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We present Surveying the Whirlpool at Arcseconds with NOEMA (SWAN), a high-resolution, high-sensitivity survey to map molecular lines in the 3 mm band in M51 (the Whirlpool galaxy). SWAN has obtained the largest high-sensitivity map (∼5 × 7 kpc²) of N₂H⁺ emission at ∼cloud-scale resolution (3″ ∼ 125 pc) in an external galaxy to date. Here, we describe the observations and data reduction of ∼214 hours of interferometric data from the Northern Extended Millimetre Array (NOEMA) and ∼55 hours of tailored new observations with the 30m telescope of the Institut de radioastronomie millimétrique (IRAM), as well as the combination of these NOEMA and new IRAM-30m observations with ∼14 hours of archival IRAM-30m observations. We detect widespread emission from nine molecular transition lines. The J = 1 - 0 transitions of the CO isotopologs ¹³CO and C¹⁸O are detected at high significance across the full observed field of view (FoV). HCN(1-0), HNC(1-0), HCO⁺(1-0), and N₂H⁺(1-0) are detected in the center, molecular ring, and spiral arms of the galaxy, while the shock tracer HNCO(4-3) and (5-4) and PDR tracer C₂H(1-0) are detected in the central ∼1 kpc and molecular ring only. For most of the lines that we detect, average line ratios with respect to ¹²CO are increased by up to a factor of ∼3 in the central 1 kpc, where an active galactic nucleus and its low-inclination outflow are present, compared to the disk. Line ratios between CO isotopologs show less variation across the SWAN FoV. Across the full SWAN FoV, ¹³CO, C¹⁸O, HCN, HNC, HCO⁺ and N₂H⁺ are $ 8\pm^2_2 $, $ 29\pm^{7}_{6} $, $ 17\pm^3_5 $, $ 37\pm^5_{10} $, $ 26\pm^5_3 $ and $ 63\pm^{38}_{10} $ times fainter than ¹²CO, respectively, in pixels where each line is significantly detected. Although we observe variations in line ratios between larger-scale environments like the center and disk of M51, the scatter within each environment also indicates the influence of smaller-scale processes. The ability to measure these effects is only possible thanks to the high resolution and high sensitivity of the SWAN dataset across multiple environments. This provides the sharpest view of these molecular transitions over the largest physical area ever captured in an external galaxy.