We present a vector magnetic field map obtained on 7 December 2003, below and around a filament located not so far from the active region NOAA 517, whose one spot is also found on the map of 240× 340 arcsec. This region was itself located near the disk center, so that the longitudinal (resp. transverse) field is nearly the vertical (resp. horizontal) one. The THEMIS telescope was used in its spectropolarimetric multiline mode MTR ("MulTiRaies"). The noise level is 5-10 Gauss in the longitudinal field and 50-100 Gauss in the transverse field, while the pixel size is 0.45 arcsec. Fundamental ambiguity is not solved, and the atmosphere is assumed to be homogeneous. The magnetic field derivation method described in this paper was validated on eight test points submitted to the UNNOFIT inversion code, and the results are found in agreement within 14% discrepancy. Two main results appear on the map: (i) a strong spatial correlation between the longitudinal and transverse field resulting in an inclined field vector (making a most probable angle of 60° or 120° with the line-of-sight in the filament region); and (ii) homogeneity of the field direction (inclination and azimuth) in the filament region. Parasitic polarities were also detected: first those located at the filament feet, as theoretically expected, on the one hand; and then weak opposite polarity regular patterns that appear between the network field (strong field at the frontiers of supergranules), on the other. The exact superimposition of the magnetic field map derived from the Fe I 6302.5 Å line and of the Hα map, which enabled association of the parasitic polarities with the filament feet, was possible because these two maps were simultaneously obtained, thanks to a unique facility available in the multiline mode of THEMIS.