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Properties

Label 11849.1180
Modulus $11849$
Conductor $11849$
Order $272$
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(11849, base_ring=CyclotomicField(272)) M = H._module chi = DirichletCharacter(H, M([27,68]))
 
Copy content gp:[g,chi] = znchar(Mod(1180, 11849))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11849.1180");
 

Basic properties

Modulus: \(11849\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11849\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(272\)
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 11849.dp

\(\chi_{11849}(173,\cdot)\) \(\chi_{11849}(278,\cdot)\) \(\chi_{11849}(337,\cdot)\) \(\chi_{11849}(401,\cdot)\) \(\chi_{11849}(483,\cdot)\) \(\chi_{11849}(583,\cdot)\) \(\chi_{11849}(606,\cdot)\) \(\chi_{11849}(624,\cdot)\) \(\chi_{11849}(870,\cdot)\) \(\chi_{11849}(975,\cdot)\) \(\chi_{11849}(1034,\cdot)\) \(\chi_{11849}(1098,\cdot)\) \(\chi_{11849}(1180,\cdot)\) \(\chi_{11849}(1280,\cdot)\) \(\chi_{11849}(1303,\cdot)\) \(\chi_{11849}(1321,\cdot)\) \(\chi_{11849}(1567,\cdot)\) \(\chi_{11849}(1672,\cdot)\) \(\chi_{11849}(1731,\cdot)\) \(\chi_{11849}(1795,\cdot)\) \(\chi_{11849}(1877,\cdot)\) \(\chi_{11849}(1977,\cdot)\) \(\chi_{11849}(2000,\cdot)\) \(\chi_{11849}(2018,\cdot)\) \(\chi_{11849}(2264,\cdot)\) \(\chi_{11849}(2369,\cdot)\) \(\chi_{11849}(2428,\cdot)\) \(\chi_{11849}(2492,\cdot)\) \(\chi_{11849}(2574,\cdot)\) \(\chi_{11849}(2674,\cdot)\) ...

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: $\Q(\zeta_{272})$
Fixed field: Number field defined by a degree 272 polynomial (not computed)

Values on generators

\((11563,6648)\) → \((e\left(\frac{27}{272}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 11849 }(1180, a) \) \(-1\)\(1\)\(e\left(\frac{49}{136}\right)\)\(e\left(\frac{231}{272}\right)\)\(e\left(\frac{49}{68}\right)\)\(e\left(\frac{63}{272}\right)\)\(e\left(\frac{57}{272}\right)\)\(e\left(\frac{37}{272}\right)\)\(e\left(\frac{11}{136}\right)\)\(e\left(\frac{95}{136}\right)\)\(e\left(\frac{161}{272}\right)\)\(e\left(\frac{9}{272}\right)\)
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)
 
\( \chi_{ 11849 }(1180,a) \;\) at \(\;a = \) e.g. 2