Time between verbal and official offer?

Two weeks ago, I received a job offer over email and I accepted. I was told the next steps would be for them to contact HR, who would send me the official letter/offer. How long should I wait before reaching out and asking about the situation?

For reference: the job is expected to begin in July (so a month out). The employer is a big hospital system, over 7,000 employees. I figured I would wait another week or maybe until mid June until emailing...?

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