You can now read this government document here (only because someone was nice enough to send it along to me. Time to grant full public access to these documents!).

Here is the abstract:

The political activities of Section 501(c)(3) organizations are often in the news, with allegations made that some groups engaged in impermissible activities. These groups are absolutely prohibited from participating in campaign activity under the Internal Revenue Code (IRC). On the other hand, they are permitted to engage in nonpartisan political activities (e.g., distributing voter guides and conducting get-out-the-vote drives) that do not support or oppose a candidate.
Determining whether an activity violates the IRC prohibition depends on the facts and circumstances of each case, and the line between impermissible and permissible activities can sometimes be difficult to discern.
Due to the IRC prohibition, Section 501(c)(3) organizations generally are not permitted to engage in the types of activities regulated by the Federal Election Campaign Act (FECA). However, the activities regulated under the IRC and FECA are not necessarily identical. An organization must comply with any applicable FECA provisions if engaging in activities regulated by FECA (e.g., making an issue advocacy communication under the IRC that constitutes an electioneering
communication under FECA).
A 2010 Supreme Court case, Citizens United v. FEC, has received considerable attention for invalidating several long-standing prohibitions in FECA on corporate and labor union campaign treasury spending. This case does not appear to significantly impact the political activities of Section 501(c)(3) organizations because they remain subject to the prohibition on such activity under the IRC.
This report examines the restrictions imposed on campaign activity by Section 501(c)(3) organizations under the tax and campaign finance laws. For a discussion limited to the ability of churches and other houses of worship to engage in campaign activity, see CRS Report RL34447, Churches and Campaign Activity: Analysis Under Tax and Campaign Finance Laws, by Erika K. Lunder and L. Paige Whitaker.