
// Make newform 7935.2.a.bc in Magma, downloaded from the LMFDB on 15 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7935_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_7935_2_a_bc();".
// 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_7935_2_a_bc();".
// 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 := [6, 16, -2, -9, 0, 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], [-1, -5, 0, 1, 0], [7, 4, -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_7935_a();"
function MakeCharacter_7935_a()
    N := 7935;
    order := 1;
    char_gens := [5291, 4762, 7411];
    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_7935_a_Hecke(Kf)
    return MakeCharacter_7935_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], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0], [-1, -1, 0, 0, -1], [2, -1, 1, 0, -1], [-1, -1, 0, 0, 1], [0, 0, 0, 0, 0], [2, 0, 1, 1, 0], [-1, 0, 1, 0, 0], [1, -3, 1, -1, -1], [-2, 1, -1, 0, 1], [4, 1, 0, -1, 1], [-1, 0, -2, 1, 0], [0, 0, 1, -1, 0], [0, 0, 1, 1, 2], [1, -1, 0, 0, -1], [-1, 3, -1, -1, -1], [2, -3, 3, -2, -1], [-1, 3, 0, 0, -1], [-2, -2, -4, 2, 0], [8, -3, 1, 2, 1], [2, 4, -4, 0, 0], [4, -1, -2, 1, 1], [-2, 1, 1, -2, 1], [6, -5, 0, -1, -1], [6, -2, 1, -3, -2], [-1, 5, 0, 0, 1], [0, 8, 1, -1, 0], [3, -3, -2, 2, 1], [8, 2, 2, 0, 0], [4, -1, -4, 1, -1], [1, 3, -3, 1, -1], [7, 0, 0, 1, 0], [2, 1, -2, -3, 1], [-7, -4, 1, 2, -2], [2, 6, -2, 2, 2], [-11, 4, 0, 1, 0], [2, -4, 2, -2, -2], [6, 3, 2, -1, 1], [10, 4, 0, -2, 0], [-3, 2, -2, 1, 2], [2, 2, -2, 0, -2], [-2, 4, -2, -2, -2], [-4, 0, -4, 2, 2], [-7, 5, -5, 1, 1], [6, 4, -2, 0, 0], [8, 8, -4, 0, 0], [2, 0, 0, 2, 0], [-6, 2, -4, -2, 0], [-6, -3, 5, 0, -1], [9, -1, 0, 0, 3], [2, 6, 2, -2, -2], [5, -8, 4, -1, -2], [6, 6, -1, -1, -2], [-2, 0, -1, 3, 4], [-15, 1, -3, 3, 1], [-6, -2, -4, 2, 2], [5, 0, 0, 3, 4], [-3, -4, 7, 2, 0], [4, -1, 3, 0, -1], [-1, 5, 0, -4, 1], [-2, -2, 2, 2, 0], [3, 5, -1, 3, -1], [9, -6, 6, -1, -4], [-1, 4, 5, -4, 0], [-14, -2, -4, 0, 4], [2, -6, 0, 4, 2], [7, -3, 1, -5, -3], [9, -2, 6, -1, -4], [7, -8, -4, -1, 0], [-1, 3, 1, 1, 3], [-4, 5, -10, -1, -1], [10, 0, -4, 4, 2], [-2, -3, -5, -2, 1], [-4, 6, -4, -4, 2], [-14, 0, -4, 0, 0], [8, 10, -6, -2, 2], [7, 9, -1, 1, 1], [-5, 2, -4, 1, 2], [-19, -3, 2, 0, -3], [-10, 0, -2, 4, 0], [-9, -4, 3, -2, -2], [6, 9, -2, 1, 1], [-11, -2, -6, 1, 6], [-6, 8, -3, -3, 0], [3, -9, 3, -3, -3], [4, 2, 0, -4, -2], [11, 1, 6, -2, 1], [-18, 10, -7, -3, 2], [9, 2, 0, -5, -2], [-3, -15, 0, 2, 1], [-12, -5, -1, 2, 1], [-3, -4, 11, 0, 0], [10, -1, -1, -4, 3], [-12, 3, 0, -7, -1], [-5, -8, -2, -1, -4], [-8, 10, 2, -2, 2], [8, 9, 6, -3, -3], [10, -8, 2, 0, -4], [-6, 0, 5, -3, -4], [4, -6, 3, -3, 2], [8, 0, 2, 4, 4], [3, 1, 6, 0, 3], [-14, -6, -8, 2, -2], [-14, -6, -4, 4, 2], [-9, 6, -8, 3, 4], [8, 4, -6, -2, 2], [-3, 0, -5, -2, 4], [3, -5, 4, 6, 3], [0, -6, 4, 2, 0], [8, 5, 3, -6, 1], [8, -8, -2, 0, 2], [-1, 3, -2, -2, -3], [-1, -4, 2, -1, -4], [-15, -3, 3, -5, 1], [7, -2, 6, -1, -2], [-1, 0, -4, -3, 6], [9, 2, 2, 1, 6], [10, 10, 0, 0, 2], [7, -5, 4, 4, -1], [2, 14, -1, 1, -2], [-1, 6, -10, -1, 2], [10, -9, 6, -1, -1], [9, -4, -2, -3, 4], [-17, 1, 6, 2, 1], [12, 4, -7, 5, -2], [-13, 6, -1, -2, -2], [5, 3, -3, 9, 1], [-9, 6, 9, 2, -2], [-8, 6, 4, -2, -2], [11, -9, 8, 4, 1], [15, -2, 1, -2, -4], [6, -8, 7, -1, 0], [3, 7, -2, -4, 3], [24, -8, 4, -2, -4], [1, 7, -1, -7, 3], [-8, -4, -9, 1, 0], [18, 9, -3, -2, 1], [-21, 7, -1, -7, -1], [12, 1, 4, -1, -3], [-2, -2, 2, -2, 6], [-8, 4, 3, 3, -4], [-7, 6, -7, -6, 2], [-10, -8, 6, -8, -4], [2, 8, 6, -6, 0], [-8, 6, 6, -6, -4], [3, -9, -5, 5, -1], [24, 14, -2, 0, -2], [-4, -10, -8, 0, 6], [-1, 12, -8, 1, 0], [5, -11, 4, -2, 1], [12, 10, 4, -2, 2], [-19, -1, -9, 1, -1], [0, -4, 0, 4, 4], [-16, 2, 2, 6, 0], [-26, 6, 1, -5, -2], [28, 9, -2, -1, -1], [-15, -4, 4, 3, 4], [6, -4, -2, 8, 0], [12, 6, 8, -4, -2], [19, -3, 2, -2, -3], [26, 2, 12, -4, -6], [22, 3, 3, -2, 3], [16, -3, 9, 2, 1], [-19, -5, -9, 3, -1], [-8, 6, -8, 0, 2], [-14, 12, -2, 0, 0], [-7, -4, 8, 7, -2], [24, -2, 2, -4, -2], [7, 15, -9, -1, -1], [22, 12, -4, 2, 0], [19, -10, 7, -4, -4], [2, 14, -4, -4, 6], [11, -8, -2, -5, 0], [8, -15, -2, 7, 5], [34, -8, 5, 1, -4], [-3, 15, 1, -5, -1], [5, -1, -8, -4, -5], [3, 1, 0, 8, -3], [-9, 14, -4, -3, 2], [-10, 8, -2, 2, -4], [1, -12, 6, 5, -2], [-38, 2, -9, 3, 2], [-27, -8, 2, -3, -4], [-3, -7, -6, -2, -3], [9, -5, -3, -5, -5], [-1, -2, -11, -4, 2], [-2, 9, 3, 2, -5], [-1, -3, -8, -2, -7], [2, -6, -6, 2, -6], [11, -1, -2, 2, 1], [-8, 3, -5, 4, 7], [12, -9, 8, -1, -5], [-10, 4, -1, 3, 0], [-1, -1, 6, 0, -5], [26, -5, 2, -5, 1], [-8, -8, -4, -2, -4], [-10, 14, 2, -2, -2], [-13, 12, -6, -1, 0], [-22, 7, -2, -5, -5], [11, 9, -6, 0, 5], [-21, -1, -4, 6, 11], [-15, 0, -12, -3, 0], [2, -6, 6, 0, 4], [-21, 3, 12, 4, 1], [-17, 4, 0, 3, -4], [20, 13, -9, 2, -3], [-8, -3, 2, -9, -7], [-2, -16, 6, 2, 0], [-12, -14, 0, 0, -6], [6, 5, 8, -11, -3], [32, 1, 4, 1, -3], [0, -6, -11, -1, 0], [11, -1, 6, 4, 11], [-12, 12, -2, -2, 4], [36, -5, 11, 0, -1], [18, 4, 7, -3, 4], [1, 2, -6, 7, 0], [-3, 6, -3, 6, 10], [16, -4, 4, -10, 2], [36, 0, -3, -5, 4], [3, 9, 1, -5, -7], [-5, -10, 12, 1, 6], [-41, 11, 1, 5, 3], [-2, -8, -8, -4, 8], [0, -3, 5, 2, -5], [4, 2, 6, 2, 10], [-10, -7, 7, -2, 3], [-27, 3, 6, 4, 3], [1, -3, 12, 8, -5], [-12, -14, 12, 0, 0], [20, 12, -7, 3, 0], [9, 13, 3, -1, 5], [4, -9, 4, 3, 3], [51, -3, 4, -4, -3], [-12, 2, 4, 8, 6], [-4, 2, -7, -5, -8], [10, 0, -9, -11, 6], [-12, -5, -3, 4, 7], [-15, 20, -15, -6, 4], [-24, 10, 2, -12, -6], [4, 9, -10, -3, 5], [-10, 6, 2, -4, -4], [6, -10, -2, 2, 0], [11, -13, 2, 4, -1], [15, -26, 8, 5, -2], [8, 4, -12, 2, -6], [22, -3, -11, 4, 1], [18, -19, 10, -3, -5], [26, 0, 12, 0, 4], [0, 9, -9, 0, -3], [16, -6, 14, -4, -6], [-3, 4, -16, 3, 2], [-32, 20, -3, -3, 2], [-9, -13, -6, 0, -9], [2, 0, -6, 14, 0], [-31, -4, 2, -3, 6], [-14, 16, -4, -4, -4], [-15, -5, 1, 3, -5], [2, -12, -1, 5, -4], [17, -8, -1, 6, 4], [-19, -12, -3, 2, 6], [8, -8, -4, 2, 2], [5, -7, 12, 8, 3], [-19, -14, -14, 7, 6], [14, 0, 6, -2, 8], [3, 12, 7, -8, -4], [-2, 4, 4, -4, 2], [5, -12, -7, 8, 0], [-45, -7, 3, 1, 1], [-25, -5, -3, 7, 5], [0, -2, -2, 2, 12], [23, 7, -2, 2, 11], [21, 13, -2, -4, 5], [-3, -10, 14, 1, 6], [13, 0, 5, -4, -4], [-7, 7, 2, -2, -5], [11, 2, 2, -1, 2], [10, -6, 8, 8, 6], [-31, -11, 6, 2, 3], [6, -8, -12, 6, 0], [6, -2, -2, -10, -6], [-9, -19, 10, -2, -3], [-4, -14, -1, 1, 4], [17, 4, 3, 4, 2], [4, 0, -12, 14, 4], [-46, -3, 2, -3, 5], [-18, -6, 12, 6, -6], [-12, -2, -4, 0, 2], [-28, -7, 7, 0, 5], [-24, 6, -4, 4, 0], [-37, 10, 8, 1, -2], [-1, -5, -2, -10, -9], [36, 10, -6, 2, 2], [33, -17, 4, 6, 1], [-21, -2, 0, 3, 0], [12, 0, -9, -9, 6], [-16, -6, -4, -6, 2], [-1, 8, 9, 0, -2], [-34, -7, -15, 10, 5], [-18, 30, -2, 0, 4], [-41, 8, 2, 9, 4], [-16, -20, 2, -4, -10], [-1, 1, -8, -10, -7], [-9, 8, 2, 5, 4], [-44, -6, 8, -2, 2], [-21, -14, -10, 7, -6], [-3, -17, 16, -2, 3], [4, 10, -14, -4, 2], [-7, -4, 8, -13, -4], [-9, -12, -2, -1, 0], [10, 16, -10, 12, 0], [-22, 3, -5, -4, 7], [17, -7, 0, 10, 5], [6, -2, -7, -7, -8], [4, 11, 1, 0, -3], [21, -10, -1, 6, -4], [-28, 3, 0, -7, -9], [-7, 23, -14, -4, 1], [13, 11, -17, 3, -3], [2, -6, -12, 10, 2], [14, -12, 6, -8, -8], [30, 8, -3, 5, 8], [27, 3, 8, 6, -1], [-21, 3, 4, -12, 1], [-21, -11, 4, -4, 5], [-8, 26, -4, -2, 2], [0, -21, 11, 8, -5], [-15, 5, -7, -7, 5], [-5, 6, 14, 1, -4], [23, 10, 11, -4, -2], [31, 0, 4, -1, 8], [-16, 8, 2, -12, -6], [8, 0, 2, 10, 4], [-12, 20, 2, 6, 8], [32, -14, 12, -4, -6], [24, -11, 10, -11, -15], [-9, -5, 7, -9, 7], [-10, 12, -10, -10, 0], [-34, 2, 12, 6, -2], [-4, -8, 16, 6, -4], [7, 25, -2, 0, 3], [2, 4, -1, 7, 16], [0, -21, 3, 6, -3], [-9, -16, 13, 2, 4], [1, 21, 2, -8, -3], [2, -24, 10, -4, -8], [8, 3, -1, 4, -5], [-29, -18, 3, 0, 10], [3, -1, -8, 4, 7], [-2, -7, -11, 14, 5], [-15, 3, -7, -5, -9], [23, -7, -2, -10, 1], [8, -26, 2, -6, 0], [-23, -18, 4, 7, -2], [22, -6, -6, 6, 2], [53, -6, -6, -5, -4], [-8, 8, 10, -2, 4], [-11, -5, -15, 9, 5], [34, -1, -1, -2, 11], [32, -10, 6, -4, -6], [-10, 0, -10, 8, 8], [-14, -12, 4, -2, 0], [-4, 14, 4, 0, 6], [-21, -6, 2, 1, 0], [-5, 9, -9, 1, 9], [-8, -22, -4, -2, -10], [-14, 6, -9, -9, -4], [-13, -24, 1, 0, -12], [18, 6, -6, 2, 2], [-18, -6, 14, 0, 2], [-20, 14, -2, 2, 8], [-37, 5, -18, 12, 9], [-8, -4, -19, -3, -2], [12, 0, 14, -6, 8], [37, 32, -9, -4, 10], [10, 0, -8, 0, 0], [12, 12, -4, -2, 4], [4, 22, -6, 2, 10], [18, 16, -20, -4, 0], [24, 2, 12, -2, 2], [-11, 6, -14, -5, -8], [32, 18, -12, 4, -4], [-18, 0, 4, -6, 4], [-23, 31, -13, -1, 7], [-9, 5, 0, -2, -7], [23, -4, 1, -2, 0], [36, -23, 5, -10, -11], [47, 3, 10, -4, -5], [-10, 0, -2, 2, -2], [16, 9, -13, 8, 3], [17, 1, 20, -2, -7], [13, 0, 4, 9, 12], [-21, 10, -9, 8, 8], [-14, 10, -6, 12, 8], [30, -4, 12, -4, -4], [-44, -10, -2, -2, 4], [30, -27, 3, -4, -7], [13, -15, 5, 1, 1], [0, 20, -4, -6, 2], [-7, 4, 7, 6, -4], [48, -7, 19, -6, -11], [-2, 39, -15, -4, 7], [37, -4, -1, 0, 0], [7, -7, 7, -5, -5], [-12, -8, 12, -6, 0], [24, -7, 8, 7, -3], [-8, 12, -2, 2, 2], [29, 1, 6, -6, 5], [-15, -22, 12, -7, -4], [-2, -12, 8, -6, 4], [-40, 6, -12, 12, 4], [-9, -2, -7, 10, 0], [46, -4, 8, 0, -8], [-2, -44, 6, 4, -6], [13, -9, -6, -6, 3], [30, 11, -15, -4, -1], [42, -2, 12, -8, -2], [29, 4, 16, -1, -8], [-44, 7, 2, -11, -7], [-3, 14, 14, 1, 6], [-20, -6, -10, -4, 0], [-23, 11, -2, 16, 11], [46, 2, 0, 2, -8], [0, -8, 8, 12, 8], [-16, -21, 5, 2, -1], [26, -1, 6, 7, 3], [-26, 4, 3, 9, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7935_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7935_2_a_bc();".
// 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_7935_2_a_bc(:prec:=5)
    chi := MakeCharacter_7935_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_7935_2_a_bc();".
// 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_7935_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7935_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-6, 16, 2, -9, 0, 1]>,<7,R![12, 40, 23, -13, -3, 1]>,<11,R![-384, 80, 118, -20, -6, 1]>],Snew);
    return Vf;
end function;