
// Make newform 2752.2.a.x in Magma, downloaded from the LMFDB on 17 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2752_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_2752_2_a_x();".
// 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_2752_2_a_x();".
// 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 := [-8, 42, 8, -13, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-5, 0, 1, 0, 0], [2, 4, -7, -1, 1], [-2, -7, 0, 1, 0]];
    Rf_basisdens := [1, 1, 1, 2, 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_2752_a();"
function MakeCharacter_2752_a()
    N := 2752;
    order := 1;
    char_gens := [2495, 517, 2497];
    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_2752_a_Hecke(Kf)
    return MakeCharacter_2752_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, -1, 0, 0, 0], [0, 0, 0, 0, 1], [-1, 0, 1, 0, -1], [0, -1, 1, -1, -1], [-2, -1, 1, -1, -1], [3, 0, 0, -1, 0], [1, 0, 1, 0, 0], [1, -1, 0, 1, 1], [0, -1, 0, -2, 0], [-1, -2, 0, 1, 0], [1, 1, 1, 0, 1], [1, 1, 2, 1, -1], [-1, 0, 0, 0, 0], [3, -1, 1, 0, -1], [2, 1, -1, 1, -1], [-4, 0, 2, 2, 0], [1, 2, -3, 0, 1], [4, -1, -1, 1, 3], [2, -4, 4, -2, -2], [-1, 2, -1, 0, -1], [1, 2, -3, 0, 0], [-2, 1, 1, -3, 1], [3, 0, 1, -2, -1], [5, -3, 2, 1, 1], [2, -3, -1, 1, -1], [-1, -1, 2, -3, 1], [-2, 4, 0, 2, 2], [4, 1, -1, -1, 3], [2, 0, 2, -2, -2], [-3, 0, 2, 1, 2], [3, 1, -1, 2, -1], [6, -4, 0, 0, 2], [10, 1, -3, 1, 3], [3, -1, -1, 0, -1], [-17, 0, 1, 0, -1], [-1, 4, -1, 2, 2], [-2, -2, 4, 0, -3], [-4, 3, -3, 3, 3], [4, 6, -2, 0, -2], [-4, -4, 4, 2, 1], [-6, -4, 2, -2, -2], [4, 0, 0, 0, -2], [3, 1, 0, 5, 3], [-10, 2, 2, 2, 2], [2, 2, 4, -2, 2], [9, 3, -1, 4, -1], [5, 4, 3, 0, -3], [-3, 6, -1, 2, 0], [-4, 5, 1, 1, -1], [-6, -2, 0, 0, -4], [-5, 6, -1, 0, 0], [-4, 0, 0, -2, 4], [-10, -1, -3, -3, 3], [21, 2, -1, -2, 1], [-2, 0, 0, -2, -2], [-4, 3, 3, -1, -3], [-9, 1, -4, -1, 5], [8, -4, 4, 0, -1], [7, 4, -4, -1, 0], [-6, 1, 3, -5, -3], [10, 2, -6, 2, 4], [4, -1, 1, -1, -7], [-6, 1, 1, 1, 3], [-3, 6, -7, 4, 7], [-6, 3, 3, -3, -1], [-5, 0, 3, 2, 3], [-15, -4, 4, -3, -4], [-2, -4, -2, 2, -3], [-8, -1, 4, 2, -4], [10, 3, -3, -1, -1], [11, 2, 4, 1, 0], [-5, -7, 1, 0, -1], [-5, -2, 5, -4, -2], [-8, -1, 1, 3, 1], [6, 0, -2, -4, -6], [-2, 0, 0, 4, 0], [-12, 0, 4, -6, -4], [5, -2, 0, 5, 0], [1, -4, 7, 2, -3], [-5, 6, -3, 0, -3], [11, 4, -9, -2, 6], [-5, -2, 2, 3, -2], [-1, 6, -3, -2, -1], [-5, -2, 2, -5, -2], [2, 4, -4, 6, 4], [-7, 8, 1, 0, -1], [7, 4, -1, -4, 1], [-10, 4, 6, -2, -2], [-7, -4, -1, -4, 3], [-1, -1, -3, 0, 1], [-11, -3, 2, 3, -5], [13, -4, 3, -2, -6], [18, 2, -4, 0, 3], [0, -1, -2, 2, 2], [-2, -2, 4, -2, -4], [-6, 3, 1, -1, -7], [3, 2, -1, 0, 5], [-2, -1, 4, -2, 0], [-20, 7, 3, -1, -3], [8, 3, -5, 1, -1], [16, -1, 1, 1, -3], [-2, 1, -3, -3, 7], [-2, 3, 1, -1, -5], [4, -3, -4, 0, 4], [1, 6, -1, -2, 3], [2, 2, 8, 2, 0], [-5, 6, -3, 6, 5], [13, 4, -8, 3, 12], [-12, -2, -8, 2, 2], [0, 0, 4, 4, 0], [-8, 2, 4, 2, -2], [13, 9, -14, 5, 15], [-4, -8, 8, 0, 0], [-2, -2, 0, 2, -8], [-5, 6, 5, -6, -1], [8, 0, 6, -6, -4], [20, 4, -4, 4, 6], [-3, -12, 1, 2, 0], [-18, -1, 5, -7, -7], [2, -1, -5, 5, 7], [-9, 4, -5, 4, 7], [3, -4, 3, -2, 0], [-20, -3, -3, -5, 5], [29, -6, 3, 0, 3], [-4, -8, 4, -2, -2], [-10, 1, 5, -1, -3], [3, -1, -5, -2, 1], [14, -4, 6, 0, -4], [1, -5, -7, 0, 5], [-11, 7, -1, 0, 3], [-16, 8, 4, 0, -2], [19, 4, -1, 2, 3], [9, -11, 5, -4, -7], [10, -8, 2, -2, 6], [11, -9, -1, 4, 1], [8, 5, 2, -4, 2], [0, 0, -2, -6, -4], [-2, 6, -10, 2, 4], [-13, -3, -9, -4, 5], [-10, 2, -12, 0, 5], [10, 1, 3, 1, -3], [2, 1, -3, -3, -1], [4, -6, 4, 0, -2], [-5, 15, -5, 4, 3], [-17, 6, -3, 4, 7], [8, 5, -7, 9, 1], [27, -2, -1, 2, 8], [7, -6, 5, -2, -8], [6, 0, -4, -2, 4], [-14, 7, 1, -1, -3], [-3, -16, 4, 1, -4], [6, -1, -1, -5, -3], [12, -2, -2, -2, 6], [-16, -5, -1, -3, -5], [-21, 0, 3, 2, 1], [5, -10, 12, -9, -4], [-20, -8, -2, 8, -6], [7, 0, 9, 2, -9], [-10, -3, 7, 5, -5], [2, 7, 1, -7, 5], [-23, 2, 9, -4, -11], [11, 7, -6, -3, 7], [2, -5, -5, -1, -1], [25, -1, 1, -4, -5], [23, 2, -3, -4, -1], [27, -14, -1, -2, -3], [2, -3, -4, 0, 4], [-6, -6, 0, -8, 4], [-16, -4, 6, 0, 4], [-11, -5, 3, -4, -5], [13, 0, -7, -2, -2], [-7, -6, 5, -6, 3], [-5, -9, 6, -1, -15], [-4, -10, 6, -10, 2], [1, -4, 7, 0, -2], [-9, -1, 3, 2, -9], [-2, 0, -14, 4, 5], [-14, 14, -8, -2, -8], [-11, -2, -1, 8, 0], [17, 6, -5, -4, 1], [4, -2, 14, -2, -4], [-7, -10, 5, 0, -11], [19, 10, -15, 6, 15], [5, -10, 5, 2, -3], [12, 6, 4, 2, 0], [-6, 8, -14, 4, 3], [8, -5, -2, 2, -10], [13, 11, -7, 0, -1], [7, -6, 3, 2, -9], [15, 0, 2, 1, -10], [12, 8, -4, 10, -3], [20, -4, -2, -2, 8], [23, 2, 5, 4, -8], [-15, -18, 7, 0, -5], [24, 16, -8, 2, 4], [-19, 7, -8, 3, 7], [-10, 7, -9, 1, -7], [3, -4, -7, 0, -2], [-17, 6, -5, 2, 3], [14, 7, -5, -7, 11], [-6, -10, 4, 2, 8], [-10, 10, -14, 2, 6], [24, -8, 6, -8, -8], [12, -5, -6, -6, 2], [13, -5, 1, 4, 1], [-8, -2, 4, 8, -10], [-22, -9, 1, 5, 3], [-10, -16, 8, 0, -2], [10, 11, -3, 5, 5], [-8, -12, 6, -12, 2], [34, -5, 3, 1, 5], [-26, -13, -1, -1, 5], [4, -7, -1, 1, 3], [3, -6, 7, -6, -13], [7, 8, -6, 9, 6], [11, -14, 3, -6, -7], [4, -2, 4, 6, -6], [25, 4, -9, 16, 9], [4, 1, -1, -1, -5], [24, -4, -4, 4, 1], [-20, -2, -6, 2, 6], [-21, 10, 7, 0, 1], [-13, 5, 0, 7, 5], [3, 2, -3, -6, 4], [-10, 11, -11, -1, 1], [32, -1, -1, -1, -3], [7, -4, -9, -6, 1], [20, 4, -6, -6, 6], [-7, 0, 5, -4, -20], [-10, 7, 6, 10, 6], [-16, -5, 5, 7, 1], [17, -5, -4, 5, 17], [-2, 10, -14, 2, 14], [12, -17, 7, -7, -13], [-23, 1, 7, -4, -7], [30, 2, 2, 10, -2], [12, 1, 11, 3, -1], [18, -2, 8, 0, -15], [15, 13, -4, 3, -5], [27, -1, -9, 2, 11], [11, 2, -3, 14, 10], [-15, 0, -5, 2, 2], [-42, -2, -2, -2, 6], [22, -4, 2, -6, -2], [11, 2, 11, -8, -2], [20, -8, 12, -8, 0], [4, 11, -7, 11, 7], [22, -5, 9, -7, -5], [-5, 11, 6, 1, 7], [32, -2, 8, -2, 2], [6, 7, -3, 3, 7], [-10, 1, 1, 5, 3], [-40, 0, 6, -4, -6], [18, -17, 12, -8, -8], [-19, 1, -7, 6, -7], [5, -2, -1, 12, 10], [6, -7, 11, -5, -5], [23, 4, -11, 2, 17], [-1, 21, 0, -3, -3], [6, -8, -4, -2, -4], [-12, -4, 0, 2, -5], [-38, -8, 4, -12, 0], [-16, 2, 0, -2, -2], [1, -10, 9, 0, 3], [3, -17, 5, -4, -3], [17, -3, 3, 6, 7], [3, 27, -18, 11, 11], [0, 10, 6, -2, -10], [-14, -17, 7, 1, -1], [6, -21, 5, -5, -9], [-24, -4, 8, -10, -13], [17, 20, -9, 2, 5], [5, 0, -13, 0, 5], [3, -13, -1, 4, 5], [31, -4, -1, 10, 5], [18, -9, 11, -5, -7], [-20, -2, -2, 10, 1], [17, -12, -12, 5, 12], [0, -2, 2, -4, -7], [21, 2, -8, 3, 8], [25, -3, -7, -6, 3], [-3, -14, 22, -3, -10], [-2, -5, 8, 8, -4], [18, -9, -14, 4, 14], [9, 18, -9, 0, 9], [19, -11, -8, 1, 9], [8, -5, -6, 4, 14], [5, 2, -7, 6, 9], [25, 7, -8, -5, -3], [-2, -9, 7, 5, -5], [-24, -3, 7, -3, -5], [27, 8, -11, 12, 6], [-3, 3, 11, 4, -1], [14, 1, -13, -7, 3], [-26, 5, 3, -7, 7], [7, 17, -8, 7, 7], [-18, -1, 13, -1, -15], [-20, -2, 6, 10, 2], [-46, -2, 10, -8, -10], [11, 0, -8, 7, 0], [-24, 14, -14, -2, -5], [6, 11, 3, 3, 1], [-22, 3, 13, 1, -5], [-16, 1, -2, -6, 6], [2, -10, 0, -4, -2], [10, -3, 9, 5, -9], [5, 6, 3, -4, 8], [47, 0, -7, 4, 9], [-21, -2, -11, 4, 2], [12, 8, -6, 10, 6], [5, 4, 7, -6, -5], [-17, -15, 8, -7, -9], [-15, 4, -11, 8, 14], [-21, -18, 6, -1, -2], [-3, 5, 14, -3, -7], [16, 2, -6, 2, -6], [-4, -6, 14, -2, -18], [-35, 4, 6, -1, 2], [-48, -2, -2, -4, 2], [-26, 19, -1, 3, 1], [-26, 4, -2, 2, 4], [11, 17, 1, -2, -7], [-2, 14, 8, 2, 4], [-10, 8, -2, 14, 6], [21, 4, 2, 3, 2], [-9, 10, -3, 4, 13], [24, -7, -9, 5, -5], [-5, 8, 1, -2, 2], [-10, -7, 0, 8, 8], [-1, 0, 17, -2, -7], [-22, 11, 9, -5, -17], [-19, 2, 1, 10, -5], [-4, 21, -20, 16, 12], [12, 18, -18, 10, 22], [-24, 3, 5, 7, 1], [10, 10, 2, -2, 10], [40, -4, -6, 0, 6], [24, -10, -4, -10, -8], [9, 22, -7, 10, 1], [20, 3, 1, 5, 5], [27, 24, -13, 4, 5], [2, -10, -6, 6, 10], [-5, -14, 11, -2, -9], [2, 17, 3, 5, -5], [-8, 5, 7, -3, 11], [-8, -1, -2, -8, 6], [-28, 2, 12, 0, 2], [-18, 6, -18, 4, -2], [16, -11, 15, -5, -13], [16, -30, 2, -2, 0], [-35, -6, 9, -4, -13], [-38, -16, 2, -6, 4], [1, 18, -1, -4, 3], [22, 4, -10, 10, 20], [27, 14, 1, 10, 1], [20, 8, -12, -4, 18], [-9, 13, 11, 2, -9], [-3, 4, 4, 1, -12], [15, 20, -14, 7, 10], [41, 1, -1, 2, -1], [-23, -14, 11, 2, -9], [-15, -18, 23, 0, -11], [27, -8, -1, 8, 6], [-22, 5, 7, 15, 9], [-8, 4, 4, -6, -2], [13, -18, 7, 8, 8], [1, -17, 15, -12, -3], [-3, -8, 7, -2, 9], [-25, -20, 11, -8, -6], [0, -15, 10, 6, -2], [-5, -14, -1, 8, 4], [-36, 4, 8, 8, -5], [-31, 14, 0, -9, -16], [-6, -2, 10, -18, -2], [11, 12, -5, 4, 10], [4, -14, 14, -10, -10], [9, 4, -8, -7, 12], [-19, 16, -7, 4, 7], [-11, -16, -5, -2, 15], [-20, 11, -11, 9, 5], [-13, 1, 8, 1, 5], [35, -10, -14, 3, 6], [-3, -16, 15, -10, -3], [0, -7, 11, 5, -3], [47, -10, 0, -7, 0], [-1, 0, -11, 4, -5], [16, 13, -19, 13, 21], [20, 2, 6, -2, -12], [-24, 3, 1, -5, -1], [-65, 12, 2, 1, 6], [18, 1, 4, -8, -4], [42, 4, -14, 8, 14], [-34, 3, 5, 17, 5], [21, -8, 1, 6, 9], [19, -20, 1, -12, -4], [-27, -12, 6, -11, 6], [1, 6, -3, 4, -9], [-34, -4, -6, -2, 21], [-46, 14, 4, -2, -9], [-4, -13, 19, 7, 1], [-7, -24, 11, 0, -8], [-25, -17, -2, -3, 7], [-19, 0, -1, -4, -2], [-19, 9, 7, 6, -1], [-7, -6, 1, 2, -3], [-13, -17, -7, 6, 1], [-9, -17, -9, -8, 5], [-5, 18, -7, -6, -2], [-40, 7, -1, 1, 1], [-15, -4, 11, 12, 6], [-12, -4, 2, -4, 6], [-1, -1, 5, -16, 9], [-6, -11, 11, -9, -1], [1, 6, 7, -6, 13], [-34, -11, 5, 3, 3], [-6, 5, -7, -15, -13], [7, 2, 7, 16, -6], [28, -10, 8, 6, 16], [31, -2, -1, -2, -1], [-40, -6, -2, 2, -1], [22, -10, 2, -18, -12], [19, -10, 3, 10, 13], [-10, 11, -11, -11, -13], [-24, 4, 14, 0, -12], [11, -10, 1, -8, 2], [-9, -5, 1, -20, -7], [-37, -10, 15, -8, -19], [26, 9, 7, 7, -15], [27, 6, -11, 12, 17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2752_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2752_2_a_x();".
// 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_2752_2_a_x(:prec:=5)
    chi := MakeCharacter_2752_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_2752_2_a_x();".
// 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_2752_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2752_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![8, 42, -8, -13, 1, 1]>,<5,R![164, 110, -26, -21, 1, 1]>,<7,R![64, 104, -24, -22, 2, 1]>],Snew);
    return Vf;
end function;