
// Make newform 2320.2.a.u in Magma, downloaded from the LMFDB on 10 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2320_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_2320_2_a_u();".
// 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_2320_2_a_u();".
// 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 := [3, 8, -3, -7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [1, -3, -1, 1, 0], [1, -5, -1, 1, 0], [3, 1, -5, -1, 1], [-8, -3, 7, 1, -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_2320_a();"
function MakeCharacter_2320_a()
    N := 2320;
    order := 1;
    char_gens := [2031, 581, 1857, 321];
    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_2320_a_Hecke(Kf)
    return MakeCharacter_2320_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], [-1, 0, 0, 1, 0], [1, 0, 0, 0, 0], [-1, 0, 1, -1, 0], [-2, 1, -1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, -1], [-2, -1, -1, 0, 0], [0, -1, 1, 0, -1], [1, 0, 0, 0, 0], [-1, 0, -1, -1, 0], [-3, 2, -2, 3, 1], [0, 0, 2, -2, 0], [-2, -1, 0, 0, 1], [-1, 0, -2, -1, -1], [-1, -1, 0, -1, -2], [-6, 2, -2, 0, 1], [0, -2, 0, 0, 1], [-4, -1, 3, -2, 0], [4, 0, 0, -4, 0], [0, -1, 2, -4, -1], [-2, -1, 3, -2, 1], [-2, -3, 1, 0, 0], [0, 4, 0, 2, 0], [-3, 0, -2, 1, -2], [-4, 0, 2, 4, -1], [-4, -1, -1, 2, 0], [-2, 3, -1, 4, 0], [2, 0, -2, 4, -2], [-5, -4, -2, -1, 0], [-5, 2, 0, -1, -1], [-2, 1, -1, 4, 2], [-9, 2, 0, -1, 0], [-5, -1, 4, 1, -2], [-2, 0, -4, 4, 2], [4, 0, 0, 0, 2], [-5, -2, 4, 1, 1], [-5, 2, 0, 3, 3], [5, 0, -3, 5, -2], [-12, 0, -4, 8, 3], [-10, -2, 2, 0, -1], [6, -4, 2, 0, -2], [9, 0, 1, 1, 0], [1, -2, 4, -3, 0], [-1, -3, -2, 3, -2], [-6, 2, 0, 6, 0], [-6, 3, -3, 0, 0], [4, -1, 1, 4, -1], [12, -5, 5, -6, -4], [3, 3, 2, -1, 2], [-14, 0, -4, 0, 0], [2, 2, 0, -6, 2], [6, 0, -6, 2, 1], [4, -5, 3, -2, 2], [-8, 0, 2, -2, 0], [-1, 4, 2, 7, -1], [-9, 1, 0, -1, -2], [2, 5, -7, 0, -2], [12, 2, 2, -6, -2], [5, -3, 4, -7, 0], [8, -3, 1, -6, 2], [-7, 0, 4, -5, -3], [-5, 2, -4, 3, 3], [16, -1, 3, -4, -3], [2, 2, -2, 4, 4], [3, -4, 10, 1, -1], [2, 5, -1, 4, 0], [0, 1, -2, 0, 1], [4, 5, -7, 6, -2], [4, 2, 0, -10, 2], [-14, 4, -2, 4, 0], [-2, 1, 1, 6, 3], [-7, -2, -4, -7, -1], [-1, 1, -2, 3, -2], [-4, -5, 5, 2, 2], [9, 0, 1, -7, 2], [-2, 6, -2, -4, 0], [-7, 7, -6, 1, -2], [0, -2, 0, 0, 3], [14, -4, 2, -8, 0], [5, 5, 0, 7, 2], [4, 6, -2, 2, 2], [-8, 0, -4, 4, -2], [1, 0, 2, -1, 3], [2, 0, -2, -2, -2], [-11, -2, -2, -9, 0], [-14, -4, -2, 8, 2], [0, 2, -6, 2, 2], [9, -1, 2, -3, -4], [2, 1, 1, -4, 6], [-4, 1, 2, -6, -1], [3, 4, -7, 11, -2], [-4, -1, -3, 0, 1], [4, -5, 3, 2, 0], [-14, 2, 6, -4, -1], [-7, 10, 0, 5, 1], [14, 4, -6, -8, 2], [1, 1, 0, -3, -4], [6, 3, -9, 4, -4], [13, 3, -10, 9, 0], [-6, 3, -1, 0, 0], [-4, 10, -2, 4, -1], [-13, -2, 6, 1, 0], [-2, -4, 10, 0, -4], [-6, -6, 6, 4, -1], [6, -7, 4, -10, -3], [8, -1, 5, 6, 4], [-4, 2, 2, -6, 0], [-11, 4, 3, 5, -2], [18, -10, 2, -8, 2], [-13, 4, 2, -5, 3], [13, -3, -2, -7, 4], [-16, 1, 6, 0, -3], [0, -3, -1, -2, 4], [-6, 0, -6, 6, 6], [8, 4, 6, -10, 0], [-18, -1, -3, 0, -4], [-2, -3, 1, 8, -2], [-15, -2, 0, 3, 5], [0, -3, -1, 10, -6], [0, 6, 2, 2, 0], [14, 2, 4, 0, 2], [9, 0, 0, -9, -1], [-26, -1, 3, -4, -4], [7, -1, -4, 5, 6], [-4, -6, 10, -2, 0], [12, -2, -2, -2, 0], [16, 0, 4, -4, 2], [-25, 0, 0, 7, 1], [7, -6, 2, -7, -3], [-14, -5, -1, 8, 6], [5, -6, 0, -7, 1], [14, 3, -1, 0, 6], [-13, 4, -8, -3, 3], [-3, 5, -2, 9, 2], [-12, 8, -4, 14, 0], [-5, 8, 2, 5, 1], [4, -3, 5, -18, 2], [-25, 4, -4, 5, 1], [-6, 2, 0, 0, 0], [25, -3, 4, -5, -6], [14, -4, 0, 4, -2], [-27, 2, -2, 5, 1], [6, -7, 8, -4, 3], [7, -11, 4, -9, -2], [20, -5, 3, -2, 0], [-1, -6, -2, -11, -1], [6, -6, 2, 4, -6], [10, -1, 11, -12, -4], [2, -1, -7, -2, -1], [6, 8, 8, 2, 1], [8, 4, 2, -10, 4], [28, 3, 3, -14, -6], [-1, -6, 8, -5, 3], [18, 1, 0, 8, -1], [-8, -3, 5, -8, -3], [12, 4, -4, 12, -4], [-19, 1, -10, -7, -2], [2, 0, 6, -8, 6], [-8, -6, 0, -6, -2], [3, 6, -6, 9, 4], [10, 8, -4, -8, 0], [-21, -2, -2, -1, -3], [-4, 9, 1, -6, 6], [-26, -6, -2, 12, -2], [-1, 4, 8, -5, 3], [-6, -2, -4, 14, 0], [13, -6, 8, -9, 1], [-1, -1, -10, -1, -2], [-7, -4, 10, -1, 1], [-16, -6, -2, -8, 0], [12, -2, 4, 2, 8], [-18, 0, 6, -2, 0], [-1, 2, 0, 3, 6], [-4, 8, -4, 12, 0], [-3, -3, 8, -15, 4], [25, 5, 8, 3, 2], [0, 4, -6, 12, 5], [-5, -8, 2, 3, 3], [-6, -8, 12, -12, 4], [-15, -4, 4, -7, -1], [-10, 1, -5, 8, 4], [5, 3, -6, -3, -2], [-4, 0, 6, 6, -2], [13, 2, -7, 9, 4], [-12, -2, 8, -10, 6], [1, -2, 10, -9, -3], [-11, 8, -4, 3, 0], [-4, 2, 2, -6, -4], [-20, -4, 4, 4, 0], [-10, -6, 0, -20, 0], [-22, 3, -8, -4, -3], [24, -7, 5, -10, -6], [3, 3, -4, -1, 0], [13, -8, 0, -13, 4], [14, -6, 4, -16, -2], [-26, -6, 10, -4, 2], [6, 10, -6, 2, 3], [-4, 4, -12, 2, 6], [-18, -5, 5, 6, -7], [-6, 4, 2, -6, 1], [24, 6, 2, -4, 0], [9, 8, -6, 7, -3], [6, 6, -12, 16, -2], [-10, 1, 5, 16, -2], [-17, -7, -2, 3, 2], [-2, 9, 9, -10, 7], [-16, 13, -13, 6, -2], [-16, -6, -4, 8, -1], [18, -5, -7, 0, -4], [-19, -4, 6, 3, -1], [10, 0, 0, -4, -8], [-15, 8, -4, 1, -5], [9, -2, 8, -15, -7], [-14, 15, -5, 10, 5], [18, 8, 10, -2, 1], [-30, -3, 15, -10, -3], [24, -2, 0, 2, 8], [-6, -3, 5, -12, -2], [-11, 2, 4, -9, -3], [-23, -7, 8, -7, 4], [2, -8, -6, -6, 0], [-22, -2, 4, 16, 0], [25, -6, -6, -7, 5], [22, 3, -3, -8, 4], [-26, -2, 0, 6, -1], [-3, 4, -10, 19, 3], [14, -11, 1, -6, -3], [5, -2, -14, 13, -3], [-7, -3, 8, -21, -2], [1, -10, 6, 7, 1], [-34, 4, -8, 12, 7], [-6, -1, -13, -4, 2], [14, 2, -6, -8, 0], [4, -5, -13, 10, -2], [5, 12, -10, 1, -5], [-14, -8, 6, 4, 0], [-5, -6, 0, -3, 7], [13, -7, 4, -9, -2], [-10, 1, 5, -20, 2], [0, -1, 6, 2, 9], [14, 0, -8, 8, -3], [-5, -6, -3, -1, 2], [-26, -12, -2, 6, 1], [8, 6, 0, 14, -2], [8, -8, 8, -12, -6], [19, 4, -7, 11, 4], [-7, -9, 10, 7, 2], [17, -3, 4, -5, 2], [33, -2, 6, -23, -3], [17, -4, -6, 3, -5], [46, 0, -4, 4, 4], [-3, -8, 2, -11, 1], [4, -2, -6, 6, 6], [-13, -6, -2, 9, -1], [-17, -1, 4, 5, -10], [24, 0, -2, -12, 1], [-15, -6, -6, -5, 0], [4, 6, 10, 0, 1], [-14, 14, -8, 16, -4], [-27, -4, -4, 1, -7], [1, 2, -2, 11, 0], [0, 10, -4, 14, 4], [25, 10, -14, 11, -5], [-33, -6, -4, 17, 5], [8, -9, 1, -10, 2], [-1, -5, 6, -5, -2], [24, 2, -8, -6, 0], [1, 6, -2, 7, 6], [0, -10, 6, 8, -1], [12, -14, 0, -14, 4], [-4, 11, -11, 26, 4], [6, 10, 4, -4, 0], [-25, -6, -5, 3, -2], [-27, 10, -8, -3, -2], [9, 2, -2, -3, -1], [-34, 7, -4, 16, 1], [16, -2, 6, -6, 4], [-26, 2, -10, -4, -6], [16, -4, 12, -12, -4], [20, -7, 3, 12, -7], [-20, 10, -2, 8, 0], [12, -5, -7, 4, -1], [6, -8, 6, 0, -6], [-13, 2, -2, -7, 4], [-3, 4, -3, -3, 0], [5, -4, 6, 1, -6], [-43, 0, -2, 7, -1], [12, -2, -2, 8, 4], [6, 0, -16, 12, 0], [46, 0, -4, 8, 0], [30, 7, -7, 16, 0], [0, 16, -4, 18, 4], [-16, 7, -1, 2, -2], [9, 0, -16, 7, 3], [0, -6, -10, -10, 2], [-10, -5, -1, 16, -6], [-3, -7, 12, -7, 2], [-18, -10, 6, 0, -5], [2, 5, 16, 0, -1], [34, 3, 10, -18, -3], [-15, -3, 4, 13, 2], [23, -10, 11, -5, 6], [25, -4, 4, -21, -2], [0, -1, 3, -14, 6], [-32, 4, 6, -2, 8], [-5, 12, 2, 7, -1], [-28, 2, -4, 2, 4], [20, -2, -2, 4, -10], [12, 4, -12, 6, -2], [3, 4, 11, 3, 6], [-21, -9, -10, -9, -2], [27, 11, 4, -13, 0], [28, 3, 1, 2, -4], [10, -3, -1, 6, -3], [25, 5, 0, 5, 12], [12, -17, 13, -18, 4], [8, 0, 12, -8, 0], [8, -4, 14, -14, -2], [-32, 6, -2, 18, -6], [12, 3, 11, 10, 0], [-18, -2, -10, 8, -4], [32, 2, -10, -10, 2], [5, 12, -12, 25, 5], [12, 10, -8, 2, -2], [-18, -2, 18, 4, -6], [-21, 1, -2, -11, -2], [42, 5, -5, -12, 2], [-45, -2, -4, 7, 3], [-18, 4, 4, 0, 8], [-32, -2, -2, -2, 2], [24, -4, 0, 0, 5], [14, 2, 4, 6, 2], [-10, -7, -2, -12, 1], [22, 5, -1, -24, 4], [2, -1, -7, 4, -8], [24, 2, 4, 2, -2], [7, 12, -14, 5, -3], [-8, 12, -2, -6, 6], [8, 3, -11, -8, 7], [15, 10, -4, 13, 5], [-8, 0, -4, 14, -2], [3, -7, 12, 19, -2], [-10, 6, -4, -2, 10], [26, -14, 6, -8, 0], [18, 1, -11, 8, -8], [-28, 6, -16, 28, 13], [-1, 0, 0, 5, -6], [28, -3, 13, -24, -5], [-31, -4, -8, -1, -1], [-6, -6, 18, -16, -3], [18, -4, 6, -12, -4], [-36, -10, 0, -12, 1], [-19, 18, -9, 17, -6], [26, 4, 8, 12, 0], [-9, -4, 6, 19, 4], [21, 14, -11, 9, -6], [-22, -8, 8, -12, 1], [13, -14, 4, -15, 8], [9, 8, -9, -15, -4], [-12, 0, -8, 6, -2], [-14, 4, -22, 6, 5], [-3, -6, 4, -3, 3], [-35, 13, -4, 7, 6], [-3, 2, -14, -5, 2], [15, 2, -8, 3, -10], [5, -10, -2, 7, -5], [2, -3, 15, -2, 5], [-8, -4, -6, 10, -14], [-64, -9, 3, 6, 4], [38, 8, 12, 8, 1], [9, -2, 5, 5, 4], [6, 2, 0, 6, 15], [12, 16, -20, 0, -6], [32, 10, -6, -4, 1], [4, 3, 17, 6, -4], [-24, -5, -13, 12, 1], [-41, 16, -2, -5, 12], [12, 2, 8, -12, 3], [30, 10, -2, -12, -6], [8, -12, 14, -16, -13], [10, -14, 6, 4, -8], [3, 6, -14, -1, -3], [7, 4, -6, 13, 17], [8, 8, -28, 4, 4], [17, 2, 9, -3, 4], [-11, -4, -19, 13, 0], [-17, -3, 12, 3, 4], [55, -2, 2, 9, -2], [31, -6, 0, -21, -1], [-12, 6, 2, -10, -6], [24, -2, 2, -18, -2], [-36, 13, -5, 14, 10], [13, -8, 0, 15, -6], [21, -8, -3, -3, -4], [34, -14, 22, -12, -4], [-4, 0, 16, -16, -10], [-6, 2, -12, -8, -8], [0, 0, 4, 4, 8], [-43, 9, -4, -3, 0], [26, 16, 0, -8, 2], [8, -11, 12, 8, 3], [-61, -6, -2, -1, -3], [-18, -6, 16, -20, 0], [32, 18, -6, -6, -4], [8, 12, 4, 12, -2], [-12, 2, -22, 0, 5], [-8, -6, -4, 6, -18], [42, 11, -4, -12, 1], [-22, -6, -14, 4, -5], [-34, 2, -6, 4, 2], [34, 8, -8, -14, 3], [14, 5, 1, 8, 0], [-39, -1, 6, -1, -6], [-43, -10, -8, -19, -6], [-8, 0, 4, -6, 10], [-2, 5, -3, 14, 3], [-36, 21, -11, 22, 4], [6, -1, 2, 2, -3], [19, -6, 14, 7, 1], [-2, -12, 6, 0, 2], [31, -4, -4, -19, 5], [3, 0, -2, -1, -7], [-36, 13, -5, 18, 0], [0, -10, -4, 6, 6], [4, -10, 10, -28, -9], [-48, -5, -1, 14, 6], [-14, 8, -26, 20, 12], [-4, 20, 4, 8, 0], [-39, -8, 15, 9, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2320_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2320_2_a_u();".
// 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_2320_2_a_u(:prec:=5)
    chi := MakeCharacter_2320_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_2320_2_a_u();".
// 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_2320_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2320_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 8, -20, -7, 3, 1]>,<7,R![-48, -160, -88, -1, 7, 1]>,<11,R![64, -224, -112, 12, 10, 1]>],Snew);
    return Vf;
end function;