8, 24], [37, -1, 16], [21, 18, 9], [24, 0, -2], [34, -22, -16], [0, 4, 0], [23, 19, 22], [11, -4, 9], [0, 12, -14], [-29, 14, 19], [-48, -4, -4], [6, -24, -10], [-19, 9, 14], [12, -8, 16], [-6, 8, 0], [14, 20, 0], [-30, -4, -12], [-20, -4, -16], [-6, 32, 16], [26, -8, -8], [18, 0, 24], [-28, -4, -16], [-58, 8, -4], [-1, -9, 14], [-34, 12, 0], [38, 8, -2], [51, 15, 14], [-34, 16, 4], [-30, -2, 4], [-22, 32, 16], [-26, 12, 0], [6, 18, 24], [2, -12, 12], [26, 8, -8], [-1, -15, -12], [-60, 16, 0], [26, 12, 20], [-30, -6, 12], [-39, 8, -5], [-21, 10, -1], [-5, 11, 18], [2, -14, -8], [-5, -21, 14], [28, 4, -18], [8, 24, 22], [54, 6, -12], [26, -20, -4], [17, 7, -16], [-34, -6, -20], [-16, -12, -4], [38, 16, 12], [-2, -22, -16], [-21, 12, -15], [-42, 0, -2], [-7, -21, 16], [6, 4, 8], [27, 4, 25], [-15, 16, 11], [-32, 12, 2], [29, 4, -5], [45, -15, -6], [-14, 16, 0], [76, -8, 0], [36, -4, 16], [6, -20, -10], [-1, -14, -21], [-31, 7, -8], [-42, -4, -10], [-30, 16, 16], [13, -10, -27], [-1, 17, 8], [13, -27, -14], [43, 21, 28], [-11, 36, 3], [0, -32, -16], [28, -28, -2], [-66, -10, 0], [-26, -8, 20], [26, 8, 32], [-2, 28, 8], [-22, -4, -14], [31, 10, -13], [43, 14, 3], [38, -18, 4], [39, -16, 1], [-6, -8, -16], [31, -25, -2], [-44, -8, -16], [14, 8, 4], [41, -10, 1], [-24, -8, 4], [-42, 22, 0], [-80, 0, -2], [42, 4, -20], [53, -1, 16], [46, -14, 0], [36, 20, -2], [-17, -25, -6], [-34, -16, -4], [18, 12, 4], [33, 15, 12], [-19, 37, 10], [69, -26, -3], [-65, -12, -19], [-62, 14, 4], [-32, -4, 10], [-6, -34, -28], [25, 3, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1856_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1856_2_a_x();".
// 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_1856_2_a_x(:prec:=3)
    chi := MakeCharacter_1856_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_1856_2_a_x();".
// 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_1856_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1856_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, -5, 2, 1]>,<5,R![-10, -3, 4, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;