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        Return (p-)new subspace of this space of Hilbert modular forms.

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        EXAMPLES::

        We make a space of level a product of 2 split primes and (2)::

            sage: from psage.modform.hilbert.sqrt5.hmf import F, HilbertModularForms
            sage: P = F.prime_above(31); Q = F.prime_above(11); R = F.prime_above(2)
            sage: H = HilbertModularForms(P*Q*R); H
            Hilbert modular forms of dimension 32, level 2*a-38 (of norm 1364=2^2*11*31) over QQ(sqrt(5))

        The full new space::

            sage: N = H.new_subspace(); N
            Subspace of dimension 22 of Hilbert modular forms of dimension 32, level 2*a-38 (of norm 1364=2^2*11*31) over QQ(sqrt(5))

        The new subspace for each prime divisor of the level::

            sage: N_P = H.new_subspace(P); N_P
            Subspace of dimension 31 of Hilbert modular forms of dimension 32, level 2*a-38 (of norm 1364=2^2*11*31) over QQ(sqrt(5))
            sage: N_Q = H.new_subspace(Q); N_Q
            Subspace of dimension 28 of Hilbert modular forms of dimension 32, level 2*a-38 (of norm 1364=2^2*11*31) over QQ(sqrt(5))
            sage: N_R = H.new_subspace(R); N_R
            Subspace of dimension 24 of Hilbert modular forms of dimension 32, level 2*a-38 (of norm 1364=2^2*11*31) over QQ(sqrt(5))
            sage: N_P.intersection(N_Q).intersection(N_R) == N
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            sage: H = psage.hilbert.sqrt5.HilbertModularForms(3); H
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        EXAMPLES::

            sage: from psage.modform.hilbert.sqrt5.hmf import HilbertModularForms, F
            sage: H = HilbertModularForms(F.prime_above(31)).elliptic_curve_factors()[0]
            sage: type(H)
            <class 'psage.modform.hilbert.sqrt5.hmf.EllipticCurveFactor'>
            sage: H.__repr__()
            'Isogeny class of elliptic curves over QQ(sqrt(5)) attached to form number 0 in Hilbert modular forms of dimension 2, level 5*a-2 (of norm 31=31) over QQ(sqrt(5))'
        sRIsogeny class of elliptic curves over QQ(sqrt(5)) attached to form number %s in %s(RtRsRY(
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        Return the base field of this elliptic curve factor.

        OUTPUT:
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        EXAMPLES::

            sage: from psage.modform.hilbert.sqrt5.hmf import HilbertModularForms, F
            sage: H = HilbertModularForms(F.prime_above(31)).elliptic_curve_factors()[0]
            sage: H.base_field()
            Number Field in a with defining polynomial x^2 - x - 1
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        OUTPUT:
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        EXAMPLES::

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        INPUT:
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        OUTPUT:
            - an integer

        EXAMPLES::

            sage: from psage.modform.hilbert.sqrt5.hmf import HilbertModularForms, F
            sage: H = HilbertModularForms(F.primes_above(31)[0]).elliptic_curve_factors()[0]
            sage: H.ap(F.primes_above(11)[0])
            4
            sage: H.ap(F.prime_above(5))
            -2
            sage: H.ap(F.prime_above(7))
            2

        We check that the ap we compute here match with those of a known elliptic curve
        of this conductor::

            sage: a = F.0; E = EllipticCurve(F, [1,a+1,a,a,0])
            sage: E.conductor().norm()
            31
            sage: 11+1 - E.change_ring(F.primes_above(11)[0].residue_field()).cardinality()
            4
            sage: 5+1 - E.change_ring(F.prime_above(5).residue_field()).cardinality()
            -2
            sage: 49+1 - E.change_ring(F.prime_above(7).residue_field()).cardinality()
            2
        it
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        with the same system of eigenvalues as self.  This is useful when
        computing a large number of `a_P`.

        If we can't find such a subspace using Hecke operators of norm
        less than B, then we raise a RuntimeError.  This should only happen
        if you set B way too small, or self is actually not new.

        INPUT:
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        Return list of traces of Frobenius for all primes P of norm
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        psage.number_fields.sqrt5.primes_of_bounded_norm(B)
        to get the corresponding primes.

        INPUT:
            - `B` -- a nonnegative integer
            - ``dual_bound`` -- default None; passed to dual_eigenvector function
            - ``algorithm`` -- 'dual' (default) or 'direct'

        OUTPUT:
             - a list of Sage integers

        EXAMPLES::

        We compute the aplists up to B=50::

            sage: from psage.modform.hilbert.sqrt5.hmf import HilbertModularForms, F
            sage: H = HilbertModularForms(F.primes_above(71)[1]).elliptic_curve_factors()[0]
            sage: v = H.aplist(50); v
            [-1, 0, -2, 0, 0, 2, -4, 6, -6, 8, 2, 6, 12, -4]

        This agrees with what we get using an elliptic curve of this
        conductor::

            sage: a = F.0; E = EllipticCurve(F, [a,a+1,a,a,0])
            sage: from psage.ellcurve.lseries.aplist_sqrt5 import aplist
            sage: w = aplist(E, 50)
            sage: v == w
            True

        We compare the output from the two algorithms up to norm 75::

            sage: H.aplist(75, algorithm='direct') == H.aplist(75, algorithm='dual')
            True
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