
// Make newform 592.2.bc.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_592_bc();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_592_bc_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_592_2_bc_d();".
// 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_592_2_bc_d();".
// 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 := [15625, 0, -15000, 0, 6600, 0, -1687, 0, 264, 0, -24, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3125, 52500, 0, -40150, 0, 13644, 0, -2568, 0, 263, 0, -12], [-7500, 125, 5100, 0, -1562, 0, 264, 0, -24, 0, 1, 0], [-7500, -125, 5100, 0, -1562, 0, 264, 0, -24, 0, 1, 0], [175000, -11875, -133000, 6600, 46040, -1687, -8880, 264, 955, -24, -45, 1], [-175000, -11875, 133000, 6600, -46040, -1687, 8880, 264, -955, -24, 45, 1], [-125000, -11875, 103125, 13000, -37605, -5719, 7560, 1393, -835, -188, 40, 12], [125000, -11875, -103125, 13000, 37605, -5719, -7560, 1393, 835, -188, -40, 12], [34375, -10000, -33000, 5975, 13240, -1687, -2905, 264, 355, -24, -20, 1], [34375, 10000, -33000, -5975, 13240, 1687, -2905, -264, 355, 24, -20, -1], [-184375, -72500, 168125, 62975, -66200, -23961, 14525, 4992, -1775, -572, 100, 28], [184375, -72500, -168125, 62975, 66200, -23961, -14525, 4992, 1775, -572, -100, 28]];
    Rf_basisdens := [1, 6250, 250, 250, 6250, 6250, 6250, 6250, 1250, 1250, 6250, 6250];
    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_592_bc();"
function MakeCharacter_592_bc()
    N := 592;
    order := 9;
    char_gens := [223, 149, 113];
    v := [9, 9, 5];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_592_bc_Hecke();"
function MakeCharacter_592_bc_Hecke(Kf)
    N := 592;
    order := 9;
    char_gens := [223, 149, 113];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, -1, 1, -1, 1, 1, 0, 1, 0], [0, -1, 0, 0, 1, 0, 0, 0, -1, -1, 0, 1], [0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 1, 1], [0, 0, -1, 0, 0, 1, -1, 1, 0, 1, 0, 0], [0, 1, 1, -1, -4, -1, 0, 3, 0, 0, -1, 0], [4, -4, -1, 0, -1, 1, -1, 1, 1, 1, 1, 0], [0, 2, 0, 0, -1, -1, 1, -1, 0, 1, -1, 0], [-4, 1, 2, -2, 1, -2, 4, -4, -1, 1, 0, 0], [0, 0, 1, -1, 1, -5, 4, 0, -1, -1, -1, 2], [-1, -1, -2, 2, 2, -2, 3, 2, -2, 0, -2, 1], [-4, 2, -2, 2, 1, 2, 0, -1, -1, 1, -1, -1], [0, 0, 1, 0, 2, -1, 5, 1, 2, 1, 1, 1], [0, -1, 0, 0, -1, 3, 0, 1, 1, 0, 1, 0], [0, 1, 0, -1, 0, -4, -1, 0, 0, 1, 0, -1], [-3, 3, 1, -2, -2, 0, -5, -3, 2, 0, 1, -1], [0, 4, 2, -1, 2, -2, 5, -1, -2, -1, -1, 1], [0, 2, 0, 0, 0, -4, 0, 0, 0, -2, 0, 0], [3, 3, 2, -2, -3, 2, 0, -2, -1, 1, -2, -2], [4, -6, -1, 0, 3, -3, -1, 1, 1, -1, 1, 1], [0, 1, -1, 1, 0, 1, 4, 0, 0, 1, 0, -1], [1, -4, 1, 2, -1, -5, 3, 1, 2, -2, 2, 3], [-8, 8, -3, 0, 2, 0, 1, 3, -1, 0, 0, -1], [-4, 4, -2, 0, 1, -5, 6, 7, 0, 0, 0, 0], [0, 4, 0, -1, 2, 4, -5, -1, 2, 3, -1, -2], [-8, 3, -3, 1, 1, -5, -2, -3, -1, 0, -3, 1], [3, -1, -3, 1, -1, -1, 2, 4, 0, -2, 0, 3], [-8, 8, 2, -2, -8, -1, -1, 0, 0, 0, 0, -2], [0, 2, 1, 1, 3, 3, 2, 1, 1, -1, 1, 2], [-4, -4, 0, 0, 4, 0, 4, 4, 0, 0, 4, 0], [0, -7, 0, 5, 1, -3, 1, -4, 3, 0, 3, -3], [-4, 0, 4, -2, 4, -4, -2, -4, 0, -4, 0, 2], [-4, 5, -3, 3, 4, 2, -3, 1, -3, 3, -2, -2], [4, -4, 1, 0, 7, -1, -3, -8, 1, -4, 0, 0], [-4, 5, -2, 2, 4, 2, -4, -3, 0, 2, 0, -2], [-4, 7, 0, -1, 2, -4, -1, 2, -2, 1, -2, -1], [-4, -4, 1, 0, -5, -1, 5, 4, 1, -4, 0, 0], [-3, 5, 1, 4, 5, -1, 5, -3, -5, 5, 0, -1], [-4, 4, -3, 3, -5, 11, -4, 3, -3, 3, -1, -1], [0, -1, 0, -4, -4, -1, -3, 0, 4, 0, 0, 0], [0, 6, -1, 1, 10, -7, 0, -4, -2, 2, -4, -4], [0, -6, 0, -1, -3, -1, 4, 4, -2, -2, 0, 2], [3, -6, -2, -1, 2, -2, 1, 0, 1, -3, 0, 2], [4, -4, 2, 0, -1, -2, 2, 3, 1, 3, 3, -2], [0, -7, 0, -1, -3, -4, 3, -1, -1, 0, -1, 1], [4, 0, 5, -5, -6, 3, 0, 2, -2, 2, 2, 2], [-4, 2, -2, 3, -6, 6, -3, -1, 2, 1, -1, -1], [0, -1, 1, -3, 1, -1, 1, 2, -1, -2, 2, 2], [0, -5, 0, -1, 1, 9, -5, 9, 3, -1, 4, -3], [-8, -3, -3, 1, 8, 3, -2, 6, 0, 3, 2, -1], [4, 1, -1, 0, 4, 5, -5, -3, -1, 5, 0, 0], [0, 7, 0, -5, -7, 0, 3, 4, -1, 3, -4, 1], [0, 1, -3, 1, 0, -1, 2, 3, -3, -2, 0, 3], [-4, 9, -1, 4, 4, 5, -5, 3, -4, 0, -1, -1], [1, -1, 0, 0, 8, 8, 4, -6, 4, 2, 2, 2], [0, 4, -1, 1, -7, -7, 8, 4, 3, 0, 0, 1], [1, 1, 3, -2, -9, 1, 0, 6, 1, 3, -2, 0], [0, 3, -1, -3, 0, 9, -4, -1, -1, 3, -1, -4], [8, -4, 0, -2, -2, 4, 2, -4, 2, 0, 0, -3], [-1, 3, -3, 4, -9, -2, 2, 12, 1, 2, 4, 0], [0, -6, 0, -1, 4, 0, -9, -5, 4, 1, 3, -4], [0, 0, 0, 0, 0, -4, 0, 4, 4, 0, 4, 0], [-5, 0, 0, 3, 5, 0, -5, 0, 0, 0, 5, -3], [9, -14, 4, 1, -4, -12, 5, -4, 5, -1, 5, 5], [-4, 1, -3, 0, -1, -9, 9, 0, -3, -3, 0, 3], [5, -2, -2, -1, -1, 3, 1, -2, 1, 0, -2, 4], [8, -8, -4, 0, -6, 4, -4, 3, 6, 3, 3, 3], [7, -7, 0, 3, 3, 0, 3, -2, 5, -3, 5, 3], [5, 3, -6, 4, -5, 6, 2, 3, 0, 6, -1, -4], [0, -4, 0, 7, 2, 0, 7, -1, -2, -1, -1, 2], [8, -9, -5, 1, 0, 13, -12, 5, -4, 0, 1, 0], [-8, -2, -1, -2, 7, -2, 5, -6, -3, -7, -2, 0], [0, -1, -1, 1, 0, 1, 0, 2, 0, 1, -6, -1], [0, 10, -2, -1, -1, 10, -11, -2, 1, 0, -2, -3], [1, 1, -3, -1, -4, -1, -5, 7, -4, -4, -1, 0], [8, -8, -1, 0, -4, 9, -14, -6, -1, 2, 2, -3], [-13, 10, 0, 0, 6, 0, 8, -12, -6, 6, -4, -4], [5, -8, 3, 1, 12, 5, -4, -7, 0, 4, 0, -3], [8, -17, -7, 2, 12, 3, -1, -4, 0, -5, 0, 7], [4, -4, 5, 0, -9, -6, -2, 4, 1, 3, 3, -2], [-12, 10, -6, 6, 6, -2, -4, -14, -6, 0, -6, -4], [0, -13, 0, -4, -10, 0, 3, 9, 3, 1, 2, -3], [-4, 0, 1, 6, -3, 3, -1, -7, -1, -3, -7, 3], [16, -7, -4, 4, -6, 8, 0, 5, 2, -2, 5, 5], [8, 0, -1, 9, 8, -7, 8, -8, 1, 0, -8, 0], [-13, 20, -3, 3, 3, 7, -17, 5, -3, 0, -3, 1], [4, 2, 2, -5, -4, -2, -10, -4, 0, -2, 0, 5], [4, 7, 3, -1, -6, -7, -2, 3, 2, 2, -1, 0], [4, -4, 1, 0, -5, -13, 9, 8, 1, -4, -4, 5], [4, 2, -1, -1, 5, -7, 6, 8, -9, -2, 0, 1], [0, 10, 3, -3, 2, 1, 12, -16, -6, 6, -4, -4], [24, -24, 4, 0, 3, -4, 4, -4, -3, -4, -4, 1], [4, 16, -3, 3, 4, -13, -4, -1, 0, 3, 3, 0], [-12, 15, 0, -2, 1, 4, -2, 11, -1, 2, -1, -2], [-5, -6, 2, 4, 5, -2, -14, 4, 0, -2, 1, -4], [12, -1, 4, -2, 1, -4, -10, -5, 2, -3, -2, 0], [-20, 15, 4, 0, 12, 16, -8, -4, -4, 3, -4, -3], [-4, 4, -4, 0, 4, -9, 13, 11, 0, -2, -2, 2], [0, -9, 0, -4, 7, -12, -12, -15, 1, -4, 5, -1], [7, 10, 6, -1, -9, 2, -3, 12, 4, 5, 0, -6], [0, -6, 0, -5, -14, 4, 7, 18, 2, 0, 2, -2], [16, -16, 0, 0, -3, -3, 4, 6, 4, 3, 3, 1], [12, 2, 4, -1, 13, -4, 3, -5, 7, 1, 7, 3], [-16, 9, -5, 5, -2, 8, 0, 10, -6, -5, -6, 0], [-12, 1, -3, -4, -15, 15, -11, 7, 3, 0, 7, 0], [-8, -1, 2, -2, -19, 18, -17, 19, -1, 1, 2, 2], [4, 14, -2, -1, -2, -2, 9, 3, 2, 9, 3, -9], [-11, 8, 6, -8, 11, -6, 6, -14, 0, -6, 2, 8], [8, -1, -2, 5, 12, -2, 3, -8, 0, -5, 0, 3], [0, 13, 3, -3, 10, 13, 0, 1, 1, 3, 1, 0], [8, 1, 5, -4, -10, -20, 16, 8, 10, 1, 0, -5], [-4, 4, 4, 0, 5, -12, 16, 7, -1, -1, -1, 0], [-4, 23, 2, 1, -18, 10, -9, 16, -2, 3, 0, -2], [1, 15, 0, 3, -4, -4, 7, 13, -8, 3, 0, 0], [8, -8, 3, 0, -7, -15, 7, 8, -1, -4, -4, 3], [-12, 11, 0, -1, -7, -16, 11, 7, -1, -6, -1, 0], [11, -9, -6, 2, 2, 2, 11, -18, -2, 0, -6, 10], [0, -11, 0, 0, 12, 0, -12, 0, 0, 1, 0, -1], [-9, 18, -4, -1, -16, 13, 7, 5, 4, 2, 5, -2], [20, -5, 2, 0, -13, -22, 10, -2, -2, 5, -2, -5], [0, -19, 0, 0, 2, -5, 1, -8, -3, 1, -4, 3], [-8, 11, 1, -4, 8, -1, 9, -12, 0, -1, -4, 4], [8, -4, -3, -1, -6, 3, -4, 8, 6, -4, 0, 3], [-4, -3, -3, 0, 6, 16, -16, -4, -6, -3, 0, 3], [-10, 3, 1, -1, 16, -17, -4, 1, 0, 0, -3, -3], [16, -21, -1, 1, -4, 9, 0, -8, 8, -1, 8, 0], [8, -16, 4, -5, -5, -8, 11, -8, 5, 0, 4, 6], [12, -12, 3, -5, 16, -15, 6, 2, -3, -6, 2, 6], [-8, 2, 6, -6, -5, -2, -20, 17, 1, -1, -3, -3], [-8, 5, 4, 2, 6, -20, 14, -10, 6, -9, 2, 0], [0, -21, 0, -1, 5, 16, -9, 10, 3, 1, 2, -3], [-11, 0, 1, -8, 11, -1, -7, -1, 0, -1, -9, 8], [4, -11, -2, 1, -6, -2, -1, -7, -3, -1, -3, -1], [0, -17, 0, -8, -15, -8, 4, 13, 3, -6, 9, -3], [1, 5, 3, -3, -3, 6, -8, -1, 3, 0, 3, -3], [-4, 4, -4, 0, 11, 20, -8, -10, 1, 6, 6, -5], [4, -2, 7, -9, -4, -7, 8, -5, 0, -7, -5, 9], [0, -4, 5, -8, 2, -5, 1, 3, -5, -5, 3, 5], [-16, 2, 4, -5, -23, -4, 15, 19, -1, -3, -5, 0], [8, -5, -6, 2, 3, -2, -4, 4, 6, 7, 4, -7], [0, -15, -1, -5, 0, 1, 10, 16, 0, 1, 8, 5], [8, -9, 9, -3, -3, -1, 6, 1, 3, 0, 9, 7], [0, 6, 1, 3, 9, -3, 4, -5, -5, -3, -5, 4], [-4, 11, 7, 0, -6, -7, 7, 0, 2, 7, 0, -7], [-17, -8, 4, -5, 14, -8, 3, -24, -7, -1, 0, -4], [0, 3, 5, -5, 5, -6, 13, -12, -4, 4, 1, 1], [0, 16, 0, -2, 4, 0, -10, -10, 4, 6, -2, -4], [20, -9, 3, -4, 4, -19, 15, 12, 4, -1, 0, -3], [0, -8, 0, 7, 3, 15, 4, 8, 0, 4, -4, 0], [-16, 7, 0, 2, -16, 16, -15, -2, 0, 6, -2, -6], [4, 3, 2, 3, 10, -6, 9, -5, -2, -2, -5, 2], [20, 6, 5, -5, -7, 7, -4, 3, -5, 5, -1, -1], [-16, 9, -9, 12, -3, -11, 19, 12, 3, 1, 12, 0], [-4, 4, 5, 0, 15, 11, 5, -19, 1, -3, -3, 4], [8, -9, -5, 5, -14, 17, -4, 11, 2, -2, 7, 7], [-4, 15, -4, -5, -5, 11, -20, 24, 5, 0, -4, 5], [-4, 7, -3, 3, -15, -5, 8, 3, 8, 0, 0, 3], [9, 1, 3, -2, 8, 9, 1, 0, 9, 2, 9, 1], [4, 10, 11, -7, 7, 1, 4, -7, -3, 7, -3, 4], [12, -1, -8, 0, -12, 8, -3, 9, 0, 8, -3, 0], [4, -26, -3, -1, -15, -1, -20, 4, 3, -2, 4, 2], [12, -15, -3, 1, -12, 3, -10, 18, 0, 3, 6, -1], [8, -1, 7, -9, -2, 0, -2, 2, 10, 9, 10, -2], [4, -10, 4, 4, -2, 8, 8, -2, 2, -4, 2, 8], [0, 2, 0, 6, 6, 4, -6, -8, 6, 6, 0, -6], [-4, 1, 0, -3, 6, -26, 23, 0, -10, -3, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_592_bc_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_592_2_bc_d();".
// 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_592_2_bc_d(:prec:=12)
    chi := MakeCharacter_592_bc();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_592_2_bc_d();".
// 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_592_2_bc_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_592_bc();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4096, 3072, 1536, 512, 384, -504, -151, 126, 24, -8, 6, -3, 1]>],Snew);
    return Vf;
end function;