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Package sage :: Package modular :: Module dirichlet :: Class DirichletCharacter |
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A DirichletCharacter is the extension of a homomorphism (Z/NZ)^* --> R^*, for some ring R, to a map Z/NZ --> R, got by sending those x with (N,x)>1 to 0. Create with DirichletCharacter(parent, values_on_gens) INPUT: parent -- DirichletGroup, a group of Dirichlet characters values_on_gens -- list of ring elements, the values of the Dirichlet character on the chosen generators of (Z/NZ)^*. OUTPUT: DirichletCharacter -- a Dirichlet character
Method Summary | |
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__init__(self,
parent,
values_on_gens)
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__call__(self,
m)
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__cmp__(self,
other)
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__hash__(self)
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__invert__(self)
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__mul__(self,
other)
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__pow__(self,
exp)
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__repr__(self)
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__rmul__(self,
left)
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Returns the base ring of the parent of self. | |
Returns the generalized Bernoulli number B_{k,eps}. | |
Returns the base extension of self to the ring R. | |
Tries to compute the Dirichlet character over R that has the same values as self, and if successful returns that character. | |
Computes and returns the conductor of self. | |
Return the decomposition of self as a product of Dirichlet characters of prime power modulus, where the prime powers exactly divide the modulus of self. | |
Returns the extension of self to a Dirichlet character modulo the multiple M of the modulus. | |
Return True if and only if self(-1) == 1. | |
Return True if and only if self(-1) != 1. | |
Returns true if self is the trivial character, i.e., has order 1. | |
Let eps : (Z/N)^* ----> Q(zeta_n) be a Dirichlet character. | |
Return a Dirichlet character that equals this one, but over as small a subfield (or subring) of the base ring as possible. | |
Returns the modulus of self. | |
Returns the order of self. | |
Returns the parent of self. | |
Returns the restriction of self to a Dirichlet character modulo the divisor M of the modulus, which must also be a multiple of the conductor of self. | |
Returns a list of the values of self on each integer between 0 and the modulus of self. | |
Returns a list of the values of self on each of the minimal generators of (Z/NZ)^*, where N is the modulus of self. |
Method Details |
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base_ring(self)Returns the base ring of the parent of self. |
bernoulli(self, k)Returns the generalized Bernoulli number B_{k,eps}. Let eps be this character (not necessarily primitive), and let k>=0 be an integer weight. This function computes the (generalized) Bernoulli number B_{k,eps}, e.g., as defined on page 44 of Diamond-Im: sum_{a=1}^{N} eps(a) t*e^(at)/(e^(N*t)-1) = sum_{k=0}^{\infty} B_{k,eps}/{k!}*t^k. where N is the modulus of eps. |
change_base(self, R)Returns the base extension of self to the ring R. |
change_base_ring(self, R)Tries to compute the Dirichlet character over R that has the same values as self, and if successful returns that character. |
conductor(self)Computes and returns the conductor of self. |
decomposition(self)Return the decomposition of self as a product of Dirichlet characters of prime power modulus, where the prime powers exactly divide the modulus of self. |
extend(self, M)Returns the extension of self to a Dirichlet character modulo the multiple M of the modulus. which must also be a |
is_even(self)Return True if and only if self(-1) == 1. |
is_odd(self)Return True if and only if self(-1) != 1. |
is_trivial(self)Returns true if self is the trivial character, i.e., has order 1. |
maximize_base_ring(self)Let eps : (Z/N)^* ----> Q(zeta_n) be a Dirichlet character. This function returns an equal Dirichlet character chi : (Z/N)^* ----> Q(zeta_m) where m is LCM(n, exponent of (Z/N)^*). |
minimize_base_ring(self)Return a Dirichlet character that equals this one, but over as small a subfield (or subring) of the base ring as possible. NOTE: This function is currently only implemented when the base ring is a number field. |
modulus(self)Returns the modulus of self. |
order(self)Returns the order of self. |
parent(self)Returns the parent of self. |
restrict(self, M)Returns the restriction of self to a Dirichlet character modulo the divisor M of the modulus, which must also be a multiple of the conductor of self. |
values(self)Returns a list of the values of self on each integer between 0 and the modulus of self. |
values_on_gens(self)Returns a list of the values of self on each of the minimal generators of (Z/NZ)^*, where N is the modulus of self. |
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