We conducted a study with introductory and upper-division physics students in a Mexican university to learn how students independently recognize the electric field's main characteristics in the electric field lines diagram and as a source or target representation in conversion processes. We used the theory of registers of semiotic representations and a phenomenographic approach as a framework to analyze data. The recognition and conversion abilities were explored through interpretation and construction tasks. We identified students' main difficulties in recognition and conversion in the interpretation and construction tasks. In conversion processes, we found that when the electric field lines diagram is the source representation, students do not interpret the field lines' density as the magnitude of the electric field. The difficulties of interpretation that arise in these conversion processes depend partially on the target representation. We also found that constructing electric field lines is especially difficult for students at both introductory and upper-division levels. Most students would prefer to draw vector field plots instead. We recommend that electricity and magnetism teachers and researchers be aware of the difficulties that the recognition and conversion in interpretation and construction tasks may represent for their students in understanding the electric field concept.