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Properties

Label 15427.26
Modulus $15427$
Conductor $15427$
Order $15426$
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(15427, base_ring=CyclotomicField(15426)) M = H._module chi = DirichletCharacter(H, M([161]))
 
Copy content gp:[g,chi] = znchar(Mod(26, 15427))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15427.26");
 

Basic properties

Modulus: \(15427\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(15427\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(15426\)
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 15427.l

\(\chi_{15427}(2,\cdot)\) \(\chi_{15427}(11,\cdot)\) \(\chi_{15427}(12,\cdot)\) \(\chi_{15427}(14,\cdot)\) \(\chi_{15427}(18,\cdot)\) \(\chi_{15427}(20,\cdot)\) \(\chi_{15427}(21,\cdot)\) \(\chi_{15427}(26,\cdot)\) \(\chi_{15427}(30,\cdot)\) \(\chi_{15427}(31,\cdot)\) \(\chi_{15427}(32,\cdot)\) \(\chi_{15427}(35,\cdot)\) \(\chi_{15427}(37,\cdot)\) \(\chi_{15427}(38,\cdot)\) \(\chi_{15427}(39,\cdot)\) \(\chi_{15427}(44,\cdot)\) \(\chi_{15427}(46,\cdot)\) \(\chi_{15427}(48,\cdot)\) \(\chi_{15427}(50,\cdot)\) \(\chi_{15427}(51,\cdot)\) \(\chi_{15427}(56,\cdot)\) \(\chi_{15427}(58,\cdot)\) \(\chi_{15427}(61,\cdot)\) \(\chi_{15427}(65,\cdot)\) \(\chi_{15427}(67,\cdot)\) \(\chi_{15427}(71,\cdot)\) \(\chi_{15427}(79,\cdot)\) \(\chi_{15427}(80,\cdot)\) \(\chi_{15427}(85,\cdot)\) \(\chi_{15427}(89,\cdot)\) ...

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: $\Q(\zeta_{7713})$
Fixed field: Number field defined by a degree 15426 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{161}{15426}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 15427 }(26, a) \) \(-1\)\(1\)\(e\left(\frac{161}{15426}\right)\)\(e\left(\frac{3919}{5142}\right)\)\(e\left(\frac{161}{7713}\right)\)\(e\left(\frac{3211}{5142}\right)\)\(e\left(\frac{5959}{7713}\right)\)\(e\left(\frac{2269}{7713}\right)\)\(e\left(\frac{161}{5142}\right)\)\(e\left(\frac{1348}{2571}\right)\)\(e\left(\frac{4897}{7713}\right)\)\(e\left(\frac{3271}{15426}\right)\)
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)
 
\( \chi_{ 15427 }(26,a) \;\) at \(\;a = \) e.g. 2