
// Make newform 1935.2.a.u in Magma, downloaded from the LMFDB on 11 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1935_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_1935_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_1935_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 := [-4, 5, 13, -7, -2, 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], [-4, 0, 1, 0, 0], [0, -6, 0, 1, 0], [1, 0, -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_1935_a();"
function MakeCharacter_1935_a()
    N := 1935;
    order := 1;
    char_gens := [1721, 1162, 46];
    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_1935_a_Hecke(Kf)
    return MakeCharacter_1935_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, -1, 0, 0, 0], [0, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [1, 0, 0, -1, 1], [1, 0, 0, -1, 0], [0, 1, -1, 0, -1], [4, -1, 1, 0, -1], [0, -2, 2, 0, -2], [0, 1, 1, 0, 1], [-2, 0, -2, -2, 2], [1, 1, -1, 1, 2], [1, 1, 1, 1, -1], [0, -2, -1, 1, -1], [-1, 0, 0, 0, 0], [0, 2, 2, 0, 0], [4, 1, -1, 0, 1], [0, 1, 0, 1, 0], [2, 2, -2, 0, -2], [6, -3, 1, 2, -1], [-2, 2, -2, -2, 4], [0, 1, -2, 1, 0], [9, -1, 1, 1, -1], [2, -3, -1, 0, -1], [-4, 2, 2, 2, -2], [6, 1, -3, 0, 1], [-1, -1, 1, 1, -2], [-2, 3, -3, 0, 3], [2, -2, 2, -2, -2], [1, 5, -1, -5, 2], [-7, 2, -2, 1, -1], [6, 3, -1, -2, 3], [0, 0, -2, -2, 0], [-8, 5, 0, -1, 2], [0, -2, 1, -3, -1], [-2, 2, -4, 2, 2], [-4, 0, 2, 2, 2], [-8, 3, 0, -3, 2], [-3, 3, -3, 1, 3], [12, -3, 1, 0, -3], [-2, 0, 2, 2, -4], [6, -2, 4, 0, 0], [4, 5, 0, 3, -2], [8, -2, 0, -2, 4], [14, -3, 1, 0, 1], [4, 4, 0, -6, 2], [-12, 4, -2, 2, 2], [10, -2, 6, 2, 0], [4, 2, 2, 4, 2], [1, 6, 2, 7, -1], [-7, 6, -4, 3, 4], [4, 3, 4, 5, 2], [-5, 2, -2, -7, -3], [4, -4, 0, 2, -4], [4, 3, 1, 2, -1], [1, 3, -1, -3, -3], [1, -3, -5, -3, 1], [-5, 8, -2, 5, 2], [0, 4, -3, 3, 3], [-1, 0, -4, 3, -1], [5, -2, 0, -5, 4], [4, 3, 5, 6, -7], [0, 2, 0, 4, -2], [14, 1, 3, -4, -1], [4, 2, -1, 3, 1], [9, -10, 2, -3, -5], [2, 1, -1, 2, 1], [-4, 2, -4, -2, 0], [4, 5, 1, 10, 1], [-3, 1, -5, 1, 1], [2, -4, 2, -2, -6], [4, -5, -3, 0, -1], [-8, 9, 3, 6, 1], [10, 0, -2, 0, 6], [-9, -6, -6, -9, 5], [11, 0, 0, 5, 0], [12, -1, -4, 3, 4], [-14, -8, 2, -2, 0], [-6, 8, -2, -2, 8], [-11, 5, 3, 3, -2], [2, -2, -2, 0, 0], [-20, 4, -2, -2, -2], [4, -8, 8, 2, -6], [16, -4, -1, 1, 1], [19, -3, 1, -1, -1], [1, -12, 4, 3, 0], [-4, -10, -4, 2, 0], [10, -6, 0, -2, 2], [8, 1, -2, -7, 8], [-7, -1, 3, 1, -3], [8, -11, -2, -5, -2], [4, -10, 2, 4, -2], [13, -3, -1, 1, 2], [2, 4, 2, 0, -6], [4, 6, 0, -2, 4], [2, 10, 4, 8, -4], [-7, 4, -4, -1, 1], [11, -7, 7, 1, -8], [-4, 2, 8, 4, -4], [-9, -1, 1, -1, -9], [11, 4, 2, -7, 2], [2, -7, 3, 8, 1], [14, -7, -3, 0, 5], [4, -1, 1, -2, -5], [-1, 15, 1, -3, 2], [2, -2, -2, 2, -6], [6, -12, 4, -4, -8], [13, 4, 8, 11, -7], [1, -1, -1, -7, -3], [7, -5, 5, 7, 2], [-8, 0, 4, 6, 0], [12, 1, 4, 5, -8], [8, 4, 6, 12, -2], [-10, -1, 3, -4, -5], [-3, -6, -2, -5, -1], [2, -8, -4, -6, 0], [-8, -14, 4, -4, -2], [2, -6, 10, 2, -6], [-11, -2, -10, 3, -1], [23, -4, 4, 11, 3], [16, 0, 5, 3, 7], [-1, -6, 0, 5, 0], [7, -5, -1, -5, 9], [17, 3, -1, -7, 1], [2, -11, -3, 2, -1], [-12, 10, 0, -6, 2], [-7, -1, -5, -7, 5], [-16, 6, -1, -7, -5], [-13, -11, 3, 3, -3], [9, -9, 5, 5, 3], [-18, 2, -6, -8, 2], [-10, 0, 2, 4, 8], [-12, -7, 0, 1, -4], [8, -8, 6, 10, 0], [-3, 2, 6, -3, -1], [24, -2, -4, 0, 4], [7, 4, 4, 3, -1], [21, -4, 8, 1, -3], [6, -12, -6, 0, -2], [12, 14, -2, -2, 2], [-8, 9, -16, 3, 6], [-18, 16, 6, 8, -2], [20, 10, -5, 1, -1], [0, -17, 9, -10, -5], [-20, 2, -6, 0, 0], [-2, 14, -4, 2, 2], [4, -4, 0, -8, 10], [-8, -3, 1, -6, -1], [18, -2, 8, 2, 0], [-8, -10, 0, 2, -4], [-11, 12, 8, 7, 1], [4, 3, -5, 2, 5], [3, -7, -5, 1, 4], [-24, 7, 1, 10, 3], [24, -2, -2, -8, 2], [18, -1, -3, 4, -3], [-10, 4, 4, -6, -2], [0, 8, 1, -5, -1], [4, 10, 4, -8, -2], [-11, 4, 0, 1, 9], [6, 5, 7, 10, -3], [-8, 7, -5, 0, 5], [17, -14, -2, 1, 3], [4, -7, -3, -8, -3], [-21, -9, -3, 3, -2], [20, -10, 10, 14, -10], [-25, -5, 9, -1, -9], [-18, -6, 0, 4, -6], [9, -1, -1, -7, -3], [4, 10, -6, 6, -2], [10, -2, 2, 4, 4], [-30, 0, -4, -2, -4], [10, -6, 10, 12, -8], [-8, -2, -12, 2, 0], [2, -1, -7, -2, -1], [-18, -4, -14, -12, 4], [5, -9, 3, 1, -11], [14, -8, 6, 0, -10], [-12, 10, -2, 10, -6], [-6, 6, 2, 2, 14], [18, -6, 2, 4, 2], [4, -7, -2, 3, -10], [-3, -18, 2, -5, -1], [-18, 6, 2, 12, 2], [19, -8, 12, 7, -5], [-5, 5, -1, -9, 9], [-26, 6, -2, -4, -2], [0, -11, 4, -5, -6], [3, 8, -8, 9, -1], [0, -13, 0, -15, 6], [-2, -18, 0, 8, -2], [-6, -5, -3, 6, -9], [8, -9, 6, 1, 6], [9, 18, -2, -1, 3], [-24, 4, -2, -8, 8], [28, -17, 2, 1, -6], [-12, 11, -10, 3, 0], [5, 2, 0, -5, 8], [-4, -7, 5, -2, -7], [0, 0, -10, -8, 14], [-8, -13, 4, -5, -4], [-19, -1, 1, -5, 8], [-38, 4, 0, 2, 4], [17, -6, -2, 9, -5], [-4, 4, 2, 8, -10], [2, 8, 12, 6, 4], [15, -9, 7, 7, -7], [-2, -14, -4, -16, 6], [0, -5, -3, -6, -3], [10, 6, 4, 6, -8], [-27, -2, -2, -13, 2], [-18, 8, -4, 0, 12], [-26, 7, 5, 6, -1], [2, -9, -9, -2, 9], [-28, 5, -9, -2, 11], [10, 10, -12, -4, 18], [8, 2, -1, 1, 3], [-7, -13, 9, 5, -1], [-28, 0, -6, -8, 0], [-24, 14, 6, 0, -2], [-10, -1, 1, -6, -1], [-10, 4, 8, -4, -8], [16, -15, -4, -7, -12], [-12, 4, 2, 4, -16], [0, 7, 1, 6, -7], [8, -11, 0, -3, 8], [-9, 4, 6, 1, -2], [16, -11, -3, -18, 3], [26, 2, 6, 10, -10], [8, 3, 6, 1, 2], [-8, 0, 12, 8, -18], [1, 17, 9, 1, 3], [-8, -2, 7, -1, 5], [-17, -11, 11, -5, -12], [-16, 4, -8, 10, 4], [4, 9, -1, 2, 15], [20, 14, -6, 0, 8], [-10, -4, -6, 6, -14], [-4, 9, 5, 2, -9], [24, -18, -6, -8, -8], [-28, -9, -3, 14, -9], [-23, -6, -6, 11, -1], [-18, -12, -8, -6, 6], [-18, 10, 6, -2, 10], [-15, 3, -1, 13, -19], [18, 14, -10, 4, -6], [-21, 1, 3, -1, -3], [-17, 17, 7, 7, 5], [27, 5, 11, 11, -8], [-11, -1, -7, -3, 4], [-4, 13, -9, -6, 15], [26, 8, -4, 0, 12], [-3, -16, 2, 11, -2], [20, -11, 5, 4, -5], [34, -12, -2, -2, -12], [0, 21, -4, -9, 6], [16, -12, 10, 6, -16], [-6, 12, 2, 14, 10], [-24, 5, -15, -8, 11], [29, -12, 16, 9, -11], [20, 3, 11, 14, -3], [-16, -2, 6, 0, -18], [12, 2, 2, 8, -18], [-1, 0, -8, 7, 3], [-16, -16, 2, -4, -2], [-3, 2, -2, -3, -1], [-24, 22, 2, 0, 6], [0, -18, -8, -16, 6], [-24, 8, 5, -3, -11], [7, 23, -3, -5, -1], [-36, -17, -3, -8, 3], [-20, 12, 5, 5, 5], [-15, -4, -4, -5, 5], [5, -1, -1, 17, -7], [-4, 20, -8, 8, 10], [-6, -2, -12, 10, 0], [12, 3, -4, -1, 4], [20, 3, 7, -4, 9], [-10, 14, -6, -4, 14], [5, -13, -7, 7, 8], [-18, 8, 4, 2, 14], [26, 13, -5, -8, 9], [-12, 9, -4, 13, 0], [-4, -9, 13, -2, -5], [-40, 10, -16, -16, 6], [-27, -6, 2, 7, -13], [-40, 22, -2, 0, 14], [-2, -3, -11, -12, 3], [-32, 3, -4, -11, 2], [14, -18, -8, -8, 2], [-1, -8, -6, -15, -10], [24, -10, 9, 11, -7], [26, -11, 3, 12, -1], [-10, 12, 14, -2, -8], [14, -6, 18, -2, 4], [-26, 3, -17, -12, 17], [-14, 17, -17, 2, 9], [-52, 3, -15, -10, 7], [-12, 27, -7, -8, 13], [-16, -20, 4, -4, -4], [-4, -12, 6, -18, -4], [0, -17, -1, 10, -3], [-4, 9, -18, 5, 0], [4, -9, 5, -2, 3], [-26, -15, 15, 0, -17], [20, 20, -4, -8, 6], [36, -21, 0, -3, -18], [-8, 7, -11, -10, 7], [10, -12, 20, -4, -2], [6, 8, -12, -2, 16], [-16, 23, -6, -1, 4], [44, -9, 10, 5, -2], [18, -10, 6, 0, 4], [7, -12, -6, -11, -2], [-24, 9, -2, 3, 2], [10, 25, -3, 10, -7], [1, -29, -7, -21, 8], [-12, -5, 4, 3, -20], [-5, 18, -14, -11, 5], [40, -3, 5, 2, -5], [-18, 10, -4, -10, -12], [20, -21, 9, 2, -9], [25, 1, 13, 1, -5], [-36, 18, -2, 2, 14], [6, -10, -4, -12, 22], [26, -14, 14, 16, -10], [12, 18, 1, 11, -11], [18, 6, 6, 2, 0], [-12, -11, 5, 0, 1], [-16, -2, 12, 2, -24], [5, -13, 15, 5, -11], [-22, -4, 6, -6, 8], [14, -17, 11, -8, -13], [18, -20, 2, 8, -6], [-29, -7, 15, -1, 5], [-21, 6, 10, -3, 1], [6, -3, 5, 22, 9], [10, 8, 0, 20, -6], [-20, 13, -4, -1, -2], [8, 14, 0, 16, 12], [0, 12, 8, -4, 4], [20, -1, 17, -8, 15], [-15, -7, 1, 9, -21], [2, -2, 4, 18, 6], [-16, -4, 8, -4, 4], [40, -11, -5, 4, 15], [16, 4, 7, -3, 5], [-4, -8, -6, 8, -2], [0, -2, -10, -12, -12], [34, -12, 16, 18, -22], [32, -29, -1, -10, -13], [35, -3, -9, -5, 21], [42, -6, 6, 4, 2], [2, -16, 0, 16, -4], [-13, 19, -11, 7, -1], [31, -8, 16, -9, 3], [18, -8, 8, 4, 8], [18, 22, -10, 6, -2], [-10, -6, -8, -8, 14], [6, -11, 3, -14, 7], [-28, 9, -9, 6, 11], [-8, -9, 8, -15, -18], [30, 8, 0, 0, 8], [-1, -8, -8, -7, -1], [12, -32, -8, 8, 2], [10, -19, -7, 14, 1], [3, -6, 10, -17, -7], [-25, 11, -1, -5, 13], [20, -12, 2, -18, 12], [50, 8, -2, 2, -4], [8, -14, -10, 0, -2], [6, -20, 10, 16, -8], [8, -22, 4, -18, 0], [-4, 12, -18, -4, 10], [-37, 6, -18, 9, 6], [3, -10, 18, -9, -3], [-2, -4, -8, -6, -8], [5, -2, 2, 7, -21], [36, 3, 1, 0, 1], [-10, -6, -8, 10, -10], [-31, 24, -20, -3, 21], [-17, -4, 2, -15, 10], [-24, 3, -5, 14, -1], [10, 15, -13, 2, 1], [17, 7, -1, 1, -3], [45, -9, -3, 1, -8], [20, -13, -9, 8, 11], [38, 2, 4, 24, 8], [24, -7, -15, -2, 17], [44, 8, -4, 4, 8], [-24, 19, -25, -4, 15], [23, -12, -10, 13, 10], [44, -7, 4, -1, -6], [-16, 8, 12, 0, 12], [14, 9, 1, -2, 13], [-22, 12, -4, 30, -2], [10, -4, -18, -2, 4], [28, 1, 15, 22, -11], [-32, 0, -6, 4, 4], [-38, 12, -10, -16, 16], [26, -8, 2, -2, 6], [-8, -4, 9, 5, -11], [4, 12, 0, 6, 10], [-42, -11, -1, -8, -9], [-30, 4, 0, -8, -4], [-32, -12, -4, 2, 6], [-12, 2, -2, -16, 8], [14, 0, 0, 16, 4], [47, 0, 8, 15, -13], [-19, 22, 6, 15, -1], [41, -4, -8, 3, 8], [-16, 10, -10, -18, -2], [12, 13, -16, -9, 22], [-29, 3, 5, -5, -1], [-12, -12, -5, -19, 9], [7, 9, -11, -5, -9], [-24, -4, 15, -5, -9], [47, -9, 9, 15, -10], [-30, 10, 4, -4, -2], [-50, 6, -20, -14, 12], [-15, -14, 14, 3, 7], [-2, 14, -4, 22, 10], [-10, 16, -8, -8, 16], [24, 7, -16, -5, -4], [-5, 0, -4, 17, -4], [7, 25, -11, -5, 11], [17, 1, -7, 5, 3], [-19, -4, -4, 7, -11], [0, -2, -4, -24, 8], [-11, 0, 0, 19, -20], [18, 4, 12, -10, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1935_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1935_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_1935_2_a_u(:prec:=5)
    chi := MakeCharacter_1935_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_1935_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_1935_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1935_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 5, -13, -7, 2, 1]>,<7,R![-160, -58, 97, -14, -5, 1]>,<11,R![12, -59, 43, 1, -6, 1]>],Snew);
    return Vf;
end function;