\section{Cubic Hecke characters and the prime coefficient estimate}
\label{sec:hecke}

Let $\eta$ be a nontrivial cubic Hecke character of $F=\Q(\zeta_3)$
satisfying $\eta^\sigma=\eta^{-1}$, where $\sigma$ is complex
conjugation. Extend $\eta$ by zero on ideals not coprime to its conductor,
and define
\begin{equation}\label{eq:hecke-coeff}
 L_F(s,\eta)=\sum_{n\geq1}\frac{\lambda_\eta(n)}{n^s},\qquad
 \lambda_\eta(n)=\sum_{\Nm\mathfrak a=n}\eta(\mathfrak a).
\end{equation}
The classical theta-series construction~\cite[\S4.8]{Miyake} associates to this series a
holomorphic cuspidal newform of weight one. Its nebentypus is
$\psi=\chi_{-3}$. Indeed, the restriction of $\eta$ to rational ideles
is both invariant under conjugation and equal to its inverse, and has
order dividing three; it is therefore trivial. Cuspidality follows from
$\eta\ne\eta^\sigma$. The induced representation is self-dual, so its
coefficients are real, including at ramified primes.

The ideal formula, or the degree-two Euler factors, gives
\begin{equation}\label{eq:hecke-bounds}
 |\lambda_\eta(n)|\leq\tau(n),\qquad
 \lambda_\eta(mn)=\lambda_\eta(m)\lambda_\eta(n)\quad((m,n)=1).
\end{equation}
At an unramified prime set
\[
 t_\eta(p)=
 \begin{cases}
 \eta(\mathfrak p)+\eta(\overline{\mathfrak p}),&p=\mathfrak p\overline{\mathfrak p}
       \text{ in }F,\\
 0,&p\text{ inert in }F.
 \end{cases}
\]
At the finitely many ramified primes put $t_\eta(p)=\lambda_\eta(p)$.
Thus $\lambda_\eta(p)=t_\eta(p)$ and $|t_\eta(p)|\leq2$ at every prime.
At an inert prime the Euler factor is $(1-p^{-2s})^{-1}$.
At a split prime with trace $2$, the coefficients at $p^j$ are $j+1$;
with trace $-1$ they are $1,-1,0,1,-1,0,\ldots$. These formulas
will allow us to keep the prime-power coefficients until their
contribution has been estimated.

\begin{lemma}\label{lem:prime-tests}
Fix $A_0,B,A,\delta,C>0$. Uniformly for Dirichlet characters $\theta$
of modulus $k\leq L^{A_0}$, intervals $J\subset[H,2H]$ with
$x^\delta\leq H\leq x^C$, and $|t|\leq x^B$, one has
\begin{equation}\label{eq:prime-tests}
 \frac1H\left|\sum_{p\in J}t_\eta(p)\theta(p)p^{it}\right|
 \ll_{\eta,A_0,B,A,\delta,C}L^{-A}.
\end{equation}
The same holds after inserting any fixed Dirichlet character.
\end{lemma}

\begin{proof}
Put $\xi=\eta(\theta\circ\Nm_{F/\Q})$. The norm pullback is
invariant under $\sigma$. If $\xi$ were principal, $\eta$ would be
invariant as well, contrary to $\eta^\sigma=\eta^{-1}\ne\eta$.
The primitive character inducing $\xi$ is thus nonprincipal and has
conductor norm $Q_\xi\ll_\eta k^2$.

The zero-free and strip-growth theorems in~\cite{QRH} imply
\begin{equation}\label{eq:log-derivative}
 \left|\frac{L_F'}{L_F}(15/16+iT,\xi)\right|
 \ll_\eta\log\bigl(Q_\xi(3+|T|)\bigr).
\end{equation}
Here is the uniformity argument. The disk centered at $2+iT$ with
radius $35/32$ lies in $\Re s>7/8$. Choose an analytic logarithm of
$L_F(s,\xi)$ on the disk, normalized by the Euler series at its center.
The strip-growth bound controls the real part of this logarithm by
$O(\log(Q_\xi(3+|T|)))$, while the value at the center is bounded.
The Borel--Carath\'eodory theorem, followed by Cauchy's estimate on two
slightly smaller disks, gives~\eqref{eq:log-derivative}, since the point
$15/16+iT$ is at distance $34/32$ from the center. The omitted Euler
factors of an imprimitive character contribute $O(\log(2k))$ there.

For a smooth function $g$ supported in a fixed positive dyadic interval,
Mellin inversion of $-L_F'/L_F$, with the oscillation carried in its
height variable, gives
\begin{align}\label{eq:smoothed-prime-ideal}
 &\sum_{\mathfrak p}\sum_{j\geq1}
 \xi(\mathfrak p)^j\log\Nm\mathfrak p\,
 g\!\left(\frac{(\Nm\mathfrak p)^j}{H}\right)
 (\Nm\mathfrak p)^{ijt}
 \notag\\[-2mm]
 &\hspace{30mm}\ll_g H^{15/16}\log\bigl(Q_\xi(3+|t|)\bigr).
\end{align}
We shifted the line of integration to $15/16$; nonprincipality and
zero-freeness ensure that no pole is crossed. The terms with residue
degree or exponent at least two contribute
$O(H^{1/2}(\log H)^2)$. The remaining terms are precisely the rational
prime sum with coefficient $t_\eta(p)\theta(p)\log p$, apart from
finitely many ramified primes.

To pass to an arbitrary interval $J$, smooth its endpoints in intervals
of length $HL^{-D}$. The derivative and Mellin norms increase by fixed
powers of $L$, and the boundary error is at most
$O(HL^{-D}\log H+\log H)$ by counting integers. Choose $D$ after $A$.
Since $H\geq x^\delta$, the factor $H^{-1/16}$ absorbs every such
logarithmic loss. Partial summation removes $\log p$ and proves the
claim. A fixed extra character only changes the conductor by a fixed
factor and hence is covered by increasing $A_0$.
\end{proof}

Combining Lemma~\ref{lem:prime-tests} with
Proposition~\ref{prop:transfer} applies the marked Type~II estimate
to the prime-supported coefficient
\[
 \alpha_m=\ind_{m\ {\rm prime}}\lambda_\eta(m)\chi(m)m^{iv},
 \qquad |v|\leq L^C,
\]
with zero comparison term. In particular, for every prescribed
logarithmic saving and sufficiently many fixed marking intervals,
\begin{equation}\label{eq:trace-type-ii}
 \sum_{m,n}\alpha_m\beta_nG(mn-1)W(mn-1)\ll xL^{-A}
\end{equation}
in the ranges of that proposition. The second coefficient may be any
allowed rough sequence of logarithmic size. The function $G$ is
invariant under multiplication by marking primes; the trace is placed
in $\alpha$, where Lemma~\ref{lem:prime-tests} supplies its cancellation.