
// Make newform 1710.2.l.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1710_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1710_l_Hecke();"
// 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_l_j();".
// 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_l_j();".
// 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 := [49, 0, 7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 7, 7];
    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_l();"
function MakeCharacter_1710_l()
    N := 1710;
    order := 3;
    char_gens := [191, 1027, 1351];
    v := [3, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1710_l_Hecke();"
function MakeCharacter_1710_l_Hecke(Kf)
    N := 1710;
    order := 3;
    char_gens := [191, 1027, 1351];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, -1, 0], [0, 0, 0, 0], [-1, 0, -1, 0], [0, 0, 0, 1], [1, 0, 0, -2], [0, 0, 0, 0], [-3, -1, -3, 0], [-2, 1, 2, 0], [0, 0, -1, 0], [0, 1, 1, 1], [2, 0, 0, -2], [3, 0, 0, -2], [-6, -1, -6, 0], [-6, 0, -6, 0], [0, -4, -2, -4], [0, 1, 2, 1], [0, 2, 0, 0], [0, 3, -1, 3], [0, -1, -7, -1], [-11, -1, -11, 0], [-11, -1, -11, 0], [0, -2, 0, 0], [-6, 0, 0, 4], [0, -5, 0, -5], [-1, 1, -1, 0], [0, -6, 4, -6], [-6, 0, 0, -3], [-7, 0, 0, -1], [3, -1, 3, 0], [3, 0, 0, -1], [0, -5, 2, -5], [-13, 2, -13, 0], [0, 0, 12, 0], [0, 2, -2, 2], [-3, -5, -3, 0], [-7, 0, 0, 5], [-3, 6, -3, 0], [-8, 0, 0, -2], [0, -4, -7, -4], [0, 3, 0, 0], [7, 0, 0, 4], [0, 1, -17, 1], [-6, 0, 0, -4], [-9, -1, -9, 0], [0, 0, 0, -1], [0, -3, 11, -3], [0, 7, 0, 0], [-4, -1, -4, 0], [1, 0, 0, 9], [20, 0, 0, 2], [10, 4, 10, 0], [24, 0, 0, 0], [0, -4, 12, -4], [0, 4, -8, 4], [0, 0, -6, 0], [-1, 2, -1, 0], [5, 3, 5, 0], [4, -6, 4, 0], [-8, 0, 0, 0], [0, 9, -4, 9], [-4, -6, -4, 0], [12, 0, 0, 1], [7, 7, 7, 0], [4, 0, 0, -8], [0, -4, 8, -4], [0, 3, -12, 3], [-16, 0, 0, 1], [-12, 2, -12, 0], [2, 4, 2, 0], [25, 0, 0, 3], [15, 0, 0, 3], [-3, 3, -3, 0], [0, -6, 14, -6], [13, 0, 0, -2], [2, 0, 0, -6], [-26, -4, -26, 0], [0, 7, -1, 7], [1, 10, 1, 0], [6, 0, 6, 0], [0, 4, -29, 4], [-17, 0, 0, 8], [11, 3, 11, 0], [0, 1, -31, 1], [0, 3, -3, 3], [11, -3, 11, 0], [0, -3, 33, -3], [20, 0, 0, 1], [15, 0, 0, 5], [-14, -8, -14, 0], [32, 0, 0, 1], [3, 0, 0, 7], [0, -3, 3, -3], [18, 0, 0, -1], [13, 10, 13, 0], [-10, 1, -10, 0], [0, -12, 5, -12], [0, -2, 8, -2], [4, 0, 0, 10], [0, 5, -27, 5], [0, 6, 2, 6], [0, -2, 24, -2], [0, 7, 6, 7], [-32, 0, 0, -4], [-8, 0, 0, -1], [0, 0, 0, -10], [-5, 0, 0, -3], [38, -2, 38, 0], [0, 1, 33, 1], [0, 7, -1, 7], [-11, 0, 0, 2], [6, 0, 0, 5], [13, 6, 13, 0], [0, 4, -8, 4], [-26, 0, 0, -3], [0, -1, 19, -1], [-16, -2, -16, 0], [-25, 9, -25, 0], [-21, 0, 0, 6], [-18, 0, 0, 3], [0, -4, 1, -4], [0, -4, -24, -4], [-22, 0, 0, 6], [28, 0, 0, 5], [-21, 0, 0, -3], [10, 0, 0, -9], [2, 16, 2, 0], [0, -3, 17, -3], [-18, 2, -18, 0], [4, -6, 4, 0], [-7, 0, 0, 2], [6, -17, 6, 0], [-31, -4, -31, 0], [0, 0, -44, 0], [3, -10, 3, 0], [2, 0, 0, -7], [0, 2, 14, 2], [0, 3, -6, 3], [-21, 0, 0, -11], [-2, 0, 0, -15], [24, 0, 0, 0], [0, -1, 30, -1], [0, 4, 10, 4], [0, 5, 18, 5], [0, 3, -39, 3], [-26, 0, 0, 10], [0, 6, 0, 0], [26, -4, 26, 0], [-12, 0, -12, 0], [0, -7, 6, -7], [-31, 0, 0, -4], [39, 2, 39, 0], [38, 0, 0, 1], [0, -11, -17, -11], [0, 10, -20, 10], [6, 4, 6, 0], [-8, 0, 0, 2], [-3, 0, 0, -5], [-24, -3, -24, 0], [0, 3, 29, 3], [0, -5, -33, -5], [-9, -13, -9, 0], [-8, -8, -8, 0], [-18, -6, -18, 0], [30, -12, 30, 0], [9, 0, 0, 3], [-33, -6, -33, 0], [9, 1, 9, 0], [0, 0, 11, 0], [13, -4, 13, 0], [0, -5, 16, -5], [45, 0, 0, -6], [-14, 14, -14, 0], [-27, -3, -27, 0], [-20, 0, 0, 6], [0, -18, -8, -18], [0, -3, 18, -3], [0, 4, 16, 4], [19, 5, 19, 0], [6, 0, 0, -9], [-9, 13, -9, 0], [0, 3, -14, 3], [-13, 0, 0, -12], [0, -12, 7, -12], [0, 22, 0, 0], [-1, 0, 0, 2], [-26, 0, 0, 2], [0, 10, 18, 10], [-11, -7, -11, 0], [32, 0, 0, -4], [11, 0, 0, 17], [0, -8, -20, -8], [0, 3, -9, 3], [6, 0, 0, -2], [-7, -1, -7, 0], [0, -6, 36, -6], [0, 2, 50, 2], [0, 6, 19, 6], [-1, 0, -1, 0], [-2, 0, 0, 12], [27, 0, 0, -6], [0, 1, 19, 1], [0, 6, -38, 6], [-49, 8, -49, 0], [-25, 0, -25, 0], [9, -12, 9, 0], [0, -9, 42, -9], [0, 17, -11, 17], [0, -5, -23, -5], [-34, -4, -34, 0], [-32, 0, 0, 7], [2, -14, 2, 0], [0, 7, -7, 7], [-10, 0, 0, 10], [0, 21, 13, 21], [-3, 0, 0, -19], [0, 12, -27, 12], [0, 8, 0, 0], [20, 0, 0, -8], [-24, 0, 0, 14], [16, 14, 16, 0], [0, 0, 2, 0], [13, 0, 0, -15], [8, -5, 8, 0], [54, 2, 54, 0], [30, 4, 30, 0], [0, 3, -31, 3], [-22, 0, 0, -8], [-27, 7, -27, 0], [46, 1, 46, 0], [-55, 0, 0, -2], [0, 10, -3, 10], [0, -21, -8, -21], [15, 0, 0, 17], [24, 0, 0, 3], [-40, 0, 0, 4], [-31, 10, -31, 0], [-15, 0, 0, 20], [38, 0, 0, 13], [-43, -8, -43, 0], [0, 14, -20, 14], [26, 10, 26, 0], [28, 0, 0, -15], [0, 6, 34, 6], [0, 10, -32, 10], [-1, 19, -1, 0], [-1, 0, 0, -7], [0, 8, 4, 8], [0, -2, -8, -2], [-32, 9, -32, 0], [0, 20, -10, 20], [13, 0, 0, -18], [30, 9, 30, 0], [-19, 0, 0, -8], [0, -4, -51, -4], [20, 9, 20, 0], [0, -4, -64, -4], [0, -1, 41, -1], [21, 0, 0, -7], [6, -15, 6, 0], [0, -17, -33, -17], [0, -15, 4, -15], [42, 12, 42, 0], [12, 0, 12, 0], [-4, 24, -4, 0], [0, 1, 3, 1], [32, 0, 0, -15], [0, 0, 0, -20], [-14, 0, 0, -19], [0, 7, 19, 7], [0, -6, 12, -6], [-9, 0, 0, 5], [8, 0, 0, 10], [43, 5, 43, 0], [7, 0, 0, -9], [0, -10, 28, -10], [-30, 11, -30, 0], [-32, 0, 0, -14], [-4, -14, -4, 0], [0, 8, 4, 8], [0, -10, 49, -10], [40, 0, 0, -10], [49, 0, 0, 7], [0, 0, 18, 0], [-4, 0, 0, -2], [-42, 10, -42, 0], [0, 6, 60, 6], [11, 0, 0, -21], [0, 8, -12, 8], [15, 13, 15, 0], [-36, 0, 0, 10], [4, 0, 0, -4], [-39, 0, 0, 1], [0, -14, 50, -14], [34, 0, 0, -16], [-10, 0, -10, 0], [-43, 0, 0, 7], [0, -8, -10, -8], [-6, 7, -6, 0], [13, -12, 13, 0], [19, 0, 0, -5], [29, -15, 29, 0], [14, -3, 14, 0], [0, 0, 72, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1710_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1710_2_l_j();".
// 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_l_j(:prec:=4)
    chi := MakeCharacter_1710_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_l_j();".
// 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_l_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1710_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-7, 0, 1]>,<11,R![-27, -2, 1]>],Snew);
    return Vf;
end function;