\section{A shifted bilinear estimate at weight one}\label{sec:spectral}

Let $\varphi=\theta_\eta$ be the weight-one theta series attached to the
cubic character $\eta$ over $F=\Q(\zeta_3)$.  Its level $N$ is fixed,
its nebentypus is the quadratic character $\psi=\chi_{-3}$, and its
Fourier expansion is
\[
 \varphi(z)=\sum_{n\geq1}\lambda_\eta(n)e(nz),
 \qquad e(t)=\exp(2\pi i t).
\]
The coefficients are real and satisfy
$|\lambda_\eta(n)|\leq\tau(n)$.  This section proves the estimate needed
when a large prime factor is separated from its small cofactor.

\begin{proposition}\label{prop:bilinear}
Let $\Phi\in C_c^\infty((0,\infty))$ be fixed, let $X\geq2$, and let
$1\leq M_1\leq M_2$.  Suppose that $|\alpha_m|,|\beta_n|\leq1$ and that
$\alpha_m$ is supported on squarefree positive integers.  For every
$\nu>0$,
\begin{equation}\label{eq:bilinear}
 \left|\sum_{M_1\leq m\leq M_2}\sum_{n\geq1}
 \alpha_m\beta_n\Phi(mn/X)\lambda_\eta(mn+1)\right|
 \ll_{\eta,\Phi,\nu}
 X^\nu\left(XM_1^{-1/2}+X^{7/8}M_2^{3/8}\right).
\end{equation}
There is no coprimality restriction on $m$ and no further restriction
on $\beta_n$.
\end{proposition}

We adapt the argument of Assing--Blomer--Li
\cite[Theorem~2.4 and Sections~8.2--8.3]{ABL}.  Their Voronoi formula
is stated for trivial nebentypus.  We first establish the formula for
$\varphi$ and calculate the character factors in its two applications.

\subsection{Voronoi summation and coefficient norms}
Suppose that $N\mid q$ and $(a,q)=1$.  Choose integers $b,d$ with
$ad-bq=1$.  Applying the modular transformation
$\left(\begin{smallmatrix}a&b\\q&d\end{smallmatrix}\right)$ to
$-d/q+i/(q^2y)$ gives
\begin{equation}\label{eq:weight-one-transform}
 \varphi(a/q+iy)
 =\psi(d)\frac{i}{qy}\,
   \varphi\bigl(-d/q+i/(q^2y)\bigr).
\end{equation}
Here $\psi(d)=\psi(a)$ because $\psi$ is quadratic.  Mellin inversion
of \eqref{eq:weight-one-transform} yields, for
$H\in C_c^\infty((0,\infty))$,
\begin{align}
 \sum_{n\geq1}\lambda_\eta(n)e(an/q)H(n)
 &=\frac{\psi(d)}q\sum_{n\geq1}\lambda_\eta(n)e(-dn/q)
       \mathcal H(n/q^2),\label{eq:weight-one-voronoi}\\
 \mathcal H(y)&=2\pi i\int_0^\infty
              H(x)J_0(4\pi\sqrt{xy})\,dx.\notag
\end{align}
Cuspidality justifies the Mellin integrals at both endpoints.
The formula has a single holomorphic Bessel transform.  The estimates
for this transform follow by the integration-by-parts argument in
\cite[Lemma~7.5]{ABL}, with $J_{k-1}=J_0$.

We also need a coefficient estimate uniform in an additive frequency.
The function $(\operatorname{Im}z)^{1/2}|\varphi(z)|$ is invariant under
$\Gamma_0(N)$ and bounded on a fundamental domain: it is bounded on
the compact part and tends to zero at each cusp.  Thus
\[
 \sup_{z\in\mathbb H}(\operatorname{Im}z)^{1/2}|\varphi(z)|<\infty.
\]
For $T\geq2$, put $y=1/T$ in the Fourier expansion and convolve over
one period with
\[
 D_T(t)=\sum_{1\leq j\leq T}\exp(2\pi j/T)e(jt).
\]
Partial summation of the Dirichlet kernel gives
$\|D_T\|_1\ll\log(2T)$.  The convolution cancels the factors
$\exp(-2\pi n/T)$ for $n\leq T$, and consequently
\begin{equation}\label{eq:wilton-weight-one}
 \sup_{\theta\in\mathbb R}
 \left|\sum_{n\leq T}\lambda_\eta(n)e(n\theta)\right|
 \ll_\eta T^{1/2}\log(2T).
\end{equation}
A smooth dyadic cutoff and a factor $n^{-it}$ may be inserted by partial
summation, at the cost of $O(1+|t|)$ and the cutoff seminorms.
The elementary bound by $\tau$ also gives
\begin{equation}\label{eq:spectral-square-norm}
 \sum_{n\leq T}|\lambda_\eta(n)|^2\ll_\nu T^{1+\nu}.
\end{equation}

For later use, fix coprime $m_3,m_4$, a positive integer $D$, and smooth
cutoffs $w_j$ supported on $k_j\asymp K_j$.  Define
\[
 A_\rho=
 \sum_{\substack{m_4k_2-m_3k_1=\rho\\
                  m_3^2k_1\equiv m_4^2k_2\pmod D}}
 \lambda_\eta(k_1)\lambda_\eta(k_2)
 k_1^{-it_1}k_2^{-it_2}w_1(k_1)w_2(k_2).
\]
Expand the congruence as a normalized average of $D$ additive
characters.  For each character, $A_\rho$ is the coefficient sequence
of a product of two trigonometric polynomials, with frequencies
$-m_3k_1$ and $m_4k_2$.  Parseval, the supremum bound
\eqref{eq:wilton-weight-one} on one factor, and
\eqref{eq:spectral-square-norm} on the other imply
\begin{equation}\label{eq:dual-coefficient-norm}
 \|A\|_2\ll_{\eta,\nu,w_1,w_2}
 (K_1K_2)^{(1+\nu)/2}(1+|t_1|+|t_2|)^2.
\end{equation}
The triangle inequality for the normalized character average introduces
no factor of $D$.

\subsection{The double Voronoi transformation}
We first restrict the small variables to be coprime to $N$.  Let
$m_1,m_2\asymp M$ be distinct squarefree integers, with
$(m_1m_2,N)=1$, and write
\[
 d=(m_1,m_2),\qquad m_1=dm_3,\quad m_2=dm_4,\qquad
 D=(m_1-m_2,d^\infty),\quad F_0=ND^2.
\]
The notation $(v,d^\infty)$ means the largest divisor of $|v|$ all of
whose prime divisors divide $d$.  Squarefreeness gives
\[
 (m_3,m_4)=1,\qquad (D,m_3m_4)=1,\qquad d\mid D,
 \qquad (F_0,m_3m_4)=1.
\]
For a smooth function $g$ supported on $[T,2T]$, consider
\[
 \mathcal T(m_1,m_2)=\sum_n
 \lambda_\eta(m_1n+1)\lambda_\eta(m_2n+1)g(n).
\]
Writing $k_i=m_in+1$ transforms the relation between the two arguments
into
\begin{equation}\label{eq:shifted-linear-relation}
 m_4(k_1-1)=m_3(k_2-1),\qquad k_1\equiv1\pmod d.
\end{equation}
Use the circle moduli
\[
 q\asymp Q,\qquad F_0\mid q,\qquad(q,m_3m_4)=1,
\]
where $Q$ is a sufficiently large fixed power of the parameters.
Their total totient weight is $Q^{2+o(1)}/F_0$, as in the circle
construction of \cite[Section~8.2]{ABL}.  In particular, both $q$ and
$q/d$ are divisible by $N$.

After detecting \eqref{eq:shifted-linear-relation}, the two additive
numerators in the Voronoi transformations are
\[
 a_1=m_4a(1+jq/d),\qquad a_2=-m_3a,
 \qquad (a,q)=1,\quad j\pmod d.
\]
Both are units modulo $q$.  Indeed, $d$ is squarefree and $d^2\mid q$,
so every prime divisor of $q$ also divides $q/d$.  Hence
$1+jq/d$ is a unit modulo $q$.  The additional factor from the two
applications of \eqref{eq:weight-one-voronoi} is
\begin{equation}\label{eq:quadratic-cancellation}
 \psi(a_1)\psi(a_2)
 =\psi(-m_3m_4)\psi(a)^2\psi(1+jq/d)
 =\psi(-m_3m_4).
\end{equation}
This scalar is independent of $a$ and $j$.  In particular, the sign
$\psi(-1)=-1$ is retained by the formula.

Write
\[
 S(b,c;q)=\sum_{a\pmod q}^{*}e\bigl((ba+c\bar a)/q\bigr)
\]
for the ordinary Kloosterman sum.  Summing over $a$ and $j$ now gives,
apart from the scalar in \eqref{eq:quadratic-cancellation}, precisely
\begin{equation}\label{eq:ordinary-kloosterman-kernel}
 S\bigl(m_3-m_4,\overline{m_3m_4}(m_4k_2-m_3k_1);q\bigr)
 \boldsymbol{1}_{m_3^2k_1\equiv m_4^2k_2\pmod D}.
\end{equation}
The factors $q^{-2}$, the normalized circle weight, and the
$d$-congruence average are unchanged.
The terms with $m_4k_2=m_3k_1$ have the Ramanujan sum
$S(m_3-m_4,0;q)$.  Since $m_3\ne m_4$, their total is
$O(Q^{-1})$ times a fixed polynomial in the other parameters, by the
same divisor estimate as in \cite[Section~8.2]{ABL}; the choice of $Q$
makes this negligible.

For the remaining terms put $\rho=m_4k_2-m_3k_1$ and divide the
$\rho$-range into dyadic intervals $|\rho|\asymp R$.  The coefficient
norm is \eqref{eq:dual-coefficient-norm}.  We may therefore apply
\cite[Theorem~2.4]{ABL} to \eqref{eq:ordinary-kloosterman-kernel}, with
its parameters
\[
 (r,s,n,m,M,C,Z)
 =\bigl(m_3m_4,F_0,|m_3-m_4|,\rho,R,Q/F_0,Z_1\bigr).
\]
Here $Z_1$ is the smoothness parameter furnished by the Bessel
transforms.  The coprimality condition $(r,s)=1$ follows from the
identities above.  The dual-length truncations and the condition on
the Bessel scale are those in \cite[Section~8.2]{ABL}, because
\eqref{eq:weight-one-voronoi} has its holomorphic transform with $k=1$.
The Kloosterman estimate includes the exceptional spectrum.  All
dependence on the original level $N$ is absorbed into the fixed-form
constant.

In particular, if $g^{(j)}\ll_j T^{-j}$, the resulting correlation
bound, specialized to the shift $1$, is
\begin{equation}\label{eq:shifted-correlation}
 |\mathcal T(m_1,m_2)|\ll_{\eta,\nu,g}
 (MT)^\nu\left\{
 (MT+1)^{1/2}M^{1/2}\frac{D}{d^{1/2}}
 +(MT+1)^{3/4}\frac{D^{1/4}}{M^{1/4}}
 \right\}.
\end{equation}
The two terms are the two contributions with nonnegative powers of
$Q$ in the calculation on the last page of
\cite[Section~8.2]{ABL}.  The arithmetic factor involving the shift
is $(1,Dm_3m_4)^{1/4}=1$.  Thus the character computation
\eqref{eq:quadratic-cancellation} and the norm estimate
\eqref{eq:dual-coefficient-norm} supply the required changes at weight
one.

\subsection{Bilinear summation}
We complete the proof of Proposition~\ref{prop:bilinear}.
First suppose $m\asymp M$, $n\asymp T$, $(m,N)=1$, and $MT\asymp X$.
Cauchy's inequality in $n$ reduces the square of the bilinear sum to
\[
 T\sum_{m_1,m_2}\alpha_{m_1}\overline{\alpha_{m_2}}
 \sum_n g(n)\lambda_\eta(m_1n+1)\lambda_\eta(m_2n+1),
\]
with a smooth majorant $g$ on a fixed enlargement of the $n$-interval.
The diagonal contributes $\ll_\nu X^\nu TX$.  To see this without
losing a factor $M$, sum first over $k=mn+1$.  Each $k$ has at most
$\tau(k-1)$ representations in the two intervals, and the bound
$|\lambda_\eta(k)|\leq\tau(k)$ gives the assertion.

For the off-diagonal terms, fix $d,D$ as above.  The relation
\[
 m_3\equiv m_4\pmod{D/d},\qquad m_3\ne m_4,
\]
implies that the number of pairs is $\ll M^2/(dD)$; if $D/d$ exceeds
a fixed multiple of $M/d$, there are no such pairs.  Substituting this
count into the first term of \eqref{eq:shifted-correlation} gives
\[
 \ll_\nu X^\nu T X^{1/2}M^{5/2},
\]
since $\sum_{d\leq2M}d^{-3/2}$ is bounded and the number of admissible
$D$ for a fixed squarefree $d$ is absorbed by $X^\nu$.  Explicitly, for
any $v>0$, Rankin's bound gives
\[
 \#\{D\leq4M:\operatorname{rad}(D)\mid d\}
 \leq (4M)^v\prod_{p\mid d}(1-p^{-v})^{-1}
 \ll_v M^v d^v;
\]
one chooses $v$ smaller than the prescribed $\nu$.
For the second term, the same count gives
\[
 \ll_\nu X^\nu T X^{3/4}M^{7/4},
\]
using $D\geq d$ and the convergent sum of $d^{-7/4}$.
These are the arithmetic sums in \cite[Proposition~8.1]{ABL}.
We have therefore obtained
\begin{equation}\label{eq:bilinear-square}
 |B|^2\ll_{\eta,\nu}X^\nu
 \left(TX+TX^{1/2}M^{5/2}+TX^{3/4}M^{7/4}\right).
\end{equation}
When $M\leq X^{1/3}$ the middle term is bounded by the last.
Taking square roots and using $T\asymp X/M$ yields
\[
 |B|\ll_{\eta,\nu}X^\nu
 \left(XM^{-1/2}+X^{7/8}M^{3/8}\right).
\]
For $M>X^{1/3}$ the same conclusion follows from the trivial bound
$|B|\ll_\nu X^{1+\nu}$.

A Mellin inversion of $\Phi(mn/X)$ separates the two variables.
On the integration line $\operatorname{Re}s=0$, the resulting factors
$m^{-s}$ and $n^{-s}$ have modulus one; the Mellin transform has rapid
decay.  Dividing $[M_1,M_2]$ into dyadic intervals and summing the two
geometric series proves \eqref{eq:bilinear} under $(m,N)=1$.

Finally, for a squarefree $m$ put
\[
 d_0=(m,\operatorname{rad}(N)),\qquad m=d_0m_0,\qquad n_0=d_0n.
\]
Then $(m_0,N)=1$ and $mn+1=m_0n_0+1$.  For each of the finitely many
$d_0\mid\operatorname{rad}(N)$ use the separate coefficients
\[
 \alpha'_{m_0}=\alpha_{d_0m_0},\qquad
 \beta'_{n_0}=\boldsymbol{1}_{d_0\mid n_0}\beta_{n_0/d_0},
\]
retaining the original interval restrictions in $\alpha'$.
The coprime case applies, and its factors depending on $d_0$ are
absorbed into the constant.  If a lower endpoint becomes less than
one, the corresponding bounded interval is covered by the trivial
bound.  Summing over $d_0$ proves the proposition.

\begin{remark}\label{rem:bilinear-residue-conditions}
An additional bounded function of $m\pmod r$ and $n\pmod r$ can be
inserted in \eqref{eq:bilinear} at a cost of at most $r^2$, by dividing
both coefficient sequences into residue classes.  In particular, a
fixed Dirichlet character $\chi(mn+1)$ costs only a fixed constant.
A restriction on $n$ alone is absorbed directly into $\beta_n$.
\end{remark}