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Properties

Label 20992.701
Modulus $20992$
Conductor $20992$
Order $640$
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(20992, base_ring=CyclotomicField(640)) M = H._module chi = DirichletCharacter(H, M([0,255,192]))
 
Copy content gp:[g,chi] = znchar(Mod(701, 20992))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20992.701");
 

Basic properties

Modulus: \(20992\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(20992\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(640\)
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 20992.hy

\(\chi_{20992}(45,\cdot)\) \(\chi_{20992}(189,\cdot)\) \(\chi_{20992}(269,\cdot)\) \(\chi_{20992}(277,\cdot)\) \(\chi_{20992}(373,\cdot)\) \(\chi_{20992}(517,\cdot)\) \(\chi_{20992}(597,\cdot)\) \(\chi_{20992}(605,\cdot)\) \(\chi_{20992}(701,\cdot)\) \(\chi_{20992}(845,\cdot)\) \(\chi_{20992}(925,\cdot)\) \(\chi_{20992}(933,\cdot)\) \(\chi_{20992}(1029,\cdot)\) \(\chi_{20992}(1173,\cdot)\) \(\chi_{20992}(1253,\cdot)\) \(\chi_{20992}(1261,\cdot)\) \(\chi_{20992}(1357,\cdot)\) \(\chi_{20992}(1501,\cdot)\) \(\chi_{20992}(1581,\cdot)\) \(\chi_{20992}(1589,\cdot)\) \(\chi_{20992}(1685,\cdot)\) \(\chi_{20992}(1829,\cdot)\) \(\chi_{20992}(1909,\cdot)\) \(\chi_{20992}(1917,\cdot)\) \(\chi_{20992}(2013,\cdot)\) \(\chi_{20992}(2157,\cdot)\) \(\chi_{20992}(2237,\cdot)\) \(\chi_{20992}(2245,\cdot)\) \(\chi_{20992}(2341,\cdot)\) \(\chi_{20992}(2485,\cdot)\) ...

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: $\Q(\zeta_{640})$
Fixed field: Number field defined by a degree 640 polynomial (not computed)

Values on generators

\((18943,4101,15873)\) → \((1,e\left(\frac{51}{128}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 20992 }(701, a) \) \(1\)\(1\)\(e\left(\frac{57}{128}\right)\)\(e\left(\frac{639}{640}\right)\)\(e\left(\frac{59}{320}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{491}{640}\right)\)\(e\left(\frac{337}{640}\right)\)\(e\left(\frac{71}{160}\right)\)\(e\left(\frac{9}{160}\right)\)\(e\left(\frac{553}{640}\right)\)\(e\left(\frac{403}{640}\right)\)
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)
 
\( \chi_{ 20992 }(701,a) \;\) at \(\;a = \) e.g. 2