We propose a finite modular symmetric inflationary model in which a Coleman-Weinberg potential is generated by heavy vector-like quarks coupled to a complex modulus $\tau$ through modular forms. The imaginary part of the modulus is identified with the inflaton, while the real part plays the role of an axion-like field. We show that the model can yield inflationary predictions consistent with current cosmological constraints, including those from Planck, and discuss the cosmological role of the axion-like degree of freedom in this framework.