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Properties

Label 20808.3241
Modulus $20808$
Conductor $289$
Order $272$
Real no
Primitive no
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(20808, base_ring=CyclotomicField(272)) M = H._module chi = DirichletCharacter(H, M([0,0,0,199]))
 
Copy content gp:[g,chi] = znchar(Mod(3241, 20808))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20808.3241");
 

Basic properties

Modulus: \(20808\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(289\)
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: no, induced from \(\chi_{289}(62,\cdot)\)
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 20808.fj

\(\chi_{20808}(73,\cdot)\) \(\chi_{20808}(505,\cdot)\) \(\chi_{20808}(649,\cdot)\) \(\chi_{20808}(721,\cdot)\) \(\chi_{20808}(793,\cdot)\) \(\chi_{20808}(1009,\cdot)\) \(\chi_{20808}(1153,\cdot)\) \(\chi_{20808}(1297,\cdot)\) \(\chi_{20808}(1729,\cdot)\) \(\chi_{20808}(1873,\cdot)\) \(\chi_{20808}(1945,\cdot)\) \(\chi_{20808}(2017,\cdot)\) \(\chi_{20808}(2233,\cdot)\) \(\chi_{20808}(2305,\cdot)\) \(\chi_{20808}(2521,\cdot)\) \(\chi_{20808}(2953,\cdot)\) \(\chi_{20808}(3097,\cdot)\) \(\chi_{20808}(3169,\cdot)\) \(\chi_{20808}(3241,\cdot)\) \(\chi_{20808}(3457,\cdot)\) \(\chi_{20808}(3529,\cdot)\) \(\chi_{20808}(3601,\cdot)\) \(\chi_{20808}(3745,\cdot)\) \(\chi_{20808}(4321,\cdot)\) \(\chi_{20808}(4393,\cdot)\) \(\chi_{20808}(4465,\cdot)\) \(\chi_{20808}(4681,\cdot)\) \(\chi_{20808}(4753,\cdot)\) \(\chi_{20808}(4825,\cdot)\) \(\chi_{20808}(4969,\cdot)\) ...

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

\((15607,10405,18497,20233)\) → \((1,1,1,e\left(\frac{199}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 20808 }(3241, a) \) \(-1\)\(1\)\(e\left(\frac{147}{272}\right)\)\(e\left(\frac{109}{272}\right)\)\(e\left(\frac{225}{272}\right)\)\(e\left(\frac{27}{68}\right)\)\(e\left(\frac{33}{136}\right)\)\(e\left(\frac{41}{272}\right)\)\(e\left(\frac{11}{136}\right)\)\(e\left(\frac{123}{272}\right)\)\(e\left(\frac{159}{272}\right)\)\(e\left(\frac{16}{17}\right)\)
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)
 
\( \chi_{ 20808 }(3241,a) \;\) at \(\;a = \) e.g. 2