
// Make newform 1710.2.a.x in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1710_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_1710_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_1710_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 := [-4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [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_1710_a();"
function MakeCharacter_1710_a()
    N := 1710;
    order := 1;
    char_gens := [191, 1027, 1351];
    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_1710_a_Hecke(Kf)
    return MakeCharacter_1710_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, 0], [1, 1], [2, 0], [4, 0], [-1, -1], [-1, 0], [0, 0], [-2, 0], [5, -1], [0, 0], [-1, 1], [2, -2], [2, -2], [4, 2], [1, 1], [4, -2], [-2, -2], [-2, -2], [6, 0], [5, -1], [11, 1], [1, 3], [-6, 0], [-10, 0], [0, 2], [6, -2], [9, 1], [-4, -2], [-12, -2], [-6, 4], [-15, 1], [0, 4], [-22, 0], [7, 1], [3, 1], [12, 0], [-8, 0], [0, -2], [-13, -1], [-3, -3], [-4, 0], [2, -4], [0, 2], [2, -2], [-20, 0], [10, 0], [-4, 4], [2, -4], [11, 3], [-10, -2], [-8, -2], [8, -2], [10, 0], [-4, -4], [-4, -6], [-6, -2], [11, 5], [7, -3], [-8, -4], [-2, 4], [-14, -2], [20, 0], [0, 6], [-14, 0], [-8, 4], [-26, 0], [21, 3], [6, 4], [-3, -7], [-2, 2], [-5, -1], [-6, 2], [-8, -4], [20, -4], [-18, 4], [9, -1], [-3, 7], [-6, 0], [-4, 2], [-11, 1], [16, 0], [-18, 0], [-15, -1], [9, 7], [-17, -3], [26, 0], [6, -4], [-19, 1], [35, 1], [4, 0], [-24, -2], [-6, 4], [-4, 0], [32, 0], [26, -4], [-7, -5], [18, -6], [-18, -4], [-6, -2], [8, -2], [-2, 10], [-37, 1], [16, -4], [0, -6], [-23, 3], [5, 5], [4, 4], [12, -6], [4, 2], [29, 3], [5, 1], [36, 0], [-4, 8], [-17, -7], [-16, 4], [-12, -4], [-6, -8], [-21, -5], [-11, 5], [6, 0], [2, -8], [20, 0], [-16, -4], [-2, 8], [14, 0], [2, -2], [19, 3], [13, -5], [0, 4], [-8, -8], [21, 7], [-7, 7], [-38, 2], [-32, -2], [40, 2], [8, -4], [-14, -4], [6, 2], [-12, 8], [14, 0], [-27, -7], [22, 2], [-7, 9], [8, 8], [3, -3], [-6, 12], [16, 4], [-12, 4], [30, -2], [-16, 4], [24, -4], [24, 8], [16, 8], [24, 8], [0, -4], [0, -4], [40, 2], [-4, -10], [11, 5], [-10, 12], [-47, 1], [19, -5], [-58, 0], [-20, 4], [19, 5], [23, -3], [-32, 6], [16, 6], [3, -1], [-3, 1], [-26, -6], [-16, -2], [-13, 1], [18, 2], [8, -12], [-6, 8], [42, 4], [-18, 8], [29, -7], [13, -7], [-8, 8], [18, 8], [-30, 6], [30, -4], [2, -6], [12, -8], [4, 2], [-34, 6], [-42, 0], [19, -3], [-24, 4], [36, 6], [15, -3], [-20, 6], [16, 2], [-5, -7], [13, -3], [18, 6], [14, -4], [-45, 1], [-32, -4], [22, 8], [26, -8], [12, 14], [26, 2], [-10, 2], [-12, 0], [62, -2], [22, 8], [10, 0], [23, 3], [12, -4], [-30, -6], [52, -2], [-43, 5], [-15, -13], [24, 8], [52, 2], [-25, 7], [9, -9], [-41, 1], [-7, 5], [-48, 4], [30, -8], [-30, 0], [24, 0], [-24, -2], [10, 0], [-25, -3], [22, 6], [-23, -9], [-19, -1], [-8, -12], [0, -8], [-4, -10], [2, -8], [-8, 10], [2, 10], [-16, 0], [4, 0], [43, 3], [-41, -5], [-40, 6], [10, -6], [-45, 7], [-41, 3], [-26, 6], [0, 0], [5, 3], [-36, 4], [-46, -2], [-52, 6], [-50, 0], [28, -6], [-24, 10], [-4, 8], [-26, 0], [4, 2], [-18, 4], [30, 2], [-12, 2], [-10, 2], [-21, -9], [24, 12], [50, 0], [20, 4], [-6, -14], [10, 16], [25, -11], [22, 10], [44, 2], [38, -2], [18, -12], [-1, 11], [13, -5], [-9, 3], [-10, 4], [14, -8], [-36, 4], [48, 8], [-64, 0], [9, -3], [-46, 6], [14, 6], [-12, 6], [36, -10], [-28, -4], [-3, 3], [-26, -4], [39, -3], [-44, -6], [-41, 11], [-14, 14], [38, 4], [9, 11], [14, 8], [-41, 3], [-18, 10], [-30, 12], [-16, -14], [36, -12], [-18, 14], [24, -12], [6, -12], [-34, -10], [-25, 7], [-8, 12], [49, -5], [22, 2], [42, -4], [53, -1], [-44, 4], [2, -2], [46, 8], [-24, 2], [-50, 6], [52, -6], [46, -6], [6, 6], [68, 2], [26, -4], [4, 10], [-37, 3], [-12, 2], [-16, -12], [-34, -14], [4, 4], [-20, -6], [40, 6], [28, -6], [8, -12], [19, 1], [0, -4], [3, 9], [7, 15], [42, -4], [46, -4], [5, -3], [-30, 2], [27, -13], [20, 2], [-17, 1], [-10, -8], [29, -3], [16, 14], [64, 0], [6, -2], [46, 8], [54, 6], [6, -4], [10, -12], [48, 10], [-39, 1], [66, -4], [44, -4], [-54, 8], [61, -3], [0, -8], [-21, -7], [10, 6], [36, 4], [-53, -9], [-36, 8], [-58, 4], [-46, -4], [52, 6], [-54, -4], [42, 0], [14, 6], [28, 4], [-8, -10], [58, -2], [31, 9], [41, 1], [44, 4], [-24, -14], [-36, -16], [18, 8], [38, 4], [7, 19], [3, 23], [-5, 3], [40, 12], [-56, 0], [-30, 2], [43, 5], [74, 6], [-76, 4], [-14, -4], [34, 12], [24, -14], [32, 4], [20, -20], [-42, 4], [63, 1], [25, -17], [-42, -2], [-30, 0], [57, -11], [31, 11], [42, 0], [18, 8], [-8, -2], [46, 6], [-13, -3], [87, -3], [-42, -2], [50, 12], [-22, 8], [-46, 8], [0, -4], [4, 0], [12, -14], [18, -12], [-8, 0], [-18, -12], [-79, 1], [-24, 0], [-30, 0], [50, 2], [70, 2], [-81, -1], [-18, -8], [78, -4], [34, 18], [48, 0], [28, -8], [64, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1710_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1710_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_1710_2_a_x(:prec:=2)
    chi := MakeCharacter_1710_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_1710_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_1710_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1710_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, -2, 1]>,<11,R![-2, 1]>,<13,R![-4, 1]>,<53,R![-52, -8, 1]>],Snew);
    return Vf;
end function;