Cremona modular symbols#

sage.libs.eclib.constructor.CremonaModularSymbols(level, sign=0, cuspidal=False, verbose=0)[source]#

Return the space of Cremona modular symbols with given level, sign, etc.

INPUT:

  • level – an integer >= 2 (at least 2, not just positive!)

  • sign – an integer either 0 (the default) or 1 or -1.

  • cuspidal – (default: False); if True, compute only the cuspidal subspace

  • verbose – (default: False): if True, print verbose information while creating space

EXAMPLES:

sage: M = CremonaModularSymbols(43); M
Cremona Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0
sage: M = CremonaModularSymbols(43, sign=1); M
Cremona Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1
sage: M = CremonaModularSymbols(43, cuspidal=True); M
Cremona Cuspidal Modular Symbols space of dimension 6 for Gamma_0(43) of weight 2 with sign 0
sage: M = CremonaModularSymbols(43, cuspidal=True, sign=1); M
Cremona Cuspidal Modular Symbols space of dimension 3 for Gamma_0(43) of weight 2 with sign 1

>>> from sage.all import *
>>> M = CremonaModularSymbols(Integer(43)); M
Cremona Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0
>>> M = CremonaModularSymbols(Integer(43), sign=Integer(1)); M
Cremona Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1
>>> M = CremonaModularSymbols(Integer(43), cuspidal=True); M
Cremona Cuspidal Modular Symbols space of dimension 6 for Gamma_0(43) of weight 2 with sign 0
>>> M = CremonaModularSymbols(Integer(43), cuspidal=True, sign=Integer(1)); M
Cremona Cuspidal Modular Symbols space of dimension 3 for Gamma_0(43) of weight 2 with sign 1

When run interactively, the following command will display verbose output:

sage: M = CremonaModularSymbols(43, verbose=1)
After 2-term relations, ngens = 22
ngens     = 22
maxnumrel = 32
relation matrix has = 704 entries...
Finished 3-term relations: numrel = 16 ( maxnumrel = 32)
relmat has 42 nonzero entries (density = 0.0596591)
Computing kernel...
time to compute kernel =  (... seconds)
rk = 7
Number of cusps is 2
ncusps = 2
About to compute matrix of delta
delta matrix done: size 2x7.
About to compute kernel of delta
done
Finished constructing homspace.
sage: M
Cremona Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0

>>> from sage.all import *
>>> M = CremonaModularSymbols(Integer(43), verbose=Integer(1))
After 2-term relations, ngens = 22
ngens     = 22
maxnumrel = 32
relation matrix has = 704 entries...
Finished 3-term relations: numrel = 16 ( maxnumrel = 32)
relmat has 42 nonzero entries (density = 0.0596591)
Computing kernel...
time to compute kernel =  (... seconds)
rk = 7
Number of cusps is 2
ncusps = 2
About to compute matrix of delta
delta matrix done: size 2x7.
About to compute kernel of delta
done
Finished constructing homspace.
>>> M
Cremona Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0

The input must be valid or a ValueError is raised:

sage: M = CremonaModularSymbols(-1)
Traceback (most recent call last):
...
ValueError: the level (= -1) must be at least 2
sage: M = CremonaModularSymbols(0)
Traceback (most recent call last):
...
ValueError: the level (= 0) must be at least 2

>>> from sage.all import *
>>> M = CremonaModularSymbols(-Integer(1))
Traceback (most recent call last):
...
ValueError: the level (= -1) must be at least 2
>>> M = CremonaModularSymbols(Integer(0))
Traceback (most recent call last):
...
ValueError: the level (= 0) must be at least 2

The sign can only be 0 or 1 or -1:

sage: M = CremonaModularSymbols(10, sign = -2)
Traceback (most recent call last):
...
ValueError: sign (= -2) is not supported; use 0, +1 or -1

>>> from sage.all import *
>>> M = CremonaModularSymbols(Integer(10), sign = -Integer(2))
Traceback (most recent call last):
...
ValueError: sign (= -2) is not supported; use 0, +1 or -1

We do allow -1 as a sign (see Issue #9476):

sage: CremonaModularSymbols(10, sign = -1)
Cremona Modular Symbols space of dimension 0 for Gamma_0(10) of weight 2 with sign -1

>>> from sage.all import *
>>> CremonaModularSymbols(Integer(10), sign = -Integer(1))
Cremona Modular Symbols space of dimension 0 for Gamma_0(10) of weight 2 with sign -1