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Properties

Label 3897.38
Modulus $3897$
Conductor $3897$
Order $432$
Real no
Primitive yes
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(3897, base_ring=CyclotomicField(432)) M = H._module chi = DirichletCharacter(H, M([72,55]))
 
Copy content gp:[g,chi] = znchar(Mod(38, 3897))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3897.38");
 

Basic properties

Modulus: \(3897\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3897\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(432\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3897.ed

\(\chi_{3897}(14,\cdot)\) \(\chi_{3897}(20,\cdot)\) \(\chi_{3897}(23,\cdot)\) \(\chi_{3897}(29,\cdot)\) \(\chi_{3897}(38,\cdot)\) \(\chi_{3897}(83,\cdot)\) \(\chi_{3897}(101,\cdot)\) \(\chi_{3897}(122,\cdot)\) \(\chi_{3897}(281,\cdot)\) \(\chi_{3897}(311,\cdot)\) \(\chi_{3897}(317,\cdot)\) \(\chi_{3897}(326,\cdot)\) \(\chi_{3897}(353,\cdot)\) \(\chi_{3897}(362,\cdot)\) \(\chi_{3897}(371,\cdot)\) \(\chi_{3897}(410,\cdot)\) \(\chi_{3897}(419,\cdot)\) \(\chi_{3897}(443,\cdot)\) \(\chi_{3897}(452,\cdot)\) \(\chi_{3897}(518,\cdot)\) \(\chi_{3897}(527,\cdot)\) \(\chi_{3897}(545,\cdot)\) \(\chi_{3897}(563,\cdot)\) \(\chi_{3897}(596,\cdot)\) \(\chi_{3897}(614,\cdot)\) \(\chi_{3897}(617,\cdot)\) \(\chi_{3897}(626,\cdot)\) \(\chi_{3897}(653,\cdot)\) \(\chi_{3897}(659,\cdot)\) \(\chi_{3897}(680,\cdot)\) ...

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: $\Q(\zeta_{432})$
Fixed field: Number field defined by a degree 432 polynomial (not computed)

Values on generators

\((434,2170)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{55}{432}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 3897 }(38, a) \) \(1\)\(1\)\(e\left(\frac{7}{72}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{415}{432}\right)\)\(e\left(\frac{59}{432}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{25}{432}\right)\)\(e\left(\frac{155}{216}\right)\)\(e\left(\frac{47}{216}\right)\)\(e\left(\frac{101}{432}\right)\)\(e\left(\frac{7}{18}\right)\)
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)
 
\( \chi_{ 3897 }(38,a) \;\) at \(\;a = \) e.g. 2