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Properties

Label 4897.61
Modulus $4897$
Conductor $4897$
Order $2378$
Real no
Primitive yes
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(4897, base_ring=CyclotomicField(2378)) M = H._module chi = DirichletCharacter(H, M([41,1914]))
 
Copy content gp:[g,chi] = znchar(Mod(61, 4897))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4897.61");
 

Basic properties

Modulus: \(4897\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4897\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2378\)
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: yes
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 4897.o

\(\chi_{4897}(10,\cdot)\) \(\chi_{4897}(11,\cdot)\) \(\chi_{4897}(23,\cdot)\) \(\chi_{4897}(30,\cdot)\) \(\chi_{4897}(31,\cdot)\) \(\chi_{4897}(33,\cdot)\) \(\chi_{4897}(37,\cdot)\) \(\chi_{4897}(38,\cdot)\) \(\chi_{4897}(40,\cdot)\) \(\chi_{4897}(44,\cdot)\) \(\chi_{4897}(61,\cdot)\) \(\chi_{4897}(65,\cdot)\) \(\chi_{4897}(69,\cdot)\) \(\chi_{4897}(70,\cdot)\) \(\chi_{4897}(77,\cdot)\) \(\chi_{4897}(90,\cdot)\) \(\chi_{4897}(92,\cdot)\) \(\chi_{4897}(93,\cdot)\) \(\chi_{4897}(99,\cdot)\) \(\chi_{4897}(106,\cdot)\) \(\chi_{4897}(109,\cdot)\) \(\chi_{4897}(111,\cdot)\) \(\chi_{4897}(113,\cdot)\) \(\chi_{4897}(114,\cdot)\) \(\chi_{4897}(120,\cdot)\) \(\chi_{4897}(124,\cdot)\) \(\chi_{4897}(131,\cdot)\) \(\chi_{4897}(132,\cdot)\) \(\chi_{4897}(142,\cdot)\) \(\chi_{4897}(148,\cdot)\) ...

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: $\Q(\zeta_{1189})$
Fixed field: Number field defined by a degree 2378 polynomial (not computed)

Values on generators

\((2657,2243)\) → \((e\left(\frac{1}{58}\right),e\left(\frac{33}{41}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4897 }(61, a) \) \(-1\)\(1\)\(e\left(\frac{1955}{2378}\right)\)\(e\left(\frac{967}{1189}\right)\)\(e\left(\frac{766}{1189}\right)\)\(e\left(\frac{993}{1189}\right)\)\(e\left(\frac{1511}{2378}\right)\)\(e\left(\frac{891}{1189}\right)\)\(e\left(\frac{1109}{2378}\right)\)\(e\left(\frac{745}{1189}\right)\)\(e\left(\frac{1563}{2378}\right)\)\(e\left(\frac{1779}{2378}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 4897 }(61,a) \;\) at \(\;a = \) e.g. 2