The fuel level sensors can wear out causing erratic or inaccurate fuel gauge readings.
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Repairpal verified 2004 Dodge Neon problems
If there is a water leak into the passenger side of the interior, under the dash panel, it may be the HVAC housing drain is plugged or restricted.
A click heard when turning the steering wheel may be solved by aligning the steering column or installing revised parts in the column (requires removal of the steering column).
A clicking or popping sound from the front of the vehicle (left or right side) while accelerating from a stop may be fixed by installing a rubber gasket on the axle shaft.
Problems with the remote starter accessory may be fixed by replacing the hood switch for the remote starter. The vehicle will typically chirp the horn and not start, or it will chirp the horn, start the car, and then turn off.
The engine may stumble occasionally during wide-open-throttle acceleration (full throttle). This may not occur frequently enough to set a Check Engine (MIL) light. The spark plugs (spark plug insulators) and spark plug wires should be inspected for deterioration and carbon tracking; resistance in the spark plug wires should be checked as well. New spark plug wires from Dodge are a revised design and any defective spark plugs need to be replaced.