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Properties

Label 4232.49
Modulus $4232$
Conductor $529$
Order $253$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(506)) M = H._module chi = DirichletCharacter(H, M([0,0,258]))
 
Copy content gp:[g,chi] = znchar(Mod(49, 4232))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4232.49");
 

Basic properties

Modulus: \(4232\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(529\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(253\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{529}(49,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 4232.y

\(\chi_{4232}(9,\cdot)\) \(\chi_{4232}(25,\cdot)\) \(\chi_{4232}(41,\cdot)\) \(\chi_{4232}(49,\cdot)\) \(\chi_{4232}(73,\cdot)\) \(\chi_{4232}(81,\cdot)\) \(\chi_{4232}(105,\cdot)\) \(\chi_{4232}(121,\cdot)\) \(\chi_{4232}(169,\cdot)\) \(\chi_{4232}(193,\cdot)\) \(\chi_{4232}(209,\cdot)\) \(\chi_{4232}(225,\cdot)\) \(\chi_{4232}(233,\cdot)\) \(\chi_{4232}(257,\cdot)\) \(\chi_{4232}(265,\cdot)\) \(\chi_{4232}(289,\cdot)\) \(\chi_{4232}(305,\cdot)\) \(\chi_{4232}(353,\cdot)\) \(\chi_{4232}(361,\cdot)\) \(\chi_{4232}(377,\cdot)\) \(\chi_{4232}(393,\cdot)\) \(\chi_{4232}(409,\cdot)\) \(\chi_{4232}(417,\cdot)\) \(\chi_{4232}(441,\cdot)\) \(\chi_{4232}(449,\cdot)\) \(\chi_{4232}(473,\cdot)\) \(\chi_{4232}(489,\cdot)\) \(\chi_{4232}(537,\cdot)\) \(\chi_{4232}(545,\cdot)\) \(\chi_{4232}(561,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{253})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 253 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((3175,2117,2121)\) → \((1,1,e\left(\frac{129}{253}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4232 }(49, a) \) \(1\)\(1\)\(e\left(\frac{40}{253}\right)\)\(e\left(\frac{129}{253}\right)\)\(e\left(\frac{196}{253}\right)\)\(e\left(\frac{80}{253}\right)\)\(e\left(\frac{237}{253}\right)\)\(e\left(\frac{222}{253}\right)\)\(e\left(\frac{169}{253}\right)\)\(e\left(\frac{199}{253}\right)\)\(e\left(\frac{109}{253}\right)\)\(e\left(\frac{236}{253}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 4232 }(49,a) \;\) at \(\;a = \) e.g. 2