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Properties

Label 45373.277
Modulus $45373$
Conductor $45373$
Order $10608$
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(45373, base_ring=CyclotomicField(10608)) M = H._module chi = DirichletCharacter(H, M([9555,2584]))
 
Copy content gp:[g,chi] = znchar(Mod(277, 45373))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45373.277");
 

Basic properties

Modulus: \(45373\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(45373\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(10608\)
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 45373.fn

\(\chi_{45373}(3,\cdot)\) \(\chi_{45373}(10,\cdot)\) \(\chi_{45373}(31,\cdot)\) \(\chi_{45373}(44,\cdot)\) \(\chi_{45373}(48,\cdot)\) \(\chi_{45373}(57,\cdot)\) \(\chi_{45373}(105,\cdot)\) \(\chi_{45373}(122,\cdot)\) \(\chi_{45373}(146,\cdot)\) \(\chi_{45373}(148,\cdot)\) \(\chi_{45373}(160,\cdot)\) \(\chi_{45373}(167,\cdot)\) \(\chi_{45373}(182,\cdot)\) \(\chi_{45373}(190,\cdot)\) \(\chi_{45373}(193,\cdot)\) \(\chi_{45373}(199,\cdot)\) \(\chi_{45373}(201,\cdot)\) \(\chi_{45373}(233,\cdot)\) \(\chi_{45373}(243,\cdot)\) \(\chi_{45373}(262,\cdot)\) \(\chi_{45373}(267,\cdot)\) \(\chi_{45373}(277,\cdot)\) \(\chi_{45373}(279,\cdot)\) \(\chi_{45373}(284,\cdot)\) \(\chi_{45373}(295,\cdot)\) \(\chi_{45373}(303,\cdot)\) \(\chi_{45373}(317,\cdot)\) \(\chi_{45373}(345,\cdot)\) \(\chi_{45373}(347,\cdot)\) \(\chi_{45373}(350,\cdot)\) ...

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: $\Q(\zeta_{10608})$
Fixed field: Number field defined by a degree 10608 polynomial (not computed)

Values on generators

\((25435,39883)\) → \((e\left(\frac{245}{272}\right),e\left(\frac{19}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 45373 }(277, a) \) \(-1\)\(1\)\(e\left(\frac{859}{1768}\right)\)\(e\left(\frac{9283}{10608}\right)\)\(e\left(\frac{859}{884}\right)\)\(e\left(\frac{5431}{10608}\right)\)\(e\left(\frac{3829}{10608}\right)\)\(e\left(\frac{1491}{3536}\right)\)\(e\left(\frac{809}{1768}\right)\)\(e\left(\frac{3979}{5304}\right)\)\(e\left(\frac{10585}{10608}\right)\)\(e\left(\frac{5701}{10608}\right)\)
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)
 
\( \chi_{ 45373 }(277,a) \;\) at \(\;a = \) e.g. 2