
// Make newform 5600.2.a.bv in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5600_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_5600_2_a_bv();".
// 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_5600_2_a_bv();".
// 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 := [-2, 7, 3, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [-3, 1, 1, 0, 0], [-2, -1, 1, 0, 0], [2, 5, -4, -2, 1], [5, -3, -6, 0, 1]];
    Rf_basisdens := [1, 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_5600_a();"
function MakeCharacter_5600_a()
    N := 5600;
    order := 1;
    char_gens := [351, 4901, 5377, 801];
    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_5600_a_Hecke(Kf)
    return MakeCharacter_5600_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, -1, 0, 0], [0, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 1, 0, 1, 0], [1, 1, -2, 0, 0], [1, 0, -1, 0, 1], [2, -1, 1, 0, -1], [2, 1, -1, 1, 1], [-2, 1, 0, 0, -1], [2, 0, 0, 0, 2], [2, -1, 1, 1, -1], [0, 2, 0, 0, 0], [-2, 0, 2, 2, -2], [2, -1, 2, 0, -1], [2, -1, -1, 1, -1], [4, 1, -1, -2, -1], [-2, -1, -1, 0, 1], [0, 0, -2, -2, 0], [0, 1, 3, -1, -1], [-2, -1, -1, 1, -1], [7, 0, -1, 0, -1], [-6, -3, 1, -2, 1], [0, 0, 0, -2, 2], [-3, -2, 1, -2, -1], [2, 1, -3, 2, -1], [6, 1, 0, 2, 1], [-4, 2, 2, 2, 2], [-2, -3, 2, -2, -1], [2, 1, -5, -1, 1], [4, 3, -5, -1, 1], [0, -3, -3, -2, -1], [4, 2, 0, 4, 2], [12, -1, -3, -2, 1], [-2, 0, 0, 4, 0], [7, -1, 0, -3, 0], [-4, 2, 2, 3, 2], [-4, 4, -2, 0, 0], [2, -3, 0, 2, 1], [-5, -1, 2, 0, -4], [8, 4, -4, 0, 4], [8, -1, -5, -2, -1], [1, -4, 1, 0, -3], [8, 1, 1, -1, -1], [-2, -5, 5, -1, -1], [10, 2, 0, 0, 0], [7, -1, 8, -3, 0], [6, -3, 0, -4, -3], [0, 0, -1, 4, 0], [2, -1, 1, 0, -3], [-8, 6, 2, 2, 2], [15, 3, 0, 1, 4], [-4, 4, -2, 2, 2], [4, -5, 1, 2, -3], [-10, 3, -1, -3, 3], [8, 0, 0, -4, 0], [8, 3, 3, 2, 3], [12, 4, -6, 6, 0], [6, 1, 5, -1, 1], [6, 3, 2, 0, 1], [-8, -2, -1, 0, 2], [7, 1, 0, -4, 6], [0, -2, 7, 0, -6], [0, -4, 10, -2, -4], [-5, -4, 5, -4, -1], [-4, -3, 7, -1, -5], [8, 2, -6, 0, -2], [10, 1, -3, -3, 1], [-10, 0, 0, 4, -2], [2, 5, 1, 2, -1], [-19, -2, 9, 4, -1], [4, 0, -8, -4, 4], [6, 3, -2, -2, 7], [0, 7, -1, 3, 5], [4, 0, 8, 2, -4], [8, -4, 8, -2, -4], [2, -3, -2, -6, -5], [3, 3, 2, 0, 0], [6, -2, 5, -3, -6], [-10, -4, 2, -8, 0], [8, 5, -9, 2, -1], [-2, -7, 6, 0, -1], [-11, -3, 4, -1, 0], [2, 1, 9, 1, 1], [14, -2, 0, 2, 0], [-14, -4, 10, 0, -2], [2, 1, -8, 2, 3], [-2, -7, 11, -1, -3], [4, -3, 7, -6, -3], [14, -6, 0, 2, -4], [-4, -4, -7, -4, 0], [-6, -6, 10, -2, 0], [-12, 1, 5, -3, 3], [-5, 1, 10, -1, 2], [1, -5, 2, 5, -2], [14, 7, -12, 0, 3], [-4, -3, 5, -6, 1], [-4, -2, -6, -2, 4], [-10, -1, 3, 6, 3], [-2, 7, 2, 0, 5], [6, -4, 10, -2, -6], [4, 3, -3, -7, 5], [-18, 5, 1, 2, -3], [-2, 1, 3, 1, -5], [12, 0, 4, 2, 4], [-9, 2, 3, -4, 5], [-18, 1, 9, 2, 1], [-5, 0, 9, 2, -5], [-7, 4, 1, 4, -3], [-14, 2, 4, 4, 2], [14, 3, 4, 0, 7], [16, 5, -9, 5, -1], [2, -1, 3, -1, -5], [0, -1, -1, -2, 5], [1, 4, 5, -4, 5], [2, -1, -11, 7, -3], [12, 8, -5, 8, 4], [8, 8, 0, 2, 0], [-24, -3, 5, 3, -1], [-7, 3, -6, -5, 6], [-8, -3, 11, 0, -3], [16, -2, 0, 0, 2], [1, 5, 6, 2, 4], [-10, 4, -6, 0, -6], [6, -3, -3, -4, -3], [-18, 1, 4, -4, -1], [-2, 7, -10, 2, 1], [-18, 0, 4, 4, -2], [-8, -2, 2, 4, -6], [-7, -5, 4, 2, 6], [9, -1, 6, 7, -6], [8, -3, 5, -3, -5], [27, 3, -8, 5, 0], [12, 5, 7, 9, -5], [12, -6, 4, 4, 0], [-2, 0, -14, 0, 0], [-27, -1, -8, 0, 6], [12, -10, 1, 0, -2], [3, -5, 2, -2, 0], [2, 6, -11, -7, 10], [16, -1, -11, -2, 5], [-22, -7, 10, 0, -5], [-10, 4, 0, -2, 2], [-24, 2, -8, -4, 6], [10, 3, 9, 2, 5], [-2, 10, -10, 2, 4], [4, -6, -2, -5, -6], [-14, -9, 11, -1, -5], [2, -1, 3, 4, -1], [6, 4, -8, -6, 2], [-4, -1, -1, -11, 1], [-26, 0, 2, -2, 0], [-10, -4, 14, 0, 2], [-8, -8, 4, -6, 0], [-2, -6, 2, 8, 4], [-8, -4, 8, -8, -4], [-5, 0, -5, 4, 3], [6, -4, -12, -2, 0], [-3, 6, -11, -4, 7], [14, -3, 9, -4, -1], [-6, 12, 4, 4, 6], [18, 2, -10, 0, 0], [-16, 1, -5, 3, 7], [-20, -5, 15, 10, -11], [12, 6, -4, 0, 6], [6, 1, 2, 0, 9], [4, 3, 9, 5, 1], [-25, 1, 0, -2, -2], [-10, -10, 13, -3, -10], [-4, 2, -14, 5, 2], [-16, -4, 8, -4, 4], [24, 1, -1, 0, 5], [-23, 2, -5, 2, -1], [6, 4, -2, -8, 2], [4, 6, 8, 6, -2], [-10, -6, 8, 0, -2], [21, 3, 6, 1, -6], [6, 3, -4, 0, 9], [-4, -6, -2, 2, -6], [2, 1, 5, 4, -1], [-2, 1, -15, 3, 5], [-2, 1, -5, 8, 1], [28, -3, 5, -5, -5], [6, 3, 3, 9, 3], [18, 1, 1, 3, 9], [-22, -1, 1, 2, -3], [2, -11, -3, -5, 1], [36, 4, -5, -4, 0], [6, -9, -4, -6, 1], [-4, -8, 6, -4, 4], [-2, 9, -11, 3, 5], [-2, -10, 4, 2, 0], [-1, -1, -8, -7, 8], [-18, -5, 5, -6, -3], [20, 8, -8, 8, 4], [-37, 8, 5, 2, 3], [-14, -4, 13, -3, 4], [-18, -5, -3, 5, -9], [25, -2, 1, 4, -5], [-10, -1, -2, -2, -5], [-2, 2, 10, -2, 2], [0, -14, 18, 0, -10], [12, -12, 4, -3, 0], [8, -6, 10, 8, -8], [-26, -9, 5, -4, -5], [-5, 5, 12, 8, -10], [-22, 2, 8, 0, 0], [18, 4, -2, 2, -2], [2, 3, -11, -5, 1], [6, -5, -3, -4, 7], [2, 5, 3, 11, 5], [28, -5, -5, -4, -5], [-18, 11, 5, -1, 3], [-24, -8, -5, -8, 0], [-4, 0, -4, -8, -4], [12, 0, 8, 0, -10], [6, 4, -8, -10, 2], [8, 4, -2, -6, 10], [-2, -2, -4, 6, -8], [26, 9, 5, 9, -3], [-26, 2, -6, 4, -6], [-8, 0, 4, 8, -8], [30, 7, -12, -6, 5], [8, 3, 5, 13, -3], [-20, -4, 3, -12, 0], [-6, 2, 18, 4, 2], [-7, -8, 15, -2, 1], [-3, 5, -12, -1, 0], [-22, 9, 4, 8, 5], [4, -6, 2, 0, 2], [4, -3, -3, -9, 11], [6, -11, 17, 0, -11], [8, 9, -17, -1, -1], [-22, -13, 16, 2, -3], [4, 0, 3, -8, 4], [10, -11, 20, -4, -7], [4, 6, -12, 0, 4], [-20, -17, 21, -7, -7], [-13, -3, 6, 11, -2], [-14, 14, -14, 4, 12], [-4, -10, 10, 4, -10], [22, 11, -1, 8, 7], [-26, -9, -2, -14, 3], [10, -6, 10, -2, -10], [7, 2, -5, -12, 5], [0, -10, 16, 2, -6], [8, -2, -6, -6, 2], [-18, 3, -13, 0, 7], [-21, -1, 8, 3, -4], [-10, 10, -20, 6, 4], [24, 5, -7, 1, 7], [10, 4, -14, -6, 12], [4, 10, 4, 0, 10], [-10, 10, -4, -4, -6], [-28, 16, -16, 3, 12], [-11, -13, 6, -3, -6], [10, -6, -2, -12, -6], [-10, 1, -3, 6, -7], [13, -5, -4, 0, 2], [-21, 2, 11, 2, -7], [-18, -11, 9, -5, 1], [8, 0, 2, -12, 0], [-20, -9, -1, -8, -1], [5, -5, 16, 4, -6], [-18, -13, 15, -1, -5], [-8, 9, -15, -6, 11], [6, -3, -2, 8, -5], [-26, 2, 12, 4, 2], [16, 4, -14, 2, -4], [16, 3, 15, 1, -15], [-14, -3, -5, -16, 3], [6, -8, 4, -4, -10], [-39, 4, 3, 2, 5], [28, 8, -12, 0, 4], [-18, 7, 11, 13, 7], [-18, 3, -6, 6, -1], [-4, 0, 8, -6, -4], [6, 6, 6, 14, 6], [14, 4, -1, 3, 8], [6, 11, 3, 4, 7], [-10, 1, 0, 12, -7], [11, 0, -9, 8, 3], [0, -8, 0, -6, 8], [40, 9, -11, 4, 5], [-4, 10, -11, -4, -6], [-23, 13, -4, -1, 4], [20, 12, -10, 6, 8], [-16, 1, 9, -7, -5], [-2, 14, -4, 4, 8], [-34, -6, -2, -2, 2], [22, -5, 0, 14, -3], [-4, 4, -11, -8, 8], [-14, -1, -3, 9, -13], [-32, -1, 11, -10, 5], [12, 1, -7, -15, -1], [6, -7, -13, -4, 11], [42, -2, 8, 2, -4], [-3, 1, -2, -12, -4], [-18, -1, 19, 8, -5], [48, 4, 3, 4, -4], [1, 10, -3, -6, 7], [-2, 3, 7, 7, 7], [5, -12, 5, -4, -7], [-20, 12, 4, 2, 0], [23, -1, 4, -9, -8], [18, 4, -16, -2, -2], [-46, -10, 8, 0, -4], [-20, -15, 3, -20, -11], [20, -8, -12, -12, -4], [-6, -5, -9, -7, 3], [-20, 12, 4, -4, 8], [6, 3, 2, -10, -3], [30, -2, -6, 6, -2], [52, 4, -12, -2, 0], [-14, 10, 0, 2, 0], [-20, 4, -8, 6, 10], [-6, 1, -11, -12, -3], [21, 2, 7, 14, -5], [13, -10, 9, 4, -9], [-6, 5, 2, 4, -1], [14, -9, 7, -4, -13], [22, 12, -4, 6, -6], [28, 7, 3, -6, -1], [14, 0, -24, 2, -2], [16, -5, 1, 3, -11], [10, -4, 16, -10, -8], [-9, -11, 2, 1, 2], [6, 1, 13, -10, -7], [22, 9, -9, -7, -3], [10, -7, -13, -5, 13], [-30, -2, -2, 2, 10], [18, 15, -15, 13, 7], [22, 8, 2, -6, 6], [12, -10, 19, -8, -2], [48, 2, -18, -2, -2], [-8, -12, 19, 4, -12], [-18, 1, -10, -2, 11], [-30, -9, 11, 13, -5], [-36, -2, 6, 4, 0], [-18, 7, -6, 0, 3], [18, 9, -11, -5, 13], [18, 9, -11, 5, -7], [30, 9, -5, 8, -5], [33, -18, -5, 2, -5], [-18, -11, 15, 7, 5], [12, 0, -8, -10, -12], [4, -3, 1, 0, 9], [-6, 0, 2, 10, 10], [-6, -4, 4, -18, 6], [-21, 9, -12, -2, -6], [-4, -9, 25, -10, -7], [20, -4, 10, 2, -8], [-2, 3, -8, 2, 13], [-26, -11, 4, -6, -3], [-18, 3, 16, 2, 5], [18, 19, -17, 1, -1], [-4, 0, 4, -6, 4], [20, 9, -19, -6, 7], [-28, 0, -18, 6, 4], [-12, -13, 27, -5, 1], [16, -9, -13, -1, 5], [6, 10, -10, 14, 2], [22, -7, -5, -11, -3], [13, 13, -16, 7, 8], [42, 9, -11, -2, 5], [-8, -13, 3, 5, 5], [31, -5, 18, 2, -4], [-2, -7, 25, 3, -3], [30, -2, -3, -3, 14], [-19, 1, -16, 7, 4], [54, -1, -7, -4, 3], [-8, -10, 16, 4, 2], [6, -5, 6, 4, 1], [40, 6, -18, -14, 2], [52, -1, 1, 1, 1], [30, 3, -1, 12, 7], [-7, -3, -12, 11, -12], [-25, 2, -13, 4, 5], [6, 6, 18, 0, -10], [-21, -2, -17, 10, 13], [24, 13, -9, -4, -7], [14, 1, -19, -15, 13], [-16, -14, 16, -12, -2], [22, 10, -8, -6, 12], [48, 3, -13, -2, 1], [10, 5, 8, 12, -7], [13, -22, 15, -6, -5], [-23, 7, -4, -4, 2], [20, -12, 20, 0, -8], [-2, 5, -2, 8, -3], [-14, 4, -5, -1, 0], [11, -7, -12, -6, -6], [-45, -1, -4, -7, -4], [38, -9, 3, 10, -5], [9, -7, 4, -9, 8], [8, 0, -20, 8, 0], [40, -4, -2, 4, -4], [0, -8, -6, 14, -6], [9, -11, 4, 15, -4], [30, 6, -2, 8, -2], [-22, -7, -7, -2, 7], [14, 4, -14, 12, 2], [6, 0, -4, -6, 6], [-35, -6, 13, -2, 3], [-24, -7, 27, 8, -11], [8, 10, 10, -8, 6], [-22, -3, -15, 7, -7], [-2, -10, 15, -13, 2], [-26, -7, 17, -2, 1], [22, -1, 7, -8, 3], [5, -2, -3, 0, -9], [-24, -3, -13, 5, -1], [22, -2, -10, 2, 0], [68, 2, -10, 2, 2], [10, 2, -18, -8, 16], [0, -11, 15, 8, -7], [-11, -6, -13, -2, 3], [-28, -12, -8, -6, 0], [6, -1, -9, -1, -9], [18, -7, -12, -4, -3], [2, 5, -14, -2, 11], [8, 4, 0, -5, -12], [24, -24, 20, -4, -8], [50, -7, 13, -4, -3], [-17, -12, 5, -12, -5], [-13, 1, -8, -8, 2], [-48, -2, -8, 6, -6], [10, -2, -10, 2, 2], [24, -7, -3, 10, -5], [-4, -6, 16, 2, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5600_2_a_bv();".
// 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_5600_2_a_bv(:prec:=5)
    chi := MakeCharacter_5600_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_5600_2_a_bv();".
// 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_5600_2_a_bv( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 18, -12, -9, 2, 1]>,<11,R![128, -248, 152, -25, -4, 1]>,<13,R![-824, 378, 80, -47, 0, 1]>,<19,R![-256, -184, 72, 30, -12, 1]>],Snew);
    return Vf;
end function;