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There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:

Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be a newfrom of half-weight $k + 1/2$ for $\Gamma_0(4N)$. Does any one know if we have

$$ \sum_{n\le X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1/2+\varepsilon} ?? \tag1 $$ As is well-known, if $f$ is a $GL_2$-newform, this is called the Wilton-type bound. It is not clear for me that this estimate holds for half-integral weight cusp forms.

On the other hand, if the estimate (1) is not true, whether or not one instead has $$ \sum_{n\le X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1-\delta} $$ for some constant $\delta>0$?

So, if any expert here know some relevant results on this question, please show some guides or corresponding references, many thanks.

Thanks in advance!

There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:

Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be a newfrom of half-weight $k + 1/2$ for $\Gamma_0(4N)$. Does any one know if we have

$$ \sum_{n\le X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1/2+\varepsilon} \;?? \label{1}\tag{1} $$ As is well-known, if $f$ is a $GL_2$-newform, this is called the Wilton-type bound. It is not clear for me that this estimate holds for half-integral weight cusp forms.

On the other hand, if the estimate \eqref{1} is not true, whether or not one instead has $$ \sum_{n\le X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1-\delta} $$ for some constant $\delta>0$?

So, if any expert here know some relevant results on this question, please show some guides or corresponding references, many thanks.

Thanks in advance!

There is a basis question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:

Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be a newfrom of half-weight $k + 1/2$ for $\Gamma_0(4N)$. Does any one know if we have

$$ \sum_{n\ge X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1/2+\varepsilon} ?? \tag1 $$ As is well-known, if $f$ is a $GL_2$-newform, this is called the Wilton-type bound. It is not clear for me that this estimate holds for half-integral weight cusp forms.

On the other hand, if the estimate (1) is not true, whether or not one instead has $$ \sum_{n\ge X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1-\delta} $$ for some constant $\delta>0$?

So, if any expert here know some relevant results on this question, please show some guides or corresponding references, many thanks.

Thanks in advance!

There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:

Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be a newfrom of half-weight $k + 1/2$ for $\Gamma_0(4N)$. Does any one know if we have

$$ \sum_{n\le X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1/2+\varepsilon} ?? \tag1 $$ As is well-known, if $f$ is a $GL_2$-newform, this is called the Wilton-type bound. It is not clear for me that this estimate holds for half-integral weight cusp forms.

On the other hand, if the estimate (1) is not true, whether or not one instead has $$ \sum_{n\le X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1-\delta} $$ for some constant $\delta>0$?

So, if any expert here know some relevant results on this question, please show some guides or corresponding references, many thanks.

Thanks in advance!

There is a basis question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:

Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be a newfrom of half-weight $k + 1/2$ for $\Gamma_0(4N)$. Does any one know if we have

$$ \sum_{n\ge X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1/2+\varepsilon} ?? \tag1 $$ As is well-known, if $f$ is a $GL_2$-newform, this is called the Wilton-type bound. It is not clear for me that this estimate holds for half-integral weight cusp forms.

On the other hand, if the estimate (1) is not true, whether or not one instead has $$ \sum_{n\ge X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1-\delta} $$ for some constant $\delta>0$?

So, if any expert here know some relevant results on this question. Please show some guides or corresponding references, many thanks.

Thanks in advance!

There is a basis question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:

Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be a newfrom of half-weight $k + 1/2$ for $\Gamma_0(4N)$. Does any one know if we have

$$ \sum_{n\ge X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1/2+\varepsilon} ?? \tag1 $$ As is well-known, if $f$ is a $GL_2$-newform, this is called the Wilton-type bound. It is not clear for me that this estimate holds for half-integral weight cusp forms.

On the other hand, if the estimate (1) is not true, whether or not one instead has $$ \sum_{n\ge X} a_f(n)e(n\alpha)\ll_{f,\varepsilon}X^{1-\delta} $$ for some constant $\delta>0$?

So, if any expert here know some relevant results on this question, please show some guides or corresponding references, many thanks.

Thanks in advance!