The next generation air traffic management system (NextGen) is required to integrate flight data from multiple sources as a fully coupled information fusion and uncertainty framework. Convective weather can develop rapidly and pose safety concerns to the controller and aircrew. Thus an integrated, efficient and accurate trajectory prediction tool for aircraft under convective weather conditions is needed. This paper proposed a recurrent neural network (RNN) model for weather-related aircraft trajectory prediction task. By modifying the recurrence of LSTM to embed convolutional layers in the loop thus to extract useful information from weather features, we are able to calibrate the last on-file flight plan prior to takeoff. The flight plan, history flight tracks, and convective weather features are obtained from the Sherlock Data Warehouse (SDW). The general idea is to calibrate the flight plan with the actual flown historical flight tracks with the weather features along with it. The experiment is conducted using three months history data over the period from Nov 1, 2018 through Feb 5, 2019 with the flights from John F. Kennedy International Airport (JFK) to Los Angeles International Airport (LAX). Training with a dataset of 2528 for 500 epochs with Adam optimizer, the out-of-sample test shows that 47.0% of the calibrated flight trajectory is able to reduce the deviation with a 3D prediction and 90.0% of the flight trajectory deviations are reduced with a 4D prediction. The overall variance is reduced by 12.3% for 3D prediction and 37.3% for 4D prediction.