
// Make newform 1716.2.a.f in Magma, downloaded from the LMFDB on 23 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1716_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_1716_2_a_f();".
// 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_1716_2_a_f();".
// 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 := [-14, -12, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-8, -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_1716_a();"
function MakeCharacter_1716_a()
    N := 1716;
    order := 1;
    char_gens := [859, 1145, 937, 925];
    v := [1, 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_1716_a_Hecke(Kf)
    return MakeCharacter_1716_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, 0, 0], [0, 1, 0], [0, 0, 1], [-1, 0, 0], [-1, 0, 0], [2, 1, 1], [0, 0, -1], [0, 0, 1], [0, 1, -1], [2, -1, 0], [4, 0, 0], [2, 0, -1], [0, -1, 1], [2, 2, 2], [2, -2, -2], [4, 2, 0], [4, -2, -2], [2, -1, 2], [-2, 0, -4], [8, -2, 1], [8, 1, 1], [-4, -2, 2], [2, -3, 0], [8, 0, 0], [-2, -1, 1], [8, 0, 0], [-8, 2, 0], [12, 2, 3], [6, 2, 0], [2, -3, -1], [4, 4, 6], [2, 1, 0], [6, -3, -3], [-2, -4, 3], [0, 0, -5], [6, 2, -5], [0, 5, -2], [-8, -4, 2], [-6, -1, 3], [-4, 2, 3], [-2, 2, 5], [4, 2, -2], [-4, -2, -1], [6, -2, -3], [0, 4, 0], [-8, 1, 3], [6, -3, -4], [-10, 4, -1], [8, -6, -2], [-12, -1, -5], [-10, -4, -3], [2, 2, -4], [0, -6, -6], [6, -4, 2], [-16, 2, 0], [6, 2, 2], [-4, -6, -7], [-2, 4, 2], [2, 2, 3], [-14, -1, 1], [10, -6, 1], [4, 0, 5], [-4, 0, 3], [-6, -4, -1], [-6, -7, 4], [-12, 5, 0], [6, -6, 4], [-10, -4, 0], [-6, 6, 4], [-4, -1, 0], [-2, 0, 3], [-4, -2, 2], [2, -6, 0], [16, 3, 6], [0, -2, 6], [-2, -6, -4], [-20, -4, -8], [-4, 3, -2], [-8, -2, -3], [-8, 2, 3], [2, 0, -6], [0, -4, -2], [6, 8, -4], [-4, -1, -3], [-20, -4, -4], [32, 1, 2], [10, 4, 2], [-2, -8, -5], [4, 5, -8], [-4, 4, -3], [-6, 8, -1], [16, 3, 6], [-2, 4, -6], [18, -1, 0], [20, 6, 4], [14, 7, 8], [14, -4, -4], [-4, -3, -1], [-6, 0, 0], [2, -1, -1], [6, 0, 3], [-8, -8, -6], [-14, -5, 7], [-20, 1, 1], [-14, -8, -14], [4, -10, -8], [26, 2, -3], [12, 2, 2], [-6, 0, -4], [10, -3, -5], [-4, 2, 1], [-20, -3, 2], [20, -9, 0], [12, 1, 4], [-2, 4, -6], [6, -11, -6], [-4, 2, -6], [-6, 0, 2], [14, -2, -6], [-14, -2, 10], [-8, 12, 0], [-24, -5, 1], [2, -12, 0], [16, 5, 4], [-2, 11, 9], [8, -6, -4], [24, -8, -4], [-8, -6, 6], [12, -6, -7], [-8, -8, -5], [20, -2, 8], [-4, -6, -2], [-14, -2, 3], [2, -6, 3], [22, 0, -4], [-6, -7, -4], [-8, 8, 7], [6, -12, -4], [-2, -1, -5], [-12, 4, 1], [10, 6, -1], [8, -2, -6], [-22, 0, 3], [18, 4, 0], [4, -12, -2], [10, 0, -6], [-12, 5, 3], [-8, -2, -14], [28, 2, -2], [-18, -2, 0], [22, -4, -8], [20, -4, -8], [-2, 6, -10], [-32, 2, 2], [32, 2, -2], [34, 1, -5], [30, 1, -2], [4, -4, -2], [10, 4, -9], [-18, -2, -8], [20, 1, 5], [8, 10, 3], [24, -4, 7], [6, 11, -4], [-12, 8, -2], [-12, 14, 4], [2, 2, 12], [16, 2, 9], [6, -12, 2], [-8, -2, -12], [8, -4, -4], [-10, -10, -12], [-10, -2, -4], [12, 8, 0], [26, 4, -8], [-4, 16, 1], [2, 1, 14], [-12, 9, 7], [-6, -2, 12], [10, 3, -12], [-28, -4, 0], [14, -14, -5], [-2, -4, -1], [-10, -2, 0], [26, 2, 10], [-28, 8, 0], [24, 3, 10], [14, 2, 0], [8, -10, -6], [-34, -6, -9], [-8, 0, 2], [16, 0, 12], [6, 7, 10], [-28, -4, 6], [14, 4, -4], [-14, -8, 4], [-26, 6, 0], [22, 6, 17], [6, 6, -6], [-14, -10, -9], [-18, 1, 7], [34, 8, -4], [8, -6, 0], [14, 2, -12], [-6, 10, 4], [44, 5, 2], [36, 8, -2], [-26, -6, -1], [-24, 4, -8], [-12, -12, 6], [-18, 0, 0], [-4, -6, -18], [-12, -4, -10], [10, 4, 7], [-20, -2, 14], [8, 7, 13], [-8, -7, -19], [-14, 8, 8], [0, -3, -10], [-14, -12, 10], [-4, 14, 1], [12, -9, 6], [22, -3, 12], [-30, 2, -4], [10, 14, -2], [-10, 14, 0], [44, 2, -6], [4, -15, -5], [-32, -2, 6], [-34, -4, -4], [12, 11, -13], [4, 4, -1], [-16, -1, -11], [20, 0, 0], [-8, 4, 12], [-10, 14, 0], [-22, 14, -5], [0, 14, 3], [0, 4, 21], [0, -6, 6], [-30, -5, -3], [24, 12, -10], [14, -14, -4], [-22, -6, 9], [-2, 18, -6], [-4, 11, -10], [-10, -2, 8], [16, -12, -9], [18, -2, 6], [10, 6, -10], [6, 8, -9], [10, -4, 8], [-6, 0, 7], [-36, -1, -9], [-26, 16, 5], [-18, 4, 9], [-20, 8, 5], [10, -14, -6], [-6, 6, -14], [18, -5, 3], [-22, -10, -8], [18, 10, 18], [-14, 14, -4], [-14, 5, 10], [-12, -4, -16], [20, -9, -8], [-4, 13, 14], [-8, 12, 9], [-48, -7, 7], [6, 10, 12], [-2, 5, 12], [-14, -6, 0], [-18, -11, -9], [-6, -6, 10], [28, -11, 4], [10, 0, -4], [12, 14, 17], [-16, 14, 3], [-10, 14, 8], [50, 8, 8], [0, -5, -6], [-66, -2, -4], [64, -2, -5], [20, -2, 13], [56, -4, 0], [-6, 24, 12], [-2, -6, -17], [20, -19, 4], [16, 15, 11], [-6, 4, 14], [-24, -4, -9], [30, 4, -7], [-14, 12, 1], [24, -10, -12], [-70, 3, 3], [40, -2, 6], [18, -14, 6], [10, -20, 6], [-8, -8, 11], [-14, -8, -6], [26, 0, 1], [-46, -9, -3], [24, -4, -15], [-44, 0, 6], [36, 8, -10], [-28, -6, -19], [34, 10, -4], [-28, 0, 6], [14, -6, 6], [36, -14, -6], [20, 13, 6], [-30, -3, 15], [-20, 10, -12], [12, 14, -14], [10, 6, 12], [-16, -12, -18], [22, -12, -13], [-2, -10, 14], [-4, -21, -11], [22, -5, -7], [-18, 12, 8], [24, -15, -3], [20, -15, -4], [6, -16, -9], [10, 18, 3], [18, 7, -4], [56, 7, 12], [-12, -14, 4], [-32, 5, -7], [8, 10, -11], [-34, 16, 2], [22, 11, -13], [-10, 20, -9], [-16, 10, -5], [20, -2, 4], [4, -8, -4], [38, -2, 1], [18, 10, 14], [10, 11, -1], [-8, 24, 0], [-30, -14, 12], [46, 12, 3], [-4, -14, 4], [-10, 23, 8], [4, -18, 10], [42, -8, -8], [-36, -11, -12], [-14, 0, -1], [-2, 0, -6], [24, 0, -3], [6, -22, -2], [-12, -11, 16], [24, 15, 19], [20, -18, -8], [-38, -9, -5], [42, -10, 4], [18, 20, 17], [6, -5, 19], [-6, -6, 6], [38, -16, -2], [44, -11, -1], [-12, -6, -18], [-38, 20, 9], [8, -20, 0], [14, -6, 17], [22, -6, -17], [-24, 18, 9], [-4, -16, -14], [40, -10, -2], [24, 12, -4], [-62, -4, -1], [-2, 13, -3], [-12, 11, -11], [-62, 8, -4], [42, 2, -6], [8, -14, -12], [44, -10, -5], [-28, -17, 1], [0, 14, -10], [-8, 3, 20], [32, 11, 18], [28, 16, 9], [-46, 12, 1], [-60, 10, 1], [62, 4, -6], [6, -11, -2], [14, -18, 12], [0, -8, -21], [8, 2, 13], [20, 12, -4], [18, 5, -4], [-18, -16, 14], [0, 4, 24], [44, 2, -12], [-20, 14, 2], [8, 0, -19], [10, -16, -8], [56, 4, 0], [18, 12, -5], [58, 4, 2], [-10, -20, -22], [44, 8, 11], [-6, -3, 14], [-38, 20, 9], [14, -9, 11], [12, -14, 4], [-2, -4, 3], [14, -3, -10], [-22, 16, -10], [-10, -4, -28], [2, 15, 17], [-32, -2, 18], [-12, -5, -5], [50, 2, -6], [6, 16, -2], [-54, -8, -7], [-12, -14, -14], [32, -19, -4], [42, 4, -12], [-50, -8, -4], [58, -10, 4], [-30, -4, 20], [20, 2, -12], [-14, -8, -18], [4, 13, 24], [16, 12, 8], [-14, -12, -29], [-48, -7, -12], [-14, 16, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1716_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1716_2_a_f();".
// 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_1716_2_a_f(:prec:=3)
    chi := MakeCharacter_1716_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_1716_2_a_f();".
// 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_1716_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1716_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-14, -12, 0, 1]>,<7,R![-4, -12, 0, 1]>],Snew);
    return Vf;
end function;