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Aug 24, 2011 at 14:42 comment added Rob Harron They certainly are not! That's a big problem with the periods of modular forms as opposed to those of elliptic curves. There has however been progress in writing down periods that canonically vary in $p$-adic families, so that one can take about congruences and $\mu$-invariants of $p$-adic $L$-functions. This is what the papers of Vatsal and Emerton–Pollack–Weston discuss.
Aug 24, 2011 at 6:39 comment added David Hansen Re. update: Those periods don't seem very canonical, depending as they do on an arbitrary choice of twisting character.
Aug 24, 2011 at 5:23 history edited Rob Harron CC BY-SA 3.0
add: weight 2 case still conditional in 1976 paper
Aug 24, 2011 at 5:04 history edited Rob Harron CC BY-SA 3.0
add ref to Shimura for exact statement you're looking for
Jun 18, 2011 at 17:34 vote accept David Loeffler
Jun 18, 2011 at 17:14 history answered Rob Harron CC BY-SA 3.0