The fundamental of the overtone series does not exist as a real note on the trumpet.  The first member of the overtone series that exists on the trumpet is the first overtone, or second harmonic.  I refer below to the open/valveless overtone series as a reference point, but these tuning tendencies apply to valved series as well.  In relation to equal temperament, the notes in the overtone series are out of tune as follows:

First overtone (low C)	in tune
Second overtone (second line G)	2 cents sharp
Third overtone (middle C)	in tune
Fourth overtone (top space E)	14 cents flat
Fifth overtone (top of the staff G)	2 cents sharp
Sixth overtone (Bb above the staff, played open)	31 cents flat
Seventh overtone (high C)	in tune

These numbers vary slightly from trumpet to trumpet, but the tendencies are the same, except for a rare exception.

Because it is so out of tune, the sixth overtone is never played open; it is played first valve.  Therefore, this is technically an alternate fingering, but has become standard.  Refer to fingering charts for how to play the sixth overtone in each overtone series.

To make it easy, here are some other notes that tend to be out of tune on the trumpet, mainly because of the presence of valves, which makes it impossible for the trumpet to be completely in tune.

Low C#, D	very sharp
Low E	sharp
Second space A	sharp
A above the staff	sharp

Here is a summary of the typically out-of-tune notes on the trumpet:

Low C#, D	very sharp
Low E	sharp
Second line G	sharp
Second space A	sharp
Fourth space C# through top space E	flat
Top line F through top of the staff G	sharp
A above the staff	sharp

Use slides to fix sharp notes whenever possible.  The third valve slide should be used for the low C# and D.  The first valve slide should be used for all other sharp notes that use the first valve (low E, second space A, top line F, and A above the staff); the first valve slide will probably only need to be moved slightly for these notes.  This leaves the flat notes and some sharp notes (second line G, top line F#, top of the staff G) that one must now “lip” in tune.  This involves directing the air stream upwards for flat notes and downwards for sharp notes.  The challenge then also becomes to play these notes with as beautiful a sound as though you are playing through the center of the horn.

Some other thoughts:

• Notes from low C below are so “moveable” that they do not have traditional pitch tendencies and typically slides are not needed. These notes tend to be flat because one is relaxing the embouchure so much to reach them. 
• Notes above the staff tend to be sharp for younger players as they tend to “squeeze” for these notes, and this pinching makes these notes sharp.
• Fatigue also affects intonation on the trumpet and will affect players differently. Some go flat when tired, some go sharp.
• Temperature affects pitch. Cold trumpets play flat.  Hot trumpets play sharp.  Temperature also affects intonation exponentially on pitches which are already out-of-tune.  For example, top space E is always flat, but as the temperature decreases, this E will get flatter by larger amounts.
• Each trumpet has slightly different intonation issues. Each student needs to learn the tendencies of their own trumpet by first putting third space C or third line B in tune, using the tuning slide, while playing through the center of the horn.  The student can then play through the center of the horn on other notes with a tuner to see where they lie and adjust accordingly. 
• When shopping for a new trumpet, one should consider how in-tune the trumpet is generally, and also how out-of-tune the typically most out-of-tune notes are (the flat notes and G on top of the staff).