The main aim of this paper is to report two new detections of tidal debris in the northern stream of the Sagittarius dwarf galaxy located at 45° and 55° from the center of the galaxy. Our observational approach is based on deep color-magnitude diagrams that provide accurate distances, surface brightness, and the properties of stellar population of the studied region of this tidal stream. The derived distances for these tidal debris wraps are 45+/-5 and 62+/-6 kpc, respectively. These detections are also strong observational evidence that the tidal stream discovered by the Sloan Digitized Sky Survey is tidally stripped material from the Sagittarius dwarf and support the idea that the tidal stream completely enwraps the Milky Way in an almost polar orbit. We also confirm these detections by running numerical simulations of the Sagittarius dwarf plus the Milky Way. This model reproduces the present position and velocity of the Sagittarius main body and presents a long tidal stream formed by tidal interaction with the Milky Way potential. The tidal streams of the model traces the last orbit of Sagittarius and confirms our observational detections. This model is also in good agreement with the available observations of the Sagittarius tidal stream. The comparison of our model with the positions and distances of two nonidentified halo overdensities discovered by the Sloan Digitized Sky Survey and the QUEST survey shows that they are actually associated with the trailing arm of the Sagittarius tidal stream. In addition, we identify the proper-motion group discovered by Arnold & Gilmore as a piece of the Sagittarius northern stream. We also present a method for estimating the shape of the Milky Way halo potential using numerical simulations. From our simulations we obtain an oblateness of the Milky Way dark halo potential of 0.85, using the current database of distances and radial velocities of the Sagittarius tidal stream. The color-magnitude diagram of the apocenter of Sagittarius shows that this region of the stream shares the complex star formation history observed in the main body of the galaxy. We present the first evidence for a gradient in the stellar population along the stream, possibly correlated with its different pericenter passages.