Kohnen and Zhang Res. Number Theory (2018) 4:1
On sign changes of Fourier coeﬃcients of
and Yichao Zhang
Department of Mathematics
and Institute of Advanced Study
in Mathematics of HIT, Harbin
Institute of Technology, 92
Xidazhi Street, Nangang District,
Harbin 150001, China
Full list of author information is
available at the end of the article
It is known that if a nonzero cusp form has real Fourier coeﬃcients, then its Fourier
coeﬃcients change signs inﬁnitely often. In this paper, we prove that there is a
codimension one subspace in the space of holomorphic modular forms of square-free
level such that all of its non-zero forms have similar sign change property.
Let f be a non-zero cusp form of weight k for a congruence subgroup of the full modular
(Z) with real Fourier coeﬃcients a(n)(n ≥ 1).Thenitiswell-known
that the sequence (a(n))
has inﬁnitely many sign changes, i.e. there are inﬁnitely many
n with a(n) > 0 and inﬁnitely many n with a(n) < 0.
On the other hand, let
be the “normalized” Eisenstein series of even integral weight k ≥ 4 for
.Hereq = e
for z in the complex upper half-plane
. Furthermore, B
is the k-th Bernoulli number
Then by deﬁnition all Fourier coeﬃcients of G
with positive index are positive.
However, if one admits levels N > 1 then there exist linear combinations of Eisenstein
series whose Fourier coeﬃcients change signs inﬁnitely often, for example the function
(pz) − G
where p and are diﬀerent primes, clearly has this property. Sign changes of Eisenstein
series that are “newforms” were studied quite generally in .
The question arises if one can explicitly decompose spaces of modular forms according
to a “sign change property” of their Fourier coeﬃcients. In this paper, using certain linear
combinations of Eisenstein series we will give a simple answer to this question if the level
N is squarefree and the weight is at least 4. A similar result can be derived also in weight 2,
the precise formulation however is slightly diﬀerent due to the slightly diﬀerent structure
of the space of Eisenstein series in weight 2. We leave the exact formulation and a detailed
work-out to the reader.