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

【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.