
// Make newform 9216.2.a.bk in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9216_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_9216_2_a_bk();".
// 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_9216_2_a_bk();".
// 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 := [1, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-2, 0, 1, 0], [0, -3, 0, 1], [0, 5, 0, -1]];
    Rf_basisdens := [1, 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_9216_a();"
function MakeCharacter_9216_a()
    N := 9216;
    order := 1;
    char_gens := [8191, 2053, 4097];
    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_9216_a_Hecke(Kf)
    return MakeCharacter_9216_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, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 1], [3, 1, 0, 0], [0, 0, -2, 1], [0, 2, 0, 0], [-3, -1, 0, 0], [0, 0, -3, 1], [0, 0, 0, -1], [0, 0, -4, 0], [0, 0, 2, 1], [0, -4, 0, 0], [-3, 3, 0, 0], [0, 0, 0, 4], [0, 0, 6, 1], [9, -1, 0, 0], [0, 0, -2, -1], [-1, -3, 0, 0], [0, 0, 3, -1], [-6, 2, 0, 0], [0, 0, -4, 4], [3, -3, 0, 0], [6, -2, 0, 0], [0, 2, 0, 0], [0, 0, -6, 3], [0, 0, 5, 1], [3, 9, 0, 0], [0, 0, 8, 3], [6, 4, 0, 0], [0, 0, 4, 0], [9, -5, 0, 0], [-12, 0, 0, 0], [5, 3, 0, 0], [0, 0, -12, -1], [0, 0, -5, -1], [0, 0, 2, 3], [15, 1, 0, 0], [0, 0, 15, -1], [0, 0, -6, -1], [3, -3, 0, 0], [0, 0, -4, -7], [0, 0, -12, 0], [-8, -6, 0, 0], [0, 0, 0, 1], [0, 0, 1, -7], [5, 3, 0, 0], [0, 0, -4, 0], [9, 7, 0, 0], [0, 0, 4, 3], [6, -2, 0, 0], [0, 0, -12, -4], [-8, 6, 0, 0], [9, 7, 0, 0], [-6, -4, 0, 0], [0, 0, 3, 7], [0, 0, 0, -3], [0, 0, 8, -4], [0, 0, -2, 3], [-6, -6, 0, 0], [7, -3, 0, 0], [0, 0, 0, -7], [5, 15, 0, 0], [0, 0, 9, 1], [-8, 12, 0, 0], [0, 0, -12, 3], [-11, -9, 0, 0], [6, -16, 0, 0], [9, -1, 0, 0], [0, 0, 8, -7], [-6, 4, 0, 0], [0, 0, 3, -9], [0, 0, -16, -4], [0, 0, 2, 7], [15, -7, 0, 0], [0, 0, 0, 8], [0, 0, 6, -1], [0, 0, -4, -9], [24, -6, 0, 0], [24, -4, 0, 0], [9, 3, 0, 0], [0, 0, 4, -13], [0, 0, -12, -4], [24, 6, 0, 0], [0, 0, 7, -9], [9, 3, 0, 0], [0, 14, 0, 0], [0, -12, 0, 0], [0, 0, -12, -7], [0, 0, -20, -4], [27, 1, 0, 0], [0, 0, 12, 8], [0, 0, -7, -7], [27, -3, 0, 0], [-11, 3, 0, 0], [0, 0, -3, -15], [0, 0, -6, 9], [-12, 0, 0, 0], [21, -1, 0, 0], [0, 0, -2, 7], [-9, 5, 0, 0], [0, 0, 6, 3], [27, -7, 0, 0], [-12, 0, 0, 0], [-25, -3, 0, 0], [-30, -8, 0, 0], [3, -7, 0, 0], [-6, -8, 0, 0], [0, 0, 9, -7], [-30, -6, 0, 0], [0, 0, -8, 8], [0, 0, -4, 11], [-6, 10, 0, 0], [-11, -9, 0, 0], [0, 0, -13, 7], [-24, 6, 0, 0], [15, -3, 0, 0], [0, 0, 3, -1], [0, 0, -24, 5], [3, 13, 0, 0], [0, 0, -20, 1], [24, 10, 0, 0], [0, 0, 30, -1], [-21, 5, 0, 0], [-3, -21, 0, 0], [0, 0, -18, 5], [0, 0, -10, -11], [0, 0, 12, -12], [0, 0, -1, -9], [0, 0, 20, -3], [23, -15, 0, 0], [0, 0, -9, -1], [0, 0, 4, -4], [0, 0, 20, 1], [0, -20, 0, 0], [8, -6, 0, 0], [0, 0, -6, -5], [-27, -5, 0, 0], [0, 0, 18, 9], [-24, 4, 0, 0], [21, 15, 0, 0], [0, 0, 6, -15], [0, 0, -1, 7], [-15, -5, 0, 0], [0, 0, -10, 7], [0, 0, 15, 7], [0, 0, -22, 7], [12, 8, 0, 0], [-9, 17, 0, 0], [0, 0, 12, -8], [0, 0, 4, 7], [18, 8, 0, 0], [45, 7, 0, 0], [0, 0, 9, 1], [-3, 19, 0, 0], [0, 0, -24, 4], [0, 0, -13, 15], [0, -2, 0, 0], [-22, 18, 0, 0], [0, 0, 24, -3], [-21, 1, 0, 0], [12, 24, 0, 0], [0, 0, -7, -7], [3, 21, 0, 0], [-48, 2, 0, 0], [0, 0, -39, 1], [0, 0, 16, 16], [0, 0, -10, -3], [24, -10, 0, 0], [0, 0, 24, 3], [9, -29, 0, 0], [0, 0, -20, -3], [0, 0, 3, -1], [-2, -18, 0, 0], [0, 0, -8, 4], [18, 18, 0, 0], [31, -15, 0, 0], [0, 0, -6, 3], [0, 0, -23, 1], [0, 0, -22, -3], [0, 0, 8, -8], [33, -1, 0, 0], [0, 0, 8, -9], [42, 10, 0, 0], [0, 0, 0, 20], [0, 0, -12, -1], [0, 0, -14, -13], [-17, 9, 0, 0], [8, 12, 0, 0], [0, 0, 0, -8], [22, 12, 0, 0], [-21, -11, 0, 0], [-11, -9, 0, 0], [0, 0, 0, -1], [-15, -17, 0, 0], [18, -14, 0, 0], [6, 8, 0, 0], [0, 0, 10, -13], [6, -24, 0, 0], [0, 0, -21, -1], [0, 0, -18, 3], [0, 0, 20, -4], [0, 0, 2, -1], [6, 4, 0, 0], [3, 1, 0, 0], [0, 0, -36, -5], [0, 0, 8, 8], [9, 31, 0, 0], [-18, 6, 0, 0], [-51, 7, 0, 0], [-16, 30, 0, 0], [0, 0, -12, -17], [0, 0, 7, -17], [-15, 19, 0, 0], [0, 0, -9, -9], [36, 8, 0, 0], [0, 0, -4, 12], [0, 18, 0, 0], [0, 0, -15, -7], [0, 0, 12, -13], [0, 0, -4, 11], [0, 0, 19, -9], [0, -34, 0, 0], [0, 0, 4, -12], [3, 9, 0, 0], [0, 0, 2, -1], [6, 14, 0, 0], [0, 0, 0, -16], [0, 0, 29, 1], [3, 13, 0, 0], [0, 0, 8, -13], [-3, 15, 0, 0], [0, 0, -8, 0], [-30, 2, 0, 0], [-3, -13, 0, 0], [0, 0, 12, -4], [24, 22, 0, 0], [0, 0, -24, 11], [9, -5, 0, 0], [0, 0, -9, -9], [-21, -19, 0, 0], [-65, -3, 0, 0], [0, 0, 29, 1], [0, 0, -2, 1], [-6, -16, 0, 0], [0, 0, -15, -15], [0, 0, 4, -16], [33, -5, 0, 0], [-11, 3, 0, 0], [0, 0, 12, -4], [0, 0, -32, 9], [-6, 36, 0, 0], [0, 0, 3, -9], [2, -6, 0, 0], [0, 0, 0, 13], [27, -3, 0, 0], [0, 0, -8, -11], [-49, -15, 0, 0], [0, 0, 18, 11], [0, -12, 0, 0], [0, 0, 32, -8], [-63, 7, 0, 0], [0, 0, -14, -7], [0, 0, 16, 17], [-24, 6, 0, 0], [-41, -3, 0, 0], [0, 0, 24, -19], [6, -34, 0, 0], [-9, 9, 0, 0], [0, 0, 30, -9], [0, 0, -10, -23], [5, -9, 0, 0], [-2, 30, 0, 0], [0, 0, -8, -16], [10, -12, 0, 0], [0, 0, -5, -1], [-15, 3, 0, 0], [0, 0, 2, 27], [-30, -6, 0, 0], [3, 25, 0, 0], [0, 0, 36, 0], [0, 0, 13, -15], [0, 0, 45, 9], [0, 0, -28, -5], [-27, -13, 0, 0], [0, 0, 24, 4], [-56, 6, 0, 0], [0, 0, -42, 9], [0, 0, -5, -17], [-42, 0, 0, 0], [0, 0, 0, 5], [-21, 25, 0, 0], [6, -18, 0, 0], [3, -31, 0, 0], [0, 0, -20, -9], [0, 0, 6, 5], [0, 0, 4, 8], [0, 0, 36, 7], [-15, 19, 0, 0], [-9, 13, 0, 0], [44, 0, 0, 0], [0, 0, 30, 3], [0, 0, 40, 12], [-45, -19, 0, 0], [-57, 13, 0, 0], [-32, 18, 0, 0], [-21, 9, 0, 0], [0, 0, -10, 17], [0, 0, -39, 1], [0, 0, -26, 5], [0, 0, 0, -12], [0, 0, -24, -5], [72, 6, 0, 0], [15, 13, 0, 0], [0, 0, -9, 15], [10, 18, 0, 0], [27, -11, 0, 0], [0, 0, -24, 8], [-6, 24, 0, 0], [18, -16, 0, 0], [-35, 27, 0, 0], [-16, -12, 0, 0], [0, 0, -12, 11], [0, 0, 32, 0], [-12, -16, 0, 0], [0, -2, 0, 0], [15, -39, 0, 0], [-33, -3, 0, 0], [0, 0, -12, -8], [0, 0, -12, -27], [0, 0, 34, -11], [0, 0, -24, -1], [0, 0, -16, 8], [-15, 3, 0, 0], [77, 3, 0, 0], [-63, -9, 0, 0], [0, 0, 16, 1], [-6, 36, 0, 0], [6, 6, 0, 0], [0, 0, 40, -12], [0, 0, 16, 5], [24, 4, 0, 0], [0, 0, 18, 19], [0, 0, -43, -7], [0, 0, 30, -3], [-15, 23, 0, 0], [0, 0, 20, 3], [-11, 39, 0, 0], [0, 0, 12, -20], [0, 0, 24, 3], [-9, -23, 0, 0], [36, -8, 0, 0], [0, 0, -18, -13], [0, 0, -4, -12], [0, 0, 16, -19], [-54, -6, 0, 0], [0, 0, 24, 24], [51, 21, 0, 0], [-24, 10, 0, 0], [0, 0, 9, -15], [0, 0, 26, 17], [-18, 38, 0, 0], [0, 0, 0, 12], [-39, 7, 0, 0], [-25, 9, 0, 0], [-18, 14, 0, 0], [0, 0, 6, -29], [0, 0, -11, 25], [36, 8, 0, 0], [9, -21, 0, 0], [21, 7, 0, 0], [0, 0, 24, 4], [0, 0, 18, 5], [0, 0, 35, -1], [0, 0, 8, 17], [27, 21, 0, 0], [0, 0, 12, 32], [22, -12, 0, 0], [24, -6, 0, 0], [56, 12, 0, 0], [0, 0, -36, -5], [72, -12, 0, 0], [0, 0, -5, -1], [-54, -20, 0, 0], [-9, 25, 0, 0], [0, 0, -21, -1], [0, 0, -8, 36], [0, 0, -14, 7], [23, -15, 0, 0], [0, 0, -24, -24], [-14, -24, 0, 0], [0, 0, 6, -33], [-21, 33, 0, 0], [45, 11, 0, 0], [0, 0, -15, -7], [0, 4, 0, 0], [0, 0, 32, -16], [12, 40, 0, 0], [-27, 27, 0, 0], [0, 0, -6, -35], [0, 0, -68, -3], [-48, -2, 0, 0], [0, 0, -44, -12], [-12, 16, 0, 0], [0, 0, -12, -11], [0, 0, -11, 9], [0, 0, 46, -7], [-48, 2, 0, 0], [-27, -29, 0, 0], [9, -53, 0, 0], [42, -12, 0, 0], [0, 0, 30, -15], [33, -1, 0, 0], [-9, -47, 0, 0], [30, 14, 0, 0], [0, 0, 18, -9], [0, 0, 24, 0], [0, 0, 37, -7], [24, 10, 0, 0], [0, 0, 9, -23], [0, 0, -6, -15], [0, 0, -38, 9], [0, 0, 24, 4], [33, 15, 0, 0], [-32, -12, 0, 0], [0, 0, 6, 27], [27, -31, 0, 0], [30, -26, 0, 0], [39, -27, 0, 0], [0, 0, -39, -15], [-20, 0, 0, 0], [-15, -1, 0, 0], [-3, 7, 0, 0], [0, 0, -36, 20], [0, 0, 10, 11], [0, -50, 0, 0], [-18, -2, 0, 0], [0, 0, 20, -15], [-81, 5, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9216_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9216_2_a_bk();".
// 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_9216_2_a_bk(:prec:=4)
    chi := MakeCharacter_9216_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(3067) - 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_9216_2_a_bk();".
// 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_9216_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9216_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-6, 0, 1]>,<7,R![16, 0, -16, 0, 1]>,<11,R![6, -6, 1]>,<13,R![4, 0, -28, 0, 1]>,<17,R![-12, 0, 1]>,<19,R![6, 6, 1]>,<67,R![-26, 2, 1]>],Snew);
    return Vf;
end function;