Specification-Driven Development (SDD) has emerged as a promising paradigm in software development. This trend is fueled by recent advances in leveraging large language models (LLMs) to generate code from user intents expressed in natural language. However, the reliance on natural language specifications introduces ambiguity and challenges in ensuring correctness. To address these problems, we present Expecto, a system that automatically extracts trustworthy formal specifications from natural language intents. Expecto employs a neuro-symbolic approach, combining the strengths of LLMs and program synthesis techniques. Our key idea is to adopt a top-down, modular specification synthesis algorithm that breaks down the complex task of specification extraction into manageable units. The specifications are written in a domain-specific language (DSL) designed to succinctly express formal specifications of functional requirements. This modular synthesis with a concise DSL reduces the reasoning complexity for LLMs and enhances the accuracy of the extracted specifications. Our results demonstrate that Expecto significantly improves the accuracy and reliability of extracted specifications compared to a monolithic, purely LLM-based approach. Furthermore, when applied to real-world buggy programs in Defects4J, Expecto successfully generates formal specifications that detect more bugs than the baselines.