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Properties

Label 2382.491
Modulus $2382$
Conductor $1191$
Order $396$
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(2382, base_ring=CyclotomicField(396)) M = H._module chi = DirichletCharacter(H, M([198,223]))
 
Copy content gp:[g,chi] = znchar(Mod(491, 2382))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2382.491");
 

Basic properties

Modulus: \(2382\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1191\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(396\)
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 \(\chi_{1191}(491,\cdot)\)
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 2382.bi

\(\chi_{2382}(5,\cdot)\) \(\chi_{2382}(59,\cdot)\) \(\chi_{2382}(77,\cdot)\) \(\chi_{2382}(89,\cdot)\) \(\chi_{2382}(101,\cdot)\) \(\chi_{2382}(143,\cdot)\) \(\chi_{2382}(155,\cdot)\) \(\chi_{2382}(161,\cdot)\) \(\chi_{2382}(197,\cdot)\) \(\chi_{2382}(215,\cdot)\) \(\chi_{2382}(227,\cdot)\) \(\chi_{2382}(233,\cdot)\) \(\chi_{2382}(239,\cdot)\) \(\chi_{2382}(245,\cdot)\) \(\chi_{2382}(251,\cdot)\) \(\chi_{2382}(263,\cdot)\) \(\chi_{2382}(299,\cdot)\) \(\chi_{2382}(317,\cdot)\) \(\chi_{2382}(323,\cdot)\) \(\chi_{2382}(347,\cdot)\) \(\chi_{2382}(359,\cdot)\) \(\chi_{2382}(377,\cdot)\) \(\chi_{2382}(419,\cdot)\) \(\chi_{2382}(425,\cdot)\) \(\chi_{2382}(443,\cdot)\) \(\chi_{2382}(449,\cdot)\) \(\chi_{2382}(455,\cdot)\) \(\chi_{2382}(491,\cdot)\) \(\chi_{2382}(509,\cdot)\) \(\chi_{2382}(605,\cdot)\) ...

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: $\Q(\zeta_{396})$
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 396 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1589,799)\) → \((-1,e\left(\frac{223}{396}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2382 }(491, a) \) \(1\)\(1\)\(e\left(\frac{25}{396}\right)\)\(e\left(\frac{107}{396}\right)\)\(e\left(\frac{67}{198}\right)\)\(e\left(\frac{347}{396}\right)\)\(e\left(\frac{43}{132}\right)\)\(e\left(\frac{8}{99}\right)\)\(e\left(\frac{50}{99}\right)\)\(e\left(\frac{25}{198}\right)\)\(e\left(\frac{73}{198}\right)\)\(e\left(\frac{5}{11}\right)\)
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)
 
\( \chi_{ 2382 }(491,a) \;\) at \(\;a = \) e.g. 2