<p>Let <i>K</i>₃ be a nonnormal cubic extension over ℚ, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({a}_{{K}_{3}}\left(n\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <msub> <mi>K</mi> <mn>3</mn> </msub> </msub> <mfenced close=")" open="("> <mi>n</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> be the coefficients of the Dedekind zeta function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\zeta }_{{K}_{3}}\left(s\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <msub> <mi>K</mi> <mn>3</mn> </msub> </msub> <mfenced close=")" open="("> <mi>s</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we investigate the asymptotic formulas of the power sums</p><p><InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\sum_{n\le x}{a}_{{K}_{3}}^{l}\left(n\right) \text{and} \sum_{{n}_{1}^{2}+{n}_{2}^{2}\le x}{a}_{{K}_{3}}^{l}\left({n}_{1}^{2}+{n}_{2}^{2}\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msubsup> <mi>a</mi> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> </mrow> <mi>l</mi> </msubsup> <mfenced close=")" open="("> <mi>n</mi> </mfenced> <mtext>and</mtext> <msub> <mo>∑</mo> <mrow> <msubsup> <mi>n</mi> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>n</mi> <mrow> <mn>2</mn> </mrow> <mn>2</mn> </msubsup> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msubsup> <mi>a</mi> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> </mrow> <mi>l</mi> </msubsup> <mfenced close=")" open="("> <msubsup> <mi>n</mi> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>n</mi> <mrow> <mn>2</mn> </mrow> <mn>2</mn> </msubsup> </mfenced> </mrow> </math></EquationSource> </InlineEquation></p><p>for <i>l</i> = 2<i>,</i> 3, and improve previous error estimates.</p>