Automatic Speech Recognition (ASR) in conversational settings presents unique challenges, including extracting relevant contextual information from previous conversational turns. Due to irrelevant content, error propagation, and redundancy, existing methods struggle to extract longer and more effective contexts. To address this issue, we introduce a novel conversational ASR system, extending the Conformer encoder-decoder model with cross-modal conversational representation. Our approach leverages a cross-modal extractor that combines pre-trained speech and text models through a specialized encoder and a modal-level mask input. This enables the extraction of richer historical speech context without explicit error propagation. We also incorporate conditional latent variational modules to learn conversational level attributes such as role preference and topic coherence. By introducing both cross-modal and conversational representations into the decoder, our model retains context over longer sentences without information loss, achieving relative accuracy improvements of 8.8% and 23% on Mandarin conversation datasets HKUST and MagicData-RAMC, respectively, compared to the standard Conformer model.