On the Spectral Large Sieve Inequality for ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathbb{C})$

In this paper, we introduce concise expressions for the complex Bessel integral that enables us to improve the spectral large sieve inequality of Watt for ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathbb{C})$. Our result is optimistic in the case when the average of spectral parameters is on a square centered at $(0,0)$.