Recoil polarization measurements for neutral pion electroproduction at<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>Q</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi fontstyle="normal">GeV</mml:mi><mml:mo>/</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mi/><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>near the Δ resonance

We measured angular distributions of differential cross section, beam analyzing power, and recoil polarization for neutral pion electroproduction at ${Q}^{2}=1.0 (\mathrm{GeV}/c){}^{2}$ in 10 bins of $1.17\ensuremath{\leqslant}W\ensuremath{\leqslant}1.35$ GeV across the \ensuremath{\Delta} resonance. A total of 16 independent response functions were extracted, of which 12 were observed for the first time. Comparisons with recent model calculations show that response functions governed by real parts of interference products are determined relatively well near the physical mass, $W={M}_{\ensuremath{\Delta}}\ensuremath{\approx}1.232$ GeV, but the variation among models is large for response functions governed by imaginary parts, and for both types of response functions, the variation increases rapidly with $W&gt;{M}_{\ensuremath{\Delta}}$. We performed a multipole analysis that adjusts suitable subsets of ${\ensuremath{\ell}}_{\ensuremath{\pi}}\ensuremath{\leqslant}2$ amplitudes with higher partial waves constrained by baseline models. This analysis provides both real and imaginary parts. The fitted multipole amplitudes are nearly model independent---there is very little sensitivity to the choice of baseline model or truncation scheme. By contrast, truncation errors in the traditional Legendre analysis of $N\ensuremath{\rightarrow}\ensuremath{\Delta}$ quadrupole ratios are not negligible. Parabolic fits to the $W$ dependence around ${M}_{\ensuremath{\Delta}}$ for the multiple analysis gives values for $\mathrm{Re}({S}_{1+}/{M}_{1+})=(\ensuremath{-}6.61\ifmmode\pm\else\textpm\fi{}0.18)%$ and $\mathrm{Re}({E}_{1+}/{M}_{1+})=(\ensuremath{-}2.87\ifmmode\pm\else\textpm\fi{}0.19)%$ for the $p{\ensuremath{\pi}}^{0}$ channel at $W=1.232$ GeV and ${Q}^{2}=1.0 (\mathrm{GeV}/c){}^{2}$ that are distinctly larger than those from the Legendre analysis of the same data. Similarly, the multipole analysis gives $\mathrm{Re}({S}_{0+}/{M}_{1+})=(+7.1\ifmmode\pm\else\textpm\fi{}0.8)%$ at $W=1.232$ GeV, consistent with recent models, while the traditional Legendre analysis gives the opposite sign because its truncation errors are quite severe.