
// Make newform 8208.2.a.bs in Magma, downloaded from the LMFDB on 11 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8208_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_8208_2_a_bs();".
// 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_8208_2_a_bs();".
// 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 := [12, 0, -10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, -8, -1, 1], [-10, 6, 3, -1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_8208_a();"
function MakeCharacter_8208_a()
    N := 8208;
    order := 1;
    char_gens := [7183, 2053, 1217, 6481];
    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_8208_a_Hecke( : Kf := Kf)
    return MakeCharacter_8208_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, 0], [0, 0, 0, 0], [0, -1, 0, 0], [0, -1, 0, 0], [1, 0, -1, 0], [-1, 1, -1, -1], [-2, 0, 0, 0], [-1, 0, 0, 0], [-1, 2, 1, -1], [1, 0, 1, 2], [-1, 1, 2, 0], [0, -1, 0, 2], [-3, 1, 1, -2], [1, -2, -1, 1], [0, 2, 0, -1], [-2, 2, -2, -1], [1, 0, 1, 1], [-1, 3, 0, 0], [-5, 1, -2, -2], [7, -1, -1, 1], [-6, 0, 1, -3], [4, -1, 1, 3], [3, -1, 3, 2], [-6, 1, 2, 3], [-3, 0, 1, -1], [-6, 1, -4, -2], [0, 2, -2, 0], [7, 0, -1, -1], [-8, -1, 2, 0], [-2, -1, 0, -1], [0, -2, -4, 0], [2, -3, -2, -2], [-4, 4, 2, 2], [-6, 4, 0, 0], [1, -3, -1, 3], [-4, 5, 5, -3], [-5, 1, -2, -2], [-9, -2, -1, 1], [1, 1, 1, 1], [-2, 2, 0, -3], [9, 0, -1, -3], [-4, -3, 0, 2], [11, 0, -1, 2], [-4, -8, 2, 4], [-8, 2, 4, 0], [-2, 2, 4, 2], [0, 4, 1, 3], [8, -5, -1, 3], [6, 2, 2, 0], [-10, -1, 3, -3], [-3, -8, 1, 5], [4, -4, 2, 5], [-1, -2, 3, 1], [12, 1, 4, 0], [14, -7, 0, 0], [3, 8, -1, -2], [-9, 2, -1, 1], [-5, -4, 1, -3], [2, 1, -3, 1], [6, 7, -2, 0], [-12, 4, 2, 0], [-5, -2, -3, -6], [-19, 0, 2, -2], [1, -6, 3, 2], [3, -3, 4, -2], [-1, -4, 1, 1], [-1, 3, -1, -1], [-5, 4, 7, 5], [-11, 0, -9, -3], [-2, 7, -1, 3], [0, -4, 2, 2], [21, -4, -3, 1], [3, 9, -3, -1], [7, -2, 5, 7], [-9, -10, 0, 6], [-19, 0, 3, -5], [13, -6, 7, 3], [-14, 1, -3, -5], [15, -2, 1, 5], [-13, 0, -3, -7], [-10, -4, 2, -5], [6, -10, -2, -2], [8, 1, 4, -2], [-3, 4, -1, 5], [6, 7, -5, -5], [-5, 5, -1, -4], [-11, -3, 3, 0], [-8, 0, -9, -3], [1, -1, -3, -5], [4, -2, 2, -2], [0, 2, 6, 3], [3, -6, -9, 3], [-7, 1, 2, 2], [11, 1, -3, 4], [-12, -2, 2, -4], [-7, -2, 5, 4], [16, 6, 0, 3], [-6, -5, -6, 0], [-9, -1, 0, 2], [-13, 7, 4, 6], [3, -5, -7, -1], [2, 7, 4, 4], [-9, -8, 1, 5], [6, 13, -6, 0], [-7, 10, 1, 5], [-3, 11, -3, -5], [-4, -7, -2, 0], [3, 2, 5, 5], [-26, 1, -2, 0], [7, 8, 3, -5], [-1, -3, 0, 4], [-8, 2, -4, -2], [15, 4, 1, -3], [-6, 6, 8, 2], [-5, 1, 7, 1], [2, -1, -10, -5], [-13, -4, 3, -1], [14, -6, 0, 1], [5, 7, 1, 1], [-5, -6, 1, 1], [17, -5, -5, -1], [4, -4, 8, 2], [-7, -4, -1, -5], [3, 6, -7, -3], [-1, -2, -1, 1], [0, 3, -8, 2], [4, -2, -2, -6], [10, 8, 0, -7], [7, 5, -3, 1], [-15, -1, 6, 0], [-3, -2, -9, -1], [-35, 7, 3, 3], [10, -10, -12, -2], [-8, -3, 1, 1], [-14, 12, 8, -2], [-3, -7, -11, -1], [-18, 0, 0, -7], [-6, 14, 3, 1], [8, -6, -4, -5], [17, 0, 1, -5], [-11, -10, 4, -4], [0, -3, -10, -4], [5, 2, -3, 3], [-1, -2, 9, 1], [-13, -9, 7, 7], [-10, -14, 4, 6], [14, -13, 5, 1], [11, 8, -1, -4], [-12, 0, -12, -2], [-3, -3, -9, 3], [-9, -6, 5, -1], [12, -14, 6, 2], [-9, -2, 1, 5], [-14, 7, 8, -4], [3, -3, 3, -1], [14, -5, 4, 12], [5, -8, -1, 5], [1, 6, 3, 3], [-2, 11, -9, -3], [27, -10, -5, -3], [18, 5, 0, -7], [14, -2, -2, 1], [11, 6, 3, -3], [-20, 4, -14, -4], [-4, -12, 6, 13], [4, -13, 2, -2], [10, 6, 2, 6], [18, -10, 0, 8], [7, -18, 5, 1], [9, -4, 1, 3], [-20, 12, 2, -2], [-16, 5, -8, -10], [-8, 12, 4, 3], [9, 14, 2, -4], [4, -2, 10, -4], [-3, 18, -3, -1], [-11, 8, -1, -5], [-27, 7, -3, 3], [-9, -9, 0, -4], [18, -15, -3, 5], [9, -11, 7, -1], [22, -12, -4, 2], [-4, -5, -2, -4], [-14, -5, 6, 0], [3, -8, 9, 11], [-7, -10, -1, -1], [-15, 6, 3, -7], [4, -8, -1, -5], [8, 10, 2, 2], [-7, -12, 1, 2], [20, -2, -8, 0], [-13, 4, 13, -5], [8, 7, -1, 3], [11, 0, 1, 1], [-26, 0, 6, 3], [-4, 1, -4, -11], [-31, 2, 6, 0], [21, 3, 2, 0], [-23, -4, 1, -7], [-30, 2, -4, -9], [7, 8, 1, -4], [33, -11, 2, 4], [28, 2, -4, -6], [20, 10, -2, 0], [5, -3, 1, -2], [-32, 7, 0, -4], [9, 5, -5, -7], [-30, -12, 2, 6], [-8, -2, -6, -4], [-18, -1, -7, 1], [1, 9, 13, 3], [4, 3, -8, 2], [9, -3, -9, -15], [-12, 0, -8, -2], [26, -11, -10, -10], [12, 6, -6, 2], [21, -3, 3, 7], [-20, 7, 12, 10], [13, -13, 1, -3], [15, 12, -5, -7], [38, 0, -4, -2], [20, 8, 2, -4], [-17, 0, 3, -12], [11, 3, 5, -1], [0, -4, 4, 10], [1, 13, 4, -10], [-24, -8, 2, 7], [2, -5, -2, 2], [4, 16, 0, -12], [11, 1, -11, 0], [-12, 5, -7, -15], [-2, 2, 8, 10], [21, -1, -2, 0], [1, -7, 7, 10], [-14, -10, 0, 6], [1, -16, 3, -7], [10, 0, -9, 1], [35, -1, 9, 1], [-12, 7, 14, -9], [0, -15, -2, 12], [-4, 2, -8, 6], [-19, 16, -3, 3], [0, 15, 0, -2], [2, -9, 4, 10], [-16, -9, 4, -5], [13, 4, 3, -7], [1, -7, -11, -3], [25, -4, -9, -7], [-4, 19, 3, -1], [-36, 8, 4, -9], [12, -11, 1, 13], [-23, 4, -1, -13], [27, -4, -7, 4], [20, -4, -2, -10], [-20, 0, -2, 0], [5, 12, -5, 1], [3, 11, 0, 0], [1, 9, -7, -15], [-13, -10, -9, 4], [6, -12, 13, 11], [14, -5, 7, 3], [12, -10, 2, 6], [40, -10, -10, 2], [0, -11, -4, 0], [-43, -4, -1, 1], [-10, 10, -3, -3], [-16, -8, -4, 1], [13, -2, 11, 13], [3, -6, 10, 2], [7, 11, 5, 11], [25, -9, -3, 1], [-29, 1, -7, -17], [-11, -10, 0, -10], [-1, 1, 11, 15], [-8, -12, -8, 12], [-8, 12, 8, -10], [38, 9, -2, -1], [-1, 7, 3, -1], [23, -2, 5, -1], [53, -4, -1, 1], [13, -17, -3, 5], [23, 9, 7, 1], [25, -6, 1, 16], [1, 6, 15, 3], [2, -6, -4, -6], [-13, -6, -11, 3], [2, -5, -1, 1], [15, -20, -3, -2], [26, -4, -6, 12], [10, -9, 10, 5], [-37, 1, -3, -3], [-12, 1, 10, 1], [-15, 1, -11, 0], [1, -14, 1, -7], [30, -1, 0, 14], [-51, -4, -3, 3], [30, 11, -7, -7], [-43, 15, 9, 1], [6, 8, 0, -12], [-12, -6, -4, 6], [2, -6, -1, -1], [45, 2, -7, 3], [9, -14, -15, 11], [37, 2, 5, -1], [4, 8, 6, -4], [-12, 16, -6, -4], [37, 0, -11, 11], [23, 1, 17, -1], [42, -8, -10, 1], [-8, 14, -4, 2], [-5, -6, 11, -6], [-46, -3, 6, -12], [-19, 25, 9, -6], [4, -11, -7, 11], [45, -2, -5, -6], [6, -10, -2, -6], [-37, -4, 5, 3], [-7, 12, 3, -17], [-15, -11, -3, 11], [-25, -12, -7, 1], [-13, 1, -14, -12], [10, 11, 11, -3], [-32, -18, 2, 9], [14, 10, -10, -18], [-23, 24, 3, -3], [-3, -4, 15, -3], [18, -5, -7, -17], [-19, 13, 7, -3], [3, 15, 5, -15], [46, -13, -8, -6], [16, -4, 4, 10], [-15, -4, -15, -2], [-9, 6, -9, -25], [50, -3, -4, -2], [-5, -6, 3, -13], [-39, -10, -7, 5], [-45, 11, -3, -9], [-7, 5, -5, -2], [-23, 6, 10, -12], [28, -1, 6, 10], [27, -23, -17, 1], [-32, 8, 6, -2], [31, -14, 17, 9], [35, -15, 4, 0], [15, 20, 11, -7], [4, -14, -12, 14], [-2, -12, -10, -4], [-15, -17, -10, 12], [16, 5, -12, -18], [-19, 15, 11, 1], [22, 2, 9, 1], [39, -14, -17, 3], [-78, -3, 2, -4], [21, -3, -8, -12], [-3, 21, -7, -7], [-4, 21, 12, -6], [-9, -4, 9, 18], [24, 6, 2, -9], [-14, -2, -16, 2], [-30, -4, 10, 18], [10, 3, 3, -5], [1, -4, -23, 3], [-36, 21, 2, 6], [10, -16, -4, -4], [-13, 2, -15, 3], [29, 2, -13, -15], [18, 4, -12, 10], [9, -3, -6, -8], [75, 2, -1, 1], [-1, 7, 1, -14], [-7, 9, -3, -7], [41, -8, -5, -10], [9, 20, 7, -18], [0, 23, -6, -4], [-57, 3, -2, 4], [25, 10, 1, -11], [-48, 10, 12, 15], [7, 4, 0, 12], [-18, 8, -18, -14], [2, -4, 2, 2], [-43, -2, -1, 7], [23, 10, -9, -1], [-20, -6, -2, -6], [-1, -18, -5, 19], [-14, -12, 6, 2], [-5, -2, -13, 1], [-2, -1, -12, 6], [36, 8, -8, 2], [-22, 10, 18, 12], [40, 6, 10, -2], [-43, 8, 9, -13], [-3, 2, 7, 2], [36, 3, 2, 12], [-7, 10, 21, 9], [-2, 18, 12, 2], [10, -18, -14, 6], [0, -18, 10, -8], [10, -7, -22, -4], [-7, 15, -7, -19], [27, 11, -7, -1], [25, -14, 3, 13], [28, 6, -16, 2], [10, -17, -17, -5], [6, -19, 2, 8], [25, -8, -9, -5], [44, -5, 16, 12], [-19, -13, -6, 2], [-50, -5, 8, -15], [-25, -2, -4, 2], [28, 9, -10, 11], [-13, 16, 9, 9], [-20, 5, 6, 8], [-3, 2, -19, -3], [-55, -4, 7, -9], [12, 6, 13, -5], [60, -6, -4, 0], [13, 28, -1, -13], [-25, -7, -18, -4], [49, 16, -9, 1], [-33, -1, 13, -1], [49, 4, 3, -2], [-36, -1, 4, 2], [-12, -6, -16, -13], [13, 6, 15, 15], [40, 8, 2, 4], [-9, -22, 11, -6], [-34, 4, -16, -18], [8, -16, -12, -2], [30, 0, -6, -4], [60, 4, -6, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8208_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8208_2_a_bs();".
// 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_8208_2_a_bs(:prec:=4)
    chi := MakeCharacter_8208_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_8208_2_a_bs();".
// 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_8208_2_a_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8208_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![12, 0, -10, 1, 1]>,<7,R![12, 0, -10, 1, 1]>,<11,R![8, 1, -13, -3, 1]>],Snew);
    return Vf;
end function;