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2014-09-20 01:52:37 +0200	asked a question	Trouble verifying modularity properties Hey all, I'm having a bit of trouble doing some numerical things with modular forms, and I simply can't figure out where I'm going wrong. The $j$ function should satisfy $j(\gamma \tau) = j(\tau)$ for every $\tau$ in the upper half plane and every $\gamma\in SL_2(\mathbb{Z})$. I wrote some code to numerically compute some values of the $j$ function (there may be a better way to do it for the $j$ function, but I hope to eventually migrate this code to work for other modular forms which I define). Here it is below. ## Starts Laurent series in q R.<q> = LaurentSeriesRing(QQ) I = CC.0 # imaginary unit precision = 75 ##evaluates a function using its q-expansion def evaluate(f,z): result = 0 coeffs = f.coefficients() exps = f.exponents() for i in range(0,len(coeffs)): result = result + coeffs[i]z^(exps[i]) return result ## computes the action of a member of the modular group on tau in the upper half plane def action(gamma,tau): return ((gamma[0]tau + gamma[1])/(gamma[2]tau + gamma[3])) ## Produce Eisenstein series with specified weight and precision of q-expansion def eis(weight,precision): t = EisensteinForms(1,weight) t.set_precision(precision) t = t.eisenstein_series() e = t[0].q_expansion() return e(1/e.list()[0]) ## gives you q which corresponds to tau def qt(tau): return exp(2piItau) ## Defining delta cusp form delta = CuspForms(1,12).0 delta = delta.q_expansion(precision) # Computes j function g2 = 60eis(4,precision)/240 j = 1728g2^3/(27delta) Now when I run the following code: `tau = 1+I gamma = [3,-1,4,-1] print(evaluate(j,qt(tau)).n()) #j(tau) print(evaluate(j,qt(action(gamma,tau))).n()) #j(gamma tau)` the values $j(\tau)$ and $j(\gamma\tau)$ are not equal! I would appreciate any help.
2014-07-11 01:08:05 +0200	commented answer	Trouble running Sage in terminal Yes, I tried to do this but when I click there nothing happens. I think I might not have enough reputation to upvote.
2014-07-10 23:33:59 +0200	commented answer	Trouble running Sage in terminal It turned out to be the first file you mentioned. I uninstalled Enthought and everything works great now. Thanks so much. I would like to upvote your answer but am not sure how to.
2014-07-10 23:24:45 +0200	commented answer	Trouble running Sage in terminal Thank you very much for your help. With regards to the second file, I'm looking at the directory in Finder (I'm on a Mac) and I cannot see a file named vector_real_double_dense.pyx. Very strange! With regards to the first file, I think I had downloaded something like Enthought for a class once. I'll try to go and get that off my machine.
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2014-07-10 22:53:38 +0200	asked a question	Trouble running Sage in terminal Hi all, So I start up Sage using the command: '/Applications/Sage-6.2.app/Contents/Resources/sage'/sage I'm trying to load this file: `#mod.sage precision = 30 m = ModularForms(Gamma0(12),10,prec = precision) p = m.basis()` Unfortunately I get a very large error when I run the following command: attach("mod.sage") However when I comment out the line p = m.basis(), the error goes away. I'm attaching the text from the error message below. Any ideas on how to fix this would be greatly appreciated. Sorry for the sloppy formatting! --------------------------------------------------------------------------- IOError Traceback (most recent call last) <ipython-input-2-909b5a17cf48> in <module>() ----> 1 sage.misc.preparser.load(sage.misc.preparser.base64.b64decode("bW9kLnNhZ2U="),globals(),True) /Applications/Sage-6.2.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/misc/preparser.pyc in load(filename, globals, attach) 1753 else: 1754 raise IOError('did not find file %r in load / attach search path' \ -> 1755 % filename) 1756 1757 if fpath.endswith('.py'): IOError: did not find file 'mod.sage' in load / attach search path sage: cd /Users/brandonrayhaun/Desktop/Code/Sage /Users/brandonrayhaun/Desktop/Code/Sage sage: attach("mod.sage") --------------------------------------------------------------------------- ImportError Traceback (most recent call last) <ipython-input-4-909b5a17cf48> in <module>() ----> 1 sage.misc.preparser.load(sage.misc.preparser.base64.b64decode("bW9kLnNhZ2U="),globals(),True) /Applications/Sage-6.2.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/misc/preparser.pyc in load(filename, globals, attach) 1767 # See Trac 11812. 1768 exec_file_is(fpath) -> 1769 execfile(preparse_file_named(fpath), globals) 1770 else: 1771 # Preparse in memory only for speed. /Users/brandonrayhaun/.sage/temp/anlwl151-059.wl.anl.gov/13637/mod.sageZRRz8A.py in <module>() 6 precision = _sage_const_30 7 ----> 8 m = ModularForms(Gamma0(_sage_const_12 ),_sage_const_10 ,prec = precision) 9 # p = m.basis() /Applications/Sage-6.2.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/modular/modform/constructor.pyc in ModularForms(group, weight, base_ring, use_cache, prec) 317 M = None 318 if arithgroup.is_Gamma0(group): --> 319 M = ambient_g0.ModularFormsAmbient_g0_Q(group.level(), weight) 320 if base_ring != rings.QQ: 321 M = ambient_R.ModularFormsAmbient_R(M, base_ring) /Applications/Sage-6.2.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/modular/modform/ambient_g0.pyc in __init__(self, level, weight) 41 <class 'sage.modular.modform.ambient_g0.ModularFormsAmbient_g0_Q_with_category'> 42 """ ---> 43 ambient.ModularFormsAmbient.__init__(self, arithgroup.Gamma0(level), weight, rings.QQ) 44 45 #################################################################### /Applications/Sage-6.2.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/modular/modform/ambient.pyc in __init__(self, group, weight, base_ring, character) 110 111 if character is None and arithgroup.is_Gamma0(group): --> 112 character = dirichlet.TrivialCharacter(group.level(), base_ring) 113 114 space.ModularFormsSpace.__init__(self, group, weight, character, base_ring) /Applications/Sage-6.2.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/modular/dirichlet.pyc in trivial_character(N, base_ring) 100 Ring of integers modulo 3 101 """ --> 102 return DirichletGroup(N, base_ring)(1) 103 104 TrivialCharacter = trivial_character /Applications/Sage-6.2.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/structure/factory.so in sage.structure.factory.UniqueFactory.__call__ (sage/structure/factory.c:1308)() /Applications/Sage-6.2.app/Contents/Resources/sage/local/lib ... (more)
2014-07-07 07:49:14 +0200	asked a question	Modular forms in Sage Hi all, I'm new to Sage and have a question about how to implement and manipulate modular forms in the language. I only need to do some simple things. $1$) Can I define my own custom q-expansion? For instance, how would I store $$f(q) = 1 + 2q^2 + 2q^3 + O(q^4)$$ in the same way that modular forms are stored in Sage? 2) Is there any program written to take the Rankin-Cohen bracket of two modular forms? If not, is it possible to write a program in Sage that implements complex differentiation of these modular forms? Thanks for your help. I've tried consulting the documentation, but just when I think I've read all of it, new stuff pops up (unfortunately, none of it relevant to what I'm trying to do).