# HG changeset patch # User Alexandru Ghitza # Date 1250674690 -36000 # Node ID 0527308b32b1f92cda53f7d8d63b7d9bfc4dd55b # Parent cc7619c5890f960ff7d691968f17d23e0c940810 trac 6402: trivial fixes diff -r cc7619c5890f -r 0527308b32b1 sage/modular/overconvergent/genus0.py --- a/sage/modular/overconvergent/genus0.py Thu Jun 25 14:12:42 2009 +0200 +++ b/sage/modular/overconvergent/genus0.py Wed Aug 19 19:38:10 2009 +1000 @@ -75,7 +75,7 @@ sage: f2 = M(CuspForms(3, 8).0); f2 3-adic overconvergent modular form of weight-character 8 with q-expansion q + 6*q^2 - 27*q^3 - 92*q^4 + 390*q^5 - 162*q^6 ... -We express this in a basis, and see that the coefficients to to zero very fast: +We express this in a basis, and see that the coefficients go to zero very fast: .. link @@ -84,7 +84,7 @@ sage: [x.valuation(3) for x in f2.coordinates(60)] [+Infinity, -1, 3, 6, 10, 13, 18, 20, 24, 27, 31, 34, 39, 41, 45, 48, 52, 55, 61, 62, 66, 69, 73, 76, 81, 83, 87, 90, 94, 97, 102, 104, 108, 111, 115, 118, 124, 125, 129, 132, 136, 139, 144, 146, 150, 153, 157, 160, 165, 167, 171, 174, 178, 181, 188, 188, 192, 195, 199, 202] -This form has more level at p, and hence is less overconvergent: +This form has more level at `p`, and hence is less overconvergent: .. link @@ -104,7 +104,7 @@ ... ValueError: Form is not overconvergent enough (form is only 1/12-overconvergent) -Let's compute some Hecke operators. Note that the coefficients of this matrix are p-adically tiny: +Let's compute some Hecke operators. Note that the coefficients of this matrix are `p`-adically tiny: .. link @@ -147,7 +147,7 @@ sage: (efuncs[2] - M(EisensteinForms(3, 8).1)).valuation() 10 -The third is a genuninely new thing (not a classical modular form at all); the +The third is a genuinely new thing (not a classical modular form at all); the coefficients are almost certainly not algebraic over `\QQ`. Note that the slope is 9, so Coleman's classicality criterion (forms of slope `< k-1` are classical) does not apply. @@ -204,7 +204,7 @@ INPUT: - ``prime`` - a prime number `p`, which must be one of the primes `\{2, 3, - 5, 7, 13\}`, or the congruence subgroup Gamma0(p) where p is one of these + 5, 7, 13\}`, or the congruence subgroup Gamma0(p) where `p` is one of these primes. - ``weight`` - an integer (which at present must be 0 or `\ge 2`), the @@ -261,7 +261,7 @@ class OverconvergentModularFormsSpace(Module): r""" A space of overconvergent modular forms of level `\Gamma_0(p)`, - where p is a prime such that `X_0(p)` has genus 0. + where `p` is a prime such that `X_0(p)` has genus 0. Elements are represented as power series, with a formal power series `F` corresponding to the modular form `E_k^\ast \times F(g)` where `E_k^\ast` @@ -331,11 +331,11 @@ def _set_radius(self, radius): r""" Set the radius of overconvergence to be `r`, where `r` is a rational - number in the interval `0 < r < \frac{p}{p+1}` + number in the interval `0 < r < \frac{p}{p+1}`. This only makes sense if the base ring contains an element of normalised valuation `\frac{12r}{p-1}`. If this valuation is an - integer, we use the appropriate power of p. Otherwise, we assume the + integer, we use the appropriate power of `p`. Otherwise, we assume the base ring has a ``uniformiser`` method and take an appropriate power of the uniformiser, raising an error if no such element exists. @@ -562,8 +562,8 @@ def prime(self): r""" - Return the residue characteristic of self, i.e. the prime p such that - this is a p-adic space. + Return the residue characteristic of self, i.e. the prime `p` such that + this is a `p`-adic space. EXAMPLES:: @@ -606,7 +606,7 @@ def prec(self): r""" Return the series precision of self. Note that this is different from - the p-adic precision of the base ring. + the `p`-adic precision of the base ring. EXAMPLE:: @@ -628,7 +628,7 @@ - elements of compatible spaces of modular forms or overconvergent modular forms - - arbitrary power series in q + - arbitrary power series in `q` - lists of elements of the base ring (interpreted as vectors in the basis given by self.gens()). @@ -639,7 +639,7 @@ EXAMPLES:: - From a q-expansion:: + From a `q`-expansion:: sage: M = OverconvergentModularForms(3, 0, 1/2, prec=5) sage: R. = QQ[[]] @@ -722,7 +722,7 @@ def _coerce_from_ocmf(self, f): r""" - Try to convert the overconvergent modular form f into an element of self. An error will be raised if this is + Try to convert the overconvergent modular form `f` into an element of self. An error will be raised if this is obviously nonsense. EXAMPLES:: @@ -890,12 +890,12 @@ Calculate the matrix of the `T_m` operator in the basis of this space, truncated to an `n \times n` matrix. Conventions are that operators act on the left on column vectors (this is the opposite of the conventions - of the sage.modules.matrix_morphism class!) Uses naive q-expansion + of the sage.modules.matrix_morphism class!) Uses naive `q`-expansion arguments if use_recurrence=False and uses the Kolberg style recurrences if use_recurrence=True. The argument "exact_arith" causes the computation to be done with - rational arithmetic, even if the base ring is an inexact p-adic ring. + rational arithmetic, even if the base ring is an inexact `p`-adic ring. This is useful as there can be precision loss issues (particularly with use_recurrence=False). @@ -998,12 +998,12 @@ - ``F`` - None, or a field over which to calculate eigenvalues. If the field is None, the current base ring is used. If the base ring is not - a p-adic ring, an error will be raised. + a `p`-adic ring, an error will be raised. - ``exact_arith`` - True or False (default True). If True, use exact - rational arithmetic to calculate the matrix of the U operator and its + rational arithmetic to calculate the matrix of the `U` operator and its characteristic power series, even when the base ring is an inexact - p-adic ring. This is typically slower, but more numerically stable. + `p`-adic ring. This is typically slower, but more numerically stable. NOTE: Try using ``set_verbose(1, 'sage/modular/overconvergent')`` to get more feedback on what is going on in this algorithm. For even more @@ -1096,7 +1096,8 @@ return [f for _,f in eigenfunctions] def recurrence_matrix(self, use_smithline=True): - r""" Return the recurrence matrix satisfied by the coefficients of U, + r""" + Return the recurrence matrix satisfied by the coefficients of `U`, that is a matrix `R =(r_{rs})_{r,s=1 \dots p}` such that `u_{ij} = \sum_{r,s=1}^p r_{rs} u_{i-r, j-s}`. Uses an elegant construction which I believe is due to Smithline. See my paper in IMRN (full citation in @@ -1145,7 +1146,8 @@ def _discover_recurrence_matrix(self, use_smithline=True): - r""" Does hard work of calculating recurrence matrix, which is cached to avoid doing this every time. + r""" + Does hard work of calculating recurrence matrix, which is cached to avoid doing this every time. EXAMPLE:: @@ -1291,7 +1293,7 @@ def prec(self): r""" Return the series expansion precision of this overconvergent modular - form. (This is not the same as the p-adic precision of the + form. (This is not the same as the `p`-adic precision of the coefficients.) EXAMPLE:: @@ -1361,7 +1363,7 @@ def q_expansion(self, prec=None): r""" - Return the q-expansion of self, to as high precision as it is known. + Return the `q`-expansion of self, to as high precision as it is known. EXAMPLE:: @@ -1377,8 +1379,8 @@ def gexp(self): r""" - Return the formal power series in g corresponding to this - overconvergent modular form (so the result is F where this modular form + Return the formal power series in `g` corresponding to this + overconvergent modular form (so the result is `F` where this modular form is `E_k^\ast \times F(g)`, where `g` is the appropriately normalised parameter of `X_0(p)`). @@ -1416,7 +1418,8 @@ return self._gexp.padded_list(prec) def prime(self): - r""" If this is a p-adic modular form, return p. + r""" + If this is a `p`-adic modular form, return `p`. EXAMPLE:: @@ -1460,7 +1463,8 @@ return True def _repr_(self): - r""" String representation of self + r""" + String representation of self. EXAMPLES:: @@ -1489,7 +1493,7 @@ def r_ord(self, r): r""" - The p-adic valuation of the norm of self on the r-overconvergent region. + The `p`-adic valuation of the norm of self on the `r`-overconvergent region. EXAMPLES:: @@ -1533,7 +1537,8 @@ return min([v(x) for x in self.gexp().list()]) def governing_term(self, r): - r""" The degree of the series term with largest norm on the r-overconvergent region. + r""" + The degree of the series term with largest norm on the `r`-overconvergent region. EXAMPLES:: @@ -1584,7 +1589,7 @@ def additive_order(self): r""" Return the additive order of this element (required attribute for all - elements deriving from sage.modules.ModuleElement) + elements deriving from sage.modules.ModuleElement). EXAMPLES:: @@ -1615,7 +1620,7 @@ def _pari_(self): r""" Return the Pari object corresponding to self, which is just the - q-expansion of self as a formal power series. At present conversion of + `q`-expansion of self as a formal power series. At present conversion of power series to Pari is only implemented if the base ring is `\QQ`.