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Properties

Label 112789.281
Modulus $112789$
Conductor $112789$
Order $18060$
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(112789, base_ring=CyclotomicField(18060)) M = H._module chi = DirichletCharacter(H, M([580,11739]))
 
Copy content gp:[g,chi] = znchar(Mod(281, 112789))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112789.281");
 

Basic properties

Modulus: \(112789\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(112789\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(18060\)
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 112789.jb

\(\chi_{112789}(23,\cdot)\) \(\chi_{112789}(24,\cdot)\) \(\chi_{112789}(38,\cdot)\) \(\chi_{112789}(53,\cdot)\) \(\chi_{112789}(99,\cdot)\) \(\chi_{112789}(146,\cdot)\) \(\chi_{112789}(160,\cdot)\) \(\chi_{112789}(267,\cdot)\) \(\chi_{112789}(268,\cdot)\) \(\chi_{112789}(272,\cdot)\) \(\chi_{112789}(281,\cdot)\) \(\chi_{112789}(282,\cdot)\) \(\chi_{112789}(358,\cdot)\) \(\chi_{112789}(404,\cdot)\) \(\chi_{112789}(455,\cdot)\) \(\chi_{112789}(496,\cdot)\) \(\chi_{112789}(511,\cdot)\) \(\chi_{112789}(525,\cdot)\) \(\chi_{112789}(526,\cdot)\) \(\chi_{112789}(541,\cdot)\) \(\chi_{112789}(572,\cdot)\) \(\chi_{112789}(573,\cdot)\) \(\chi_{112789}(582,\cdot)\) \(\chi_{112789}(633,\cdot)\) \(\chi_{112789}(740,\cdot)\) \(\chi_{112789}(755,\cdot)\) \(\chi_{112789}(756,\cdot)\) \(\chi_{112789}(769,\cdot)\) \(\chi_{112789}(826,\cdot)\) \(\chi_{112789}(830,\cdot)\) ...

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: $\Q(\zeta_{18060})$
Fixed field: Number field defined by a degree 18060 polynomial (not computed)

Values on generators

\((59171,83206)\) → \((e\left(\frac{29}{903}\right),e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 112789 }(281, a) \) \(-1\)\(1\)\(e\left(\frac{453}{6020}\right)\)\(e\left(\frac{8417}{9030}\right)\)\(e\left(\frac{453}{3010}\right)\)\(e\left(\frac{4499}{9030}\right)\)\(e\left(\frac{19}{2580}\right)\)\(e\left(\frac{1853}{2580}\right)\)\(e\left(\frac{1359}{6020}\right)\)\(e\left(\frac{3902}{4515}\right)\)\(e\left(\frac{10357}{18060}\right)\)\(e\left(\frac{691}{1204}\right)\)
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)
 
\( \chi_{ 112789 }(281,a) \;\) at \(\;a = \) e.g. 2