
// Make newform 9016.2.a.x in Magma, downloaded from the LMFDB on 11 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9016_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_9016_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_9016_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;
    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_9016_a();"
function MakeCharacter_9016_a()
    N := 9016;
    order := 1;
    char_gens := [2255, 4509, 1473, 1569];
    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_9016_a_Hecke(Kf)
    return MakeCharacter_9016_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]];
    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], [2, 0], [0, 0], [0, 2], [0, 1], [0, 0], [-6, 0], [-1, 0], [-2, 3], [6, -1], [-6, 0], [-2, 3], [-4, 4], [10, -1], [-10, 0], [4, -4], [-2, 4], [12, -2], [0, -1], [6, -3], [8, -6], [-2, 6], [12, 0], [-4, 4], [0, -2], [4, 4], [8, -6], [-2, 2], [2, 6], [16, -3], [-8, 5], [-2, -6], [-12, -1], [2, -8], [0, 7], [10, -2], [0, -3], [2, -2], [8, -6], [-8, -7], [14, -4], [-8, -2], [18, -5], [10, -1], [4, -2], [-4, 0], [6, -6], [18, -6], [-2, -6], [-22, -3], [-16, -1], [8, -2], [-2, 10], [2, -1], [-8, 10], [12, 3], [-6, -2], [-14, 3], [14, 6], [-30, 0], [-6, -6], [-8, 16], [10, 3], [0, 4], [-10, -4], [8, 5], [6, 2], [-20, 8], [-4, -5], [-2, 9], [-8, -2], [4, 0], [18, 0], [24, -4], [-16, -8], [6, -6], [0, 1], [-2, -8], [10, 1], [14, 2], [-18, -6], [8, -12], [12, -2], [10, 9], [-8, 3], [-18, 8], [-30, 2], [-12, 11], [-8, -8], [18, 4], [-4, -4], [-4, 9], [24, -7], [8, 3], [16, -18], [4, -13], [20, -8], [34, -2], [2, -9], [-8, -11], [10, -10], [-10, -14], [6, 12], [4, -8], [-2, -7], [12, 5], [38, -4], [0, -16], [-2, -3], [-22, -6], [38, -2], [-34, 0], [-2, 6], [16, -12], [-2, 14], [14, -20], [30, 3], [14, -3], [4, 4], [6, 8], [18, -11], [-14, 0], [0, 9], [-32, 4], [-22, -6], [-14, 2], [18, -10], [20, -2], [18, -18], [-24, 9], [24, -12], [8, 16], [14, -4], [-14, -7], [0, -14], [-2, 18], [-10, 14], [14, 0], [-2, 4], [0, 1], [14, -4], [24, -1], [8, 8], [4, -14], [0, 8], [-44, -2], [6, 7], [4, -3], [-16, -3], [10, -20], [0, -12], [-4, 24], [14, -5], [24, 10], [32, -2], [24, 0], [-2, 19], [-8, 14], [2, 6], [24, -5], [-6, -4], [-36, -3], [-2, -8], [-42, -4], [8, 2], [-4, -4], [12, -26], [26, 12], [-32, -7], [-8, 14], [-12, 27], [8, 20], [-26, 14], [-6, 9], [-4, -4], [0, 17], [46, -3], [-16, -18], [-6, -6], [0, 21], [-26, 14], [-26, -1], [-38, 4], [-16, -10], [-2, -6], [-2, 0], [-26, -4], [-30, 7], [-18, 2], [18, -21], [-20, 16], [40, -6], [20, 2], [-28, -2], [4, -8], [-2, -20], [-42, 6], [-24, -4], [14, -23], [38, -14], [2, 22], [-40, 11], [24, -14], [26, -10], [8, -21], [32, -10], [16, 7], [-2, -11], [-4, -13], [-50, 1], [-52, 9], [-8, 20], [-6, 20], [-66, 2], [16, -12], [44, -5], [2, -23], [8, 2], [-58, 1], [38, -15], [16, -26], [-2, -13], [40, -20], [48, -11], [-26, 17], [-4, 10], [-8, 5], [-4, 2], [36, 8], [-30, 0], [-10, -10], [-32, 18], [-26, -4], [42, -2], [-12, 20], [0, 0], [22, -20], [32, -1], [2, -7], [28, 16], [32, 5], [-10, 31], [-38, -3], [-26, 20], [-30, -5], [46, 11], [14, -4], [28, 14], [16, -22], [-26, 14], [4, 2], [16, -18], [-16, 16], [48, 6], [-32, -7], [-50, 4], [-34, 0], [40, -9], [-46, 13], [-16, 24], [-36, -12], [-28, -6], [-18, -6], [34, 0], [30, -8], [2, 20], [36, -12], [-36, -14], [-26, 11], [20, -5], [-8, -22], [4, 24], [-16, -20], [30, -15], [6, 11], [12, 0], [-10, 20], [2, 20], [-30, 30], [-6, -23], [-16, 14], [-40, -2], [6, 12], [8, -32], [12, -36], [-6, -12], [42, 12], [54, 10], [10, 8], [26, -16], [26, 2], [-30, -8], [30, -8], [-38, -1], [-14, 4], [0, -8], [32, -7], [60, 11], [-48, 3], [4, 8], [6, 20], [-8, -12], [-24, 21], [44, -12], [-34, -3], [4, 24], [6, 2], [72, 2], [-26, 4], [-42, 7], [54, -20], [62, -18], [-48, 13], [8, 26], [12, 28], [4, 7], [-16, 15], [36, 20], [-10, -3], [10, -6], [46, 2], [20, -11], [-8, -8], [-54, 16], [-24, 0], [-52, 22], [44, -21], [-32, -10], [-30, 2], [-30, 11], [-26, 17], [-2, -11], [0, -1], [4, -18], [36, -5], [34, 0], [-4, 12], [-46, 31], [8, -28], [-54, -11], [46, -6], [-12, 14], [-8, -22], [22, 16], [20, 8], [40, 9], [-76, 0], [48, -18], [-38, -8], [-40, -19], [6, 27], [14, -40], [16, 18], [2, 2], [-30, 4], [-32, 0], [-6, 36], [18, 23], [-32, 12], [-10, 24], [6, -11], [-60, 9], [12, -6], [56, -17], [-46, -5], [-32, 37], [80, 2], [34, 20], [0, -11], [8, 4], [12, -21], [-30, -10], [12, 24], [2, -31], [52, -5], [64, 6], [-32, -14], [52, 12], [-18, -29], [30, -32], [58, -12], [-4, -16], [-10, -27], [-18, 22], [58, -15], [80, -7], [-28, -18], [-4, 2], [48, -16], [22, -22], [-60, -2], [64, 7], [16, -20], [-16, -4], [-10, 44], [-24, -12], [-20, -24], [32, -12], [-6, -20], [12, 13], [82, -1], [0, 20], [4, 8], [-56, -11], [42, -1], [-46, -4], [18, 15], [-38, -14], [-48, -4], [18, -17], [62, -3], [-4, -18], [48, 10], [38, -16], [20, 15], [32, -16], [38, -10], [-52, -8], [8, 24], [-6, -18], [18, 18], [32, 0], [20, 0], [-22, -23], [-12, 25], [-32, 14], [26, 4], [-36, 37], [2, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9016_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_9016_2_a_x(:prec:=2)
    chi := MakeCharacter_9016_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_9016_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_9016_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -1, 1]>,<5,R![-2, 1]>,<11,R![-16, -2, 1]>,<13,R![-4, -1, 1]>],Snew);
    return Vf;
end function;