Trac #11631: L-series attached to cusp forms are broken

{{{
sage: C = CuspForms(43,2)
sage: N = C.newforms('a')
sage: f = N[1]
sage: L = f.cuspform_lseries()
Boom!
}}}

There are two issues:

  (1) the code doesn't work at all when the degree of the form is > 1,
which is the main interesting case,

  (2) the name "cuspform_lseries" is bad, since f is already a cuspform,
and we use the name "lseries" in all other places (e.g., elliptic
curves, abelian varieties), so it is hard to find.  In fact, I didn't
even think to look for cuspform_lseries, instead only finding this via
lots of grepping and reading source code.

So to fix this issue, I think (1) the bug should get fixed, and (2) the
name should be changed (actually *deprecate* the old name as explained
in the developers guide and introduce the name lseries).

See also #12015.

The part about deprecating cuspform_lseries() is moved to #16917.

URL: http://trac.sagemath.org/11631
Reported by: was
Ticket author(s): Gonzalo Tornaría
Reviewer(s): Michael Neururer

src/sage/modular/modform/element.py

@@ -685,7 +685,7 @@ def period(self, M, prec=53):
                      + mu_dN ** n * (mu_d ** (n * b) - eps * mu_d ** (n * c)))
                    for n in range(1, numterms + 1))
 
-    def cuspform_lseries(self, prec=53,
+    def cuspform_lseries(self, conjugate=0, prec=53,
                          max_imaginary_part=0,
                          max_asymp_coeffs=40):
         r"""
@@ -697,6 +697,8 @@ def cuspform_lseries(self, prec=53,
 
         INPUT:
 
+        - ``conjugate`` - (default: 0), integer between 0 and degree-1
+
         - ``prec`` - integer (bits precision)
 
         - ``max_imaginary_part`` - real number
@@ -716,6 +718,16 @@ def cuspform_lseries(self, prec=53,
            sage: L(0.5)
            0.0296568512531983
 
+        For non-rational newforms we can specify a conjugate::
+
+           sage: f = Newforms(43, names='a')[1]
+           sage: L = f.cuspform_lseries(conjugate=0)
+           sage: L(1)
+           0.620539857407845
+           sage: L = f.cuspform_lseries(conjugate=1)
+           sage: L(1)
+           0.921328017272472
+
         Consistency check with delta_lseries (which computes coefficients in pari)::
 
            sage: delta = CuspForms(1,12).0
@@ -773,7 +785,10 @@ def cuspform_lseries(self, prec=53,
         # Find out how many coefficients of the Dirichlet series are needed
         # in order to compute to the required precision
         num_coeffs = L.num_coeffs()
-        s = 'coeff = %s;'%self.q_expansion(num_coeffs+1).padded_list()
+        coeffs = self.q_expansion(num_coeffs+1).padded_list()
+        # compute the requested embedding
+        emb = self.base_ring().embeddings(rings.ComplexField(prec))[conjugate]
+        s = 'coeff = %s;'% map(emb, coeffs)
         L.init_coeffs('coeff[k+1]',pari_precode = s,
                       max_imaginary_part=max_imaginary_part,
                       max_asymp_coeffs=max_asymp_coeffs)