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Properties

Label 2159.1067
Modulus $2159$
Conductor $2159$
Order $84$
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(2159, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([21,26]))
 
Copy content gp:[g,chi] = znchar(Mod(1067, 2159))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2159.1067");
 

Basic properties

Modulus: \(2159\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2159\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 2159.bq

\(\chi_{2159}(89,\cdot)\) \(\chi_{2159}(132,\cdot)\) \(\chi_{2159}(259,\cdot)\) \(\chi_{2159}(421,\cdot)\) \(\chi_{2159}(548,\cdot)\) \(\chi_{2159}(574,\cdot)\) \(\chi_{2159}(701,\cdot)\) \(\chi_{2159}(795,\cdot)\) \(\chi_{2159}(922,\cdot)\) \(\chi_{2159}(1067,\cdot)\) \(\chi_{2159}(1118,\cdot)\) \(\chi_{2159}(1194,\cdot)\) \(\chi_{2159}(1220,\cdot)\) \(\chi_{2159}(1245,\cdot)\) \(\chi_{2159}(1347,\cdot)\) \(\chi_{2159}(1407,\cdot)\) \(\chi_{2159}(1424,\cdot)\) \(\chi_{2159}(1534,\cdot)\) \(\chi_{2159}(1551,\cdot)\) \(\chi_{2159}(1832,\cdot)\) \(\chi_{2159}(1959,\cdot)\) \(\chi_{2159}(1985,\cdot)\) \(\chi_{2159}(2112,\cdot)\) \(\chi_{2159}(2121,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1652,511)\) → \((i,e\left(\frac{13}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2159 }(1067, a) \) \(-1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{67}{84}\right)\)
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)
 
\( \chi_{ 2159 }(1067,a) \;\) at \(\;a = \) e.g. 2