If you do not carry your residence card with you, you can be fined an amount not exceeding 200,000 yen, so be sure to report your loss to the nearest police station as soon as possible. After you report to the police, go to the Immigration Bureau and apply for a reissuance.

【Necessary documents】

• Application form for Reissuance of a Residence Card
• Documents that can prove your loss（ex. certification of report of lost property issued by police）
• Your passport

* Also take your student ID with you, in case you are asked to present it.

For further details, see here.