
// Make newform 2752.2.a.bb in Magma, downloaded from the LMFDB on 13 June 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_bb();".
// 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_bb();".
// 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 := [13, 20, -3, -9, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [-4, 0, 1, 0, 0], [4, 1, -1, 0, 0], [-2, -5, 0, 1, 0], [7, 4, -6, -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_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]];
    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, -1, -1, 0, 0], [0, 0, -1, -1, 0], [-1, 0, 0, -1, -1], [2, 2, 1, 1, -1], [0, 0, 1, -1, -1], [0, 0, -1, 0, -1], [3, -2, -1, -2, 1], [1, -1, -1, -1, 1], [-3, -3, 1, 0, 0], [-2, 0, -1, 0, -1], [0, 3, 3, 3, -1], [-3, -2, 0, -1, -2], [-1, 0, 0, 0, 0], [0, -2, 0, -1, 2], [2, -3, -2, -1, 2], [6, -1, -1, -2, 1], [1, -2, -2, 1, -1], [6, 4, -1, 3, -1], [0, 2, 4, 0, -2], [3, 6, 0, 5, -1], [-5, -1, -2, 0, 2], [4, 1, 4, 1, 0], [3, 2, -2, -1, 3], [-7, 1, 5, 3, 3], [2, 3, 4, 3, 0], [5, 4, -2, 3, 0], [-2, -5, 3, -2, -3], [2, 0, -5, 1, -3], [0, 4, 2, -2, 0], [0, -7, 0, -2, 0], [0, -1, 3, 7, 1], [-2, -2, 4, -2, -2], [10, 3, 4, 1, 0], [0, -3, -1, -5, -1], [3, 2, -2, -1, 1], [11, 2, 3, 2, -1], [2, -2, -3, -3, 6], [14, -3, -4, 3, 0], [-8, 4, 4, 4, 2], [0, -2, -3, -1, 0], [-6, -1, -7, -2, 3], [4, 10, 12, 2, -2], [1, 1, 5, -1, -1], [-4, 5, 3, -6, -1], [-6, -2, 0, 0, 0], [-6, 1, 1, -7, -3], [1, 2, 0, 7, 5], [-1, 8, 3, 8, -1], [-4, -4, -1, -3, 5], [-8, -6, -2, -4, 0], [-3, 5, 6, 0, -10], [-6, -4, 4, -6, 0], [0, 2, 3, -5, -5], [-1, 0, 2, 3, 7], [8, -6, -4, 0, -2], [2, -6, -5, -7, -1], [-17, -2, 4, -3, 2], [4, -4, -9, -1, 2], [2, -1, -10, 0, -2], [4, -1, 6, 3, 4], [-2, 13, 9, -2, 1], [-2, -3, -10, -11, 0], [14, 0, 5, -5, 7], [5, 2, 2, 1, 5], [14, 8, 1, 11, -1], [5, 2, 4, 5, 3], [-2, -3, -2, 0, 6], [-2, -12, -9, 1, 6], [5, -5, 1, 2, 4], [0, 4, 9, 3, -1], [0, -2, 1, 2, 7], [-8, 4, 2, 9, 0], [3, 0, 1, 2, 7], [-2, 6, 3, 3, 3], [14, 4, 6, 8, 0], [-2, -2, -12, 2, -4], [-4, -7, -1, 4, -1], [10, 1, 2, 6, -2], [-11, -2, 10, -7, 1], [5, -8, -12, 3, 3], [11, 8, 3, 4, -1], [-8, 7, 0, 4, 4], [1, 4, 4, -5, -9], [-12, -10, -1, -10, 5], [6, -17, -11, -8, 1], [-9, 8, 6, 11, -1], [-7, -4, -2, 5, -7], [8, 1, 3, 4, 7], [-19, -8, 2, -1, 5], [-4, -3, -1, -3, 5], [-3, -9, -11, 5, -1], [-13, -4, 1, 4, -9], [-6, 10, 5, 1, 0], [3, 5, -1, 10, 2], [10, 4, -10, 8, -4], [-10, -5, -2, 5, 2], [11, 6, -4, 1, -3], [5, -9, -1, -6, -4], [-14, -3, 6, -9, -4], [-16, 9, 12, -1, 0], [-4, -12, -3, -3, 9], [12, -12, -11, -11, -3], [4, -1, 8, 1, -8], [7, 1, 11, 6, 2], [3, 12, 0, -3, -1], [2, 6, 6, 2, -8], [13, -6, -2, -1, 7], [-2, 5, 2, 8, -4], [-8, -16, -10, 0, 2], [0, -6, -8, -2, -4], [0, 9, 1, 0, 7], [-3, -5, -15, 1, 1], [10, 4, 0, -10, 6], [2, 8, 2, 8, 8], [13, -2, 0, 5, -1], [2, -1, 11, 2, -3], [0, -10, -8, -6, 2], [1, 6, 3, -12, 5], [14, 0, -1, -5, 3], [-8, -11, -10, -9, 0], [1, -16, -12, -13, -3], [-5, -4, 5, 6, 11], [8, -8, 5, -9, 3], [7, 12, 2, 11, -9], [-2, 19, 3, 8, -1], [-4, 12, 11, 7, -5], [0, -13, -13, -7, 3], [-6, -10, -6, -10, 10], [2, 13, 1, 3, -5], [10, -9, -1, 5, -5], [8, 10, 12, 2, 6], [-15, -8, -14, -7, 1], [-4, 1, -7, 3, 1], [-4, -8, -2, -10, 0], [-10, -10, -4, -17, 2], [9, 3, 7, 0, 10], [0, -17, -17, 0, -1], [-2, -7, -7, 6, 9], [-4, 0, 6, 3, 6], [22, 10, -5, -1, 0], [6, 0, 3, -5, 7], [2, -22, -17, -1, 1], [-2, 6, -4, -6, 0], [4, 1, 3, -5, 7], [-27, 8, 0, 7, 1], [-16, 13, 10, -3, -12], [-1, 12, 7, -2, -11], [9, 0, 3, 14, -5], [12, 10, 16, -4, 8], [4, 11, -2, 11, 8], [6, -1, -8, 4, 4], [6, -6, -7, -15, 7], [28, 14, 12, 10, -8], [-4, -11, -12, -1, -4], [21, 8, -12, 7, 1], [-8, -1, 0, 12, 0], [-12, 6, 0, 12, -2], [-5, 18, 6, 11, -5], [-16, -8, -11, -9, -3], [4, -13, -2, -1, -6], [-27, -16, -4, -7, 9], [-1, 8, 8, -5, -14], [10, 12, 5, -1, -11], [-24, -12, 2, -3, 2], [17, 14, 6, 7, -9], [3, 8, 8, 15, 7], [7, 3, 3, -2, -8], [28, -8, -6, -2, 8], [4, 4, -6, 12, -6], [6, 7, 7, 9, -5], [-3, 0, 17, 4, 5], [13, 10, -10, 5, -9], [17, 9, 3, -7, 1], [-16, 14, 8, 10, 4], [-5, 11, 6, 0, -4], [26, 13, 1, 7, -3], [14, -16, -15, -17, 2], [-4, -7, 11, 4, -3], [33, -2, -3, -2, -7], [19, 18, 20, 3, -1], [28, 7, -1, 14, 3], [5, -2, 16, -5, 1], [-15, -18, -20, -11, 11], [-3, -8, -10, -21, 5], [4, 2, 2, 18, 4], [6, -12, 3, 1, -10], [-21, -3, 7, -10, -2], [6, 0, -6, 3, -8], [7, 16, 6, -3, -9], [-8, 11, 12, -2, -6], [16, -16, -15, 1, -4], [-14, 5, 9, 14, 11], [-5, -6, 5, 8, 5], [15, 8, 2, 3, -5], [-6, 12, -6, 8, 6], [37, 13, -5, 9, 1], [-26, -14, -27, -15, 5], [-15, -5, 0, 12, 2], [3, 18, 18, -1, -1], [22, 9, -2, 13, 2], [-22, 16, 12, 10, -10], [30, -4, -6, 2, -2], [32, 4, 6, 0, -6], [7, -11, -5, -10, 6], [-14, -19, -11, -5, 3], [2, 6, 14, 8, 6], [4, 13, 14, 5, 4], [2, 8, 8, 16, -10], [24, 9, 0, 23, 4], [-8, -4, -12, -6, 2], [-12, 1, 12, -11, -12], [34, 19, -2, -1, -6], [2, 8, -5, -3, 7], [-21, -20, -12, -5, 9], [-16, -20, -23, -4, 9], [7, -16, -8, -5, 7], [8, 8, 4, 18, -4], [37, 6, 4, -1, 5], [-18, 14, 1, 5, -1], [8, 10, 15, 1, -8], [-8, 17, 3, 8, -3], [-9, -6, -10, -1, 15], [7, -7, -1, -9, -9], [1, -20, -5, 6, -1], [-24, -5, -2, -5, 8], [10, -10, -1, 9, 3], [5, -18, -12, -19, -3], [10, 14, 16, 18, 2], [23, 10, -9, -6, -3], [-9, -3, 3, 0, -8], [-10, 5, 6, 5, 4], [33, 1, -3, 7, 7], [-6, -28, -16, -8, 0], [36, 5, 0, 27, 4], [6, 18, -4, 11, 4], [2, -2, -14, 8, 6], [-6, -1, -10, -9, 6], [-22, -10, 17, -7, -6], [9, -25, -9, -5, -9], [22, -7, 7, -11, -1], [-7, -26, 1, -10, -5], [21, -28, -15, -14, 5], [10, 5, -21, 8, 9], [-8, -4, 12, -12, 10], [-11, -19, -8, -22, 6], [-24, 6, 22, -12, -6], [34, 25, 12, 15, -12], [-16, -10, -13, -17, 7], [17, 16, -4, 5, 0], [6, -1, -1, 14, 11], [2, 24, 21, 17, -15], [2, -19, -6, -3, 18], [-4, -12, 4, -6, -18], [-7, -3, -11, 10, 8], [18, 21, 11, 11, -13], [1, -4, -3, 12, 3], [-20, -8, 17, -5, -5], [3, -6, -6, -5, 13], [-7, -14, 0, -13, -6], [16, 18, 2, 2, -18], [0, 20, 15, 11, 0], [42, 0, 4, -4, 4], [-12, -1, -5, -4, -15], [25, 0, -10, 7, -3], [20, 3, -11, -11, 11], [-10, -3, -3, -1, -17], [19, 16, 12, -5, -12], [-8, 4, 14, 2, -18], [16, -12, -23, 5, 1], [-14, -31, -14, -5, 12], [-8, 0, 11, -9, 4], [-43, 10, 14, -11, -3], [-3, -16, -6, -13, 11], [8, 28, 14, -7, 2], [1, 8, 4, 19, -11], [-46, -6, 1, -7, 3], [12, 6, -5, 5, 22], [14, -25, -16, -12, 4], [44, 18, 5, 11, -2], [30, 12, -11, -8, -15], [2, 9, -11, 5, 11], [-20, 9, 16, 8, 4], [-7, -9, -11, -8, -4], [-1, 7, 29, 0, 12], [-25, -2, 10, 1, -3], [-5, 12, 10, 1, -14], [31, 21, 13, 0, -14], [-19, -14, -20, -17, -9], [19, -20, -4, -1, 4], [-22, 3, -12, 9, -16], [40, 7, -6, 13, 10], [17, 25, 2, -12, -4], [2, -17, -19, -5, 25], [-28, -13, -8, -15, -2], [-38, -10, 3, -15, -3], [25, 0, -8, -11, -14], [10, 4, 11, -13, 13], [14, 17, 3, 6, -1], [6, 18, 26, -4, -16], [-12, -24, -25, -4, 19], [-8, -38, -17, -15, 8], [-18, -3, -12, 7, 0], [0, 7, 0, -3, -6], [-3, -13, 3, -4, 12], [32, 16, -2, 2, 2], [32, 0, -19, 17, 5], [-15, -8, 5, 20, 1], [-9, 28, 24, 23, 5], [3, 6, -15, 0, -9], [14, -12, 6, -2, -4], [1, -2, -4, -17, 9], [-9, -25, -17, -11, 19], [11, 4, -1, 4, 19], [-42, -12, 5, -28, 1], [-21, 14, 18, -1, 4], [60, 1, 7, 8, -3], [6, -21, -15, 2, 5], [22, 21, 28, 36, -4], [14, -34, -24, -8, 8], [-6, -24, -21, -15, 1], [-24, 0, 0, 16, -4], [-20, 5, 13, 9, 5], [18, -7, -5, -6, 11], [24, 14, 10, 24, 4], [-28, -22, -3, 0, 13], [-47, -4, 2, -7, -3], [6, 16, -1, -3, -1], [21, -8, -5, 4, 13], [11, -5, 7, 18, -8], [35, -22, -6, -13, 9], [18, -10, -17, 5, -1], [-3, 28, 18, 19, -5], [27, 5, -15, -4, -4], [6, -3, 3, -10, -13], [18, 23, 4, 5, 2], [10, 18, 0, -2, 8], [16, 8, -2, -8, -12], [4, 10, 8, -4, -10], [33, -2, -20, 3, 3], [-40, -25, 0, -5, 2], [9, -6, -4, 13, 19], [-38, 30, 18, 16, -6], [27, 14, 0, -7, -23], [68, 0, -7, -5, -7], [12, -7, -12, -19, 0], [-5, -1, 7, 24, 0], [-50, -6, -12, -6, 0], [-56, -22, -2, -2, 6], [24, 15, -2, 21, -12], [12, -7, 5, -6, 13], [25, -28, -16, -11, -5], [32, 3, 11, 10, -11], [-21, -14, -8, -1, 1], [-34, -17, -1, 0, 3], [-35, 2, 2, 11, -5], [24, 24, 10, 22, -16], [-20, -15, 3, 9, 21], [14, 38, 27, 18, -11], [-12, -12, -11, 12, 3], [-22, 9, -3, 7, -17], [-21, -16, -6, -1, 1], [11, 10, 10, -7, 5], [-31, 6, -1, 4, 11], [-6, -26, -19, -1, 15], [-14, -6, -10, -14, 0], [21, -10, -17, -4, -9], [20, 6, -8, 11, 6], [5, 12, 14, -13, 5], [9, -8, -15, -28, -15], [-7, -27, -11, 0, -6], [11, 21, 18, 18, 0], [0, 16, 13, 5, 20], [6, 18, 19, 28, -5], [30, -8, -16, -4, -12], [-11, -8, -9, -30, -5], [24, 35, 21, 4, -9], [-48, -25, -16, -2, 14], [15, 0, 6, 11, -9], [13, 36, 2, 31, -1], [-34, 9, 10, -3, -6], [33, 20, 26, 7, 0], [-14, -19, -14, 2, 16], [21, -4, -10, 7, 25], [4, 34, 19, 1, -1], [-30, -23, -16, -26, -2], [-57, -2, 10, -1, 1], [-12, -2, 7, 15, -13], [-66, 3, -9, 0, 9], [34, -10, -3, 9, 7], [-16, 25, 14, 10, 8], [-1, -13, 9, 16, -10], [-22, -18, -2, -10, -8], [6, -25, -10, 5, 0], [-29, -26, -6, -15, -1], [33, 7, 2, 10, -2], [-30, -15, -12, -16, -6], [53, 10, 8, -13, 15], [-2, -2, -21, -5, 10], [10, -2, -11, -7, -6], [-12, 3, -4, -15, -8], [41, 25, -10, 16, -4], [45, 26, 14, 5, -14], [-11, 11, -14, 2, -10], [22, -11, -11, 3, -5], [5, 22, 18, 13, 9], [26, -9, -9, 21, -3], [24, 8, -14, -15, -18], [5, 12, 17, -4, -21], [28, -15, -22, 17, 8], [45, 37, -4, 10, -16], [-8, -34, -8, -20, 2], [24, -1, -13, 23, -7], [14, 37, 14, 15, -20], [-5, 22, 10, -3, -9], [-46, -25, 12, 1, -2], [8, 18, 11, 5, -9], [19, -3, -24, 4, -8], [-36, 17, 23, 10, 3], [31, 2, -14, 19, 7], [32, 32, 15, 15, -4], [-6, -7, -11, -14, -15], [39, -6, -12, -3, -1], [-16, -4, -27, -13, 9], [0, -18, 10, -18, -6], [7, 18, 3, 14, 9], [12, -21, -5, -1, 1], [11, 28, 10, -7, -9], [-16, 9, 32, 7, -2], [-27, -8, 8, -29, -1]];
    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_bb();".
// 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_bb(: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_bb();".
// 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_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2752_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-22, -8, 20, 1, -5, 1]>,<5,R![34, 40, -12, -17, 1, 1]>,<7,R![88, 16, -58, -14, 4, 1]>],Snew);
    return Vf;
end function;