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Properties

Label 2366.185
Modulus $2366$
Conductor $1183$
Order $78$
Real no
Primitive no
Minimal yes
Parity odd

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(2366, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([13,2]))
 
Copy content gp:[g,chi] = znchar(Mod(185, 2366))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2366.185");
 

Basic properties

Modulus: \(2366\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1183\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(78\)
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 \(\chi_{1183}(185,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2366.by

\(\chi_{2366}(3,\cdot)\) \(\chi_{2366}(61,\cdot)\) \(\chi_{2366}(185,\cdot)\) \(\chi_{2366}(243,\cdot)\) \(\chi_{2366}(367,\cdot)\) \(\chi_{2366}(425,\cdot)\) \(\chi_{2366}(549,\cdot)\) \(\chi_{2366}(607,\cdot)\) \(\chi_{2366}(731,\cdot)\) \(\chi_{2366}(789,\cdot)\) \(\chi_{2366}(913,\cdot)\) \(\chi_{2366}(971,\cdot)\) \(\chi_{2366}(1095,\cdot)\) \(\chi_{2366}(1153,\cdot)\) \(\chi_{2366}(1277,\cdot)\) \(\chi_{2366}(1335,\cdot)\) \(\chi_{2366}(1459,\cdot)\) \(\chi_{2366}(1517,\cdot)\) \(\chi_{2366}(1641,\cdot)\) \(\chi_{2366}(1699,\cdot)\) \(\chi_{2366}(1823,\cdot)\) \(\chi_{2366}(2063,\cdot)\) \(\chi_{2366}(2187,\cdot)\) \(\chi_{2366}(2245,\cdot)\)

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: $\Q(\zeta_{39})$
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 78 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((339,2199)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 2366 }(185, a) \) \(-1\)\(1\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{5}{78}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{71}{78}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{1}{26}\right)\)
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)
 
\( \chi_{ 2366 }(185,a) \;\) at \(\;a = \) e.g. 2