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Properties

Label 93025.1242
Modulus $93025$
Conductor $93025$
Order $3660$
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(93025, base_ring=CyclotomicField(3660)) M = H._module chi = DirichletCharacter(H, M([2379,1036]))
 
Copy content gp:[g,chi] = znchar(Mod(1242, 93025))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("93025.1242");
 

Basic properties

Modulus: \(93025\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(93025\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3660\)
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 93025.jw

\(\chi_{93025}(12,\cdot)\) \(\chi_{93025}(42,\cdot)\) \(\chi_{93025}(83,\cdot)\) \(\chi_{93025}(137,\cdot)\) \(\chi_{93025}(147,\cdot)\) \(\chi_{93025}(408,\cdot)\) \(\chi_{93025}(423,\cdot)\) \(\chi_{93025}(727,\cdot)\) \(\chi_{93025}(988,\cdot)\) \(\chi_{93025}(1053,\cdot)\) \(\chi_{93025}(1113,\cdot)\) \(\chi_{93025}(1242,\cdot)\) \(\chi_{93025}(1277,\cdot)\) \(\chi_{93025}(1297,\cdot)\) \(\chi_{93025}(1398,\cdot)\) \(\chi_{93025}(1428,\cdot)\) \(\chi_{93025}(1537,\cdot)\)

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: $\Q(\zeta_{3660})$
Fixed field: Number field defined by a degree 3660 polynomial (not computed)

Values on generators

\((22327,70701)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{259}{915}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 93025 }(1242, a) \) \(-1\)\(1\)\(e\left(\frac{683}{732}\right)\)\(e\left(\frac{563}{1220}\right)\)\(e\left(\frac{317}{366}\right)\)\(e\left(\frac{361}{915}\right)\)\(e\left(\frac{3259}{3660}\right)\)\(e\left(\frac{195}{244}\right)\)\(e\left(\frac{563}{610}\right)\)\(e\left(\frac{12}{305}\right)\)\(e\left(\frac{1199}{3660}\right)\)\(e\left(\frac{1261}{3660}\right)\)
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)
 
\( \chi_{ 93025 }(1242,a) \;\) at \(\;a = \) e.g. 2