
// Make newform 5610.2.a.cj in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5610_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_5610_2_a_cj();".
// 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_5610_2_a_cj();".
// 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, 7, 6, -6, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [1, 2, -4, -2, 1], [5, -6, -4, 2, 0], [-6, -6, 6, 2, -1], [12, -4, -10, 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_5610_a();"
function MakeCharacter_5610_a()
    N := 5610;
    order := 1;
    char_gens := [1871, 3367, 1531, 3301];
    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_5610_a_Hecke(Kf)
    return MakeCharacter_5610_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 := [[1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [1, 0, 0, 0, 0], [2, -1, 0, 0, 0], [1, 0, 0, 0, 0], [-1, 0, 0, 0, 1], [2, 0, -1, -1, 0], [-1, 0, 1, -1, -1], [-1, 0, 0, 1, 0], [5, 0, 0, 1, 0], [2, 1, 1, -1, 0], [-1, -1, 0, 1, 0], [2, 0, 0, 2, 0], [-1, 0, -1, 0, 0], [4, -1, 1, 0, -1], [-1, 0, 2, 1, 0], [1, -1, -1, 0, 0], [-1, 1, 0, -1, -2], [2, 1, 1, 0, -1], [3, 0, -1, 0, 2], [4, -2, -1, 0, 1], [0, -2, 0, 2, 0], [4, 1, 0, -1, -1], [-5, -1, 0, -2, 1], [-2, 0, 1, 0, 1], [1, 0, -1, 1, -1], [3, 0, 0, -1, 0], [2, -2, 0, -1, 1], [-2, 2, 0, -2, 0], [-6, 2, -1, 2, 1], [3, 2, 0, -3, -2], [-5, 0, 1, -3, -1], [-1, -5, 1, 2, 0], [1, -2, 0, -1, 0], [-3, -1, 0, 2, 3], [-4, 2, -2, 0, 2], [1, -2, -1, -1, 1], [0, -2, 1, 2, -1], [2, -3, 0, 0, 2], [10, 0, -2, 0, 2], [3, 2, -3, -1, 1], [-2, 3, -1, 0, -3], [-4, -2, 1, -2, -1], [7, 0, 0, 1, 0], [1, 2, 1, 1, -3], [0, 0, 1, 0, -1], [-7, 4, -1, -1, 1], [8, 0, 3, 2, 1], [-2, -4, 0, 2, 2], [-2, 0, 0, 0, 2], [3, -6, 0, 2, 1], [-2, -1, 2, -2, -2], [-1, -2, 3, 3, 1], [0, 0, 1, 0, -1], [-4, -4, 1, 0, 1], [-3, 2, -1, -1, -3], [1, 0, 1, 1, -1], [-10, -2, -2, 0, -2], [-9, 2, 1, 1, -1], [-6, 4, -2, 0, -2], [-5, -1, 0, 3, 2], [-13, 3, 2, 1, -2], [-4, 5, -2, -1, 1], [8, 0, -2, 0, 0], [4, -2, 2, -2, 0], [-6, 5, 1, 0, -1], [-3, 4, -1, -1, -3], [6, -2, 0, -3, -1], [-1, -2, 2, -3, 0], [-2, -2, -2, 2, 2], [-5, 5, 0, -2, -1], [2, -3, -1, 0, 1], [-5, 8, 0, -3, 0], [-18, 2, -2, -2, 2], [1, 1, 0, 3, 2], [1, 0, -1, 3, 3], [-1, 7, -1, 1, -1], [4, 0, -2, 4, 4], [0, -2, 2, 2, 0], [-4, -4, -1, 0, 3], [-15, -1, 0, -2, 1], [-12, 6, 0, 0, -2], [5, 4, 1, 6, 0], [-10, 4, -2, 1, -1], [-11, -1, -1, 1, 1], [-1, 0, -2, -1, 2], [-15, -3, 0, 0, 1], [-14, 4, 3, -2, -3], [4, 2, 0, 5, 3], [-11, 1, -1, -4, 0], [-5, 7, 2, -4, -1], [-9, 3, 4, 0, -3], [-5, 2, 1, -1, -3], [1, 0, -3, 3, -1], [-5, -3, 0, -3, -2], [7, -5, 3, -3, -1], [-9, 1, -2, 1, 2], [9, -4, 3, 1, -3], [11, -6, 0, 5, 0], [0, 6, -2, 0, 0], [-6, -2, -1, -2, -3], [4, 4, -3, -2, -5], [-23, 4, -1, -1, 1], [-2, 2, -2, -2, 0], [-14, 0, 2, 1, -1], [-12, 0, -2, -2, 2], [11, 2, 2, 5, -2], [-5, -4, -2, 2, 3], [0, 5, -4, -4, 0], [10, -3, 3, -8, -3], [10, -4, -2, -1, 3], [-1, -6, 0, 5, 4], [10, -4, 0, 4, 2], [-17, -1, 6, 3, -2], [-4, -2, 2, -2, 0], [-8, -6, 4, 2, -2], [6, 0, 2, 0, -2], [-25, 3, 0, 0, 1], [14, 2, -3, 2, 1], [-2, 7, 3, -4, -3], [0, 2, 0, 0, 2], [-4, 0, 3, -2, -1], [11, -10, 0, 3, 2], [5, 1, 0, -4, 1], [12, 0, 2, 2, 2], [15, 5, 0, -7, -4], [-8, -8, 0, 0, 0], [2, -7, -4, 2, 2], [15, -2, 0, -6, -1], [-27, 2, -1, -3, -1], [1, 6, -1, -7, -1], [-1, -5, 4, 6, 1], [-18, 8, -5, 0, 1], [8, 0, 0, -6, 0], [-17, 0, -2, 0, 1], [3, 0, -4, 3, 6], [15, -6, -2, 0, 1], [-8, -3, 3, -1, 0], [5, 0, -5, 1, -1], [-15, -2, -3, -1, 1], [-9, 1, -2, 0, -5], [15, 2, -1, 5, -3], [2, -6, 3, 0, 1], [-13, -5, 0, 1, 0], [-3, -4, 1, 5, -1], [-8, 4, 3, 0, -1], [8, 6, -2, -2, 0], [-6, -4, -2, 4, 0], [-11, 8, 1, -3, -1], [-11, -1, 4, 3, 2], [-3, -7, -2, -2, -1], [-11, -4, 1, -3, 5], [4, 2, 2, -6, -4], [-7, -7, 0, 1, 2], [-9, 10, -2, -3, -2], [1, -3, 5, -1, 1], [11, 8, 2, -5, -2], [-11, -10, -1, 3, -1], [0, -6, 2, -4, -2], [6, 0, 2, 4, 6], [4, 4, 2, 2, -4], [-16, 1, -4, 0, 0], [7, 6, -2, 1, 0], [-6, -10, 2, 3, 1], [23, 0, -2, 3, 0], [13, 2, 1, -7, -5], [-17, 6, 1, -2, 0], [8, 4, 0, 4, 2], [-25, 5, 2, -2, 1], [12, 0, 3, -2, 1], [7, 3, 2, 2, 5], [-3, -8, 2, 1, -2], [18, 4, -2, 0, 0], [13, -3, -5, 5, 1], [-3, 6, 3, -5, -3], [-12, 6, -4, -6, 4], [-2, -4, 6, 4, -4], [20, 4, -2, -1, 1], [12, -4, -1, -2, -1], [-20, -2, 3, -2, -1], [-7, 4, -4, 5, 4], [-26, 8, -4, 0, 0], [-16, -8, 0, 0, 4], [19, 5, -2, -3, 0], [9, 2, 1, 5, 3], [-7, 9, -6, -4, 1], [-11, 12, 1, -6, -2], [-7, -3, 1, 0, -2], [4, 1, -5, -3, 6], [21, 0, 3, 5, 3], [26, -2, -6, 2, 6], [9, 3, -4, -7, 0], [-1, -2, -1, 5, 5], [-4, 6, -6, -3, -1], [11, 0, 2, 6, 3], [-41, 0, 2, -1, -2], [-12, 2, -3, -2, -5], [-22, 0, -5, 4, -1], [21, -2, 3, 7, -1], [-27, 2, 1, 0, -2], [3, -7, 2, 2, -5], [22, -4, -2, -4, 2], [8, -1, 6, -4, 2], [23, -6, -3, -2, 4], [22, -6, 4, 4, -6], [6, 4, 5, -6, -1], [-18, 7, -1, -5, 2], [2, -2, 2, -2, 2], [24, 1, -3, 4, 9], [-15, 7, 4, -1, -6], [7, 9, -2, -6, -7], [-4, 8, -2, -3, -3], [3, 3, 1, 2, 2], [6, 6, 2, 2, -8], [-18, -2, 2, -8, -4], [-10, -2, -2, 0, -2], [-18, 0, 0, -3, 5], [7, 4, -1, -6, -2], [-10, 4, 0, 3, 1], [29, 2, -1, 6, 2], [14, 4, 2, 0, -6], [-35, 3, 8, 0, -5], [-18, -2, 8, 2, -4], [26, 4, -4, -1, 5], [-10, -6, -2, 3, 3], [19, 5, -2, -1, 4], [-8, -2, 8, -2, -2], [-24, -4, 5, 2, -3], [29, -1, 0, 4, 1], [-1, -14, -1, 3, -1], [11, -4, -2, -9, 0], [-16, 4, 5, 2, -5], [1, -13, -4, 8, 3], [-10, 6, 4, -4, 2], [-14, 2, 2, -6, -4], [-34, 6, 2, -2, -4], [6, -10, -2, 8, 0], [11, -6, 0, 1, -4], [4, -6, -6, 6, 0], [25, -6, 0, 4, 3], [-26, -2, 1, 0, 1], [38, 2, -1, 6, 3], [-14, 4, -8, -2, 2], [15, -5, 2, -8, -1], [25, -1, -2, 0, 1], [-14, 4, 2, 2, -2], [-17, -6, -1, 0, 10], [-8, 6, -2, 0, -6], [30, 7, 2, 0, 2], [-28, -3, -1, 1, 6], [6, 2, 0, 0, 6], [-4, 2, 2, 2, 2], [4, -14, -2, 4, 0], [9, 1, 0, 8, -1], [12, -4, -2, 2, 6], [3, -6, 4, 1, 0], [22, -12, 4, -2, -2], [21, 4, 3, 7, -1], [26, 5, 0, 0, -2], [-2, -4, -2, 2, 6], [11, -8, 4, 5, -2], [-33, 8, -5, -1, 1], [-20, 2, 4, -1, -5], [30, 0, 6, 0, 0], [-9, 1, 2, -3, 0], [24, -4, 4, 2, 4], [-15, -8, -2, 7, 2], [-18, -6, 2, -4, 2], [12, -8, -6, 6, 6], [-17, 1, -3, -8, -4], [26, 2, 2, 2, -4], [11, -2, -6, 1, 2], [-9, 4, -4, -5, 0], [6, -2, -1, 2, -7], [30, 4, 0, 5, 1], [18, -6, -8, -5, 5], [8, 0, 4, 4, 2], [-21, 7, -6, 2, 1], [12, 12, 4, -2, 2], [-1, -2, 0, -1, 2], [17, 0, -5, -5, 1], [14, -4, -4, -5, 5], [-31, -4, 0, 1, -2], [17, -2, -5, -5, 1], [0, 5, 8, -5, -1], [34, 6, 3, 8, -3], [24, 8, 4, 4, 0], [-2, -8, -5, 0, 1], [8, 0, 3, 8, -5], [-8, 2, -6, -2, -4], [4, -16, -4, -2, 4], [-24, 10, -7, -2, -3], [9, -2, -5, 5, 1], [21, -4, 3, -7, -3], [-6, 6, 0, 2, 4], [-11, -10, -2, 2, 9], [2, -4, 1, 2, -1], [-17, -7, -6, 1, -2], [37, 2, 2, 1, -2], [8, 2, 2, 4, -8], [-1, -4, 1, 2, 2], [0, 8, -1, 4, -7], [23, 10, 2, -1, -8], [-8, 3, 2, 7, 1], [12, 8, 0, -11, -5], [-14, 16, -2, -4, -2], [-9, 4, -5, 9, 5], [-18, 3, -2, -6, -4], [-21, 2, -7, -1, -3], [30, 0, -5, 6, 7], [-30, -3, 3, 3, 2], [-27, -7, 5, 2, -4], [-14, 18, 0, -6, -4], [58, -4, 2, -2, 0], [20, -1, 1, 8, -1], [-33, -7, 1, -2, 2], [16, 9, 3, 1, -2], [12, -8, -1, -6, 5], [7, 4, 3, 5, -7], [-12, -1, -8, -5, 3], [19, -5, 3, -10, -2], [-18, -8, 7, 4, -1], [-4, 4, -11, -6, -1], [52, 4, 1, 4, 1], [1, -4, -3, -3, 5], [10, -10, 8, 6, -4], [-5, 8, 5, 1, -9], [4, -10, 3, 6, -3], [5, -4, -2, 5, -6], [-8, 4, 0, 4, 2], [35, 6, 5, 10, 2], [-28, -4, -4, 0, 4], [5, -4, -5, -3, 3], [26, 0, -5, 6, 9], [24, 8, 3, 2, 1], [0, -8, -3, -2, -1], [-20, 0, 4, 0, 2], [10, -16, 2, 4, 4], [21, 1, 6, 4, 3], [-3, 2, -2, 7, 4], [-39, 10, 3, -5, -1], [-8, -2, 4, 12, 0], [9, 4, -6, 0, 3], [-2, -8, 10, 6, -2], [-8, 4, 2, -6, 6], [-39, 5, 2, 0, -3], [17, -1, 4, 0, 3], [-8, 12, 0, -4, 2], [11, 2, 4, 2, -1], [-28, 5, 2, 1, 3], [23, 5, 4, -3, -2], [6, 3, -5, -6, -3], [44, -2, -2, -5, -3], [-42, -8, 2, 4, -2], [33, -6, 3, -9, -3], [0, 8, -8, 2, 2], [16, -6, -1, 6, -3], [-2, 2, 0, -6, -12], [9, -2, 3, 1, -3], [-12, 9, 3, 1, -6], [18, -12, 4, 0, 6], [34, -4, -3, -2, -1], [-5, -1, 3, 8, 2], [42, -2, -10, -2, 4], [14, -2, 1, 0, -5], [0, 6, -6, 2, 0], [51, 4, -1, 0, 4], [12, -11, 0, 0, 8], [1, 10, 3, -13, -7], [-5, -4, 1, -17, -3], [-7, 11, -9, 2, 8], [-1, 14, 4, -9, -2], [17, -10, 4, -1, -6], [56, -4, 0, 6, -2], [17, -5, 3, -2, 6], [15, 14, -2, 3, -2], [20, -12, 5, 6, 3], [5, 4, 4, 9, 4], [8, 4, -7, 0, -3], [11, 2, 10, 9, 0], [-10, 3, -9, -8, 3], [34, 4, 0, 0, -8], [-29, -10, 1, -1, -3], [16, 6, 4, 9, 3], [44, 2, -4, 2, 6], [-17, 3, 0, 10, 1], [-39, 18, 7, 1, -9], [-2, -2, -6, 6, -2], [18, -2, -4, 2, 2], [26, -4, 6, 2, 4], [2, 5, 2, -8, -8], [-33, 1, -3, 8, 4], [41, 4, 4, -3, -2], [-26, 13, 5, -10, 1], [29, 4, 1, 7, 3], [30, 4, 2, -4, -10], [-9, 12, -8, 3, 6], [21, -1, 1, 10, -6], [25, -4, -1, 1, 9], [-7, 10, 5, -3, -5], [-8, -2, -6, 0, 2], [34, -2, 2, -5, -5], [29, 1, -7, 2, 2], [5, 12, 5, -10, -4], [-13, 11, 0, 2, -3], [-26, -12, 0, 6, 6], [29, -9, -5, -4, 4], [-10, -2, -4, 2, 0], [-30, -1, 5, 2, 1], [10, -12, -6, 8, 10], [2, 20, 2, -11, -5], [-5, 4, -3, 1, 7], [-15, -10, 0, -7, 4], [32, 6, 7, 0, 7], [11, -9, 6, 6, -7], [-11, -1, -6, 10, 7], [-2, -14, 2, -5, 7], [8, -2, 9, 16, -1], [-9, 11, -2, -5, 6], [25, 8, 5, -7, -5], [12, -20, 3, 11, -2], [27, -7, 4, 10, -1], [16, 13, 2, 3, 5], [35, 10, -10, -14, -5], [-14, 0, -2, 2, 2], [8, -15, 3, 5, 2], [-2, -6, -4, -6, 0], [17, 1, -10, -4, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5610_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5610_2_a_cj();".
// 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_5610_2_a_cj(:prec:=5)
    chi := MakeCharacter_5610_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_5610_2_a_cj();".
// 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_5610_2_a_cj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-64, 64, 32, -24, -1, 1]>,<13,R![-16, -48, 56, 8, -9, 1]>,<19,R![2240, 640, -256, -64, 5, 1]>,<23,R![-10816, 832, 704, -56, -11, 1]>],Snew);
    return Vf;
end function;