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Properties

Label 68890.501
Modulus $68890$
Conductor $6889$
Order $3403$
Real no
Primitive no
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(68890, base_ring=CyclotomicField(6806)) M = H._module chi = DirichletCharacter(H, M([0,482]))
 
Copy content gp:[g,chi] = znchar(Mod(501, 68890))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("68890.501");
 

Basic properties

Modulus: \(68890\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6889\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3403\)
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 \(\chi_{6889}(501,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 68890.s

\(\chi_{68890}(11,\cdot)\) \(\chi_{68890}(21,\cdot)\) \(\chi_{68890}(31,\cdot)\) \(\chi_{68890}(41,\cdot)\) \(\chi_{68890}(51,\cdot)\) \(\chi_{68890}(61,\cdot)\) \(\chi_{68890}(81,\cdot)\) \(\chi_{68890}(111,\cdot)\) \(\chi_{68890}(121,\cdot)\) \(\chi_{68890}(131,\cdot)\) \(\chi_{68890}(151,\cdot)\) \(\chi_{68890}(191,\cdot)\) \(\chi_{68890}(231,\cdot)\) \(\chi_{68890}(241,\cdot)\) \(\chi_{68890}(261,\cdot)\) \(\chi_{68890}(341,\cdot)\) \(\chi_{68890}(361,\cdot)\) \(\chi_{68890}(381,\cdot)\) \(\chi_{68890}(391,\cdot)\) \(\chi_{68890}(431,\cdot)\) \(\chi_{68890}(441,\cdot)\) \(\chi_{68890}(451,\cdot)\) \(\chi_{68890}(501,\cdot)\) \(\chi_{68890}(521,\cdot)\) \(\chi_{68890}(531,\cdot)\) \(\chi_{68890}(561,\cdot)\) \(\chi_{68890}(591,\cdot)\) \(\chi_{68890}(611,\cdot)\) \(\chi_{68890}(621,\cdot)\) \(\chi_{68890}(651,\cdot)\) ...

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: $\Q(\zeta_{3403})$
Fixed field: Number field defined by a degree 3403 polynomial (not computed)

Values on generators

\((27557,6891)\) → \((1,e\left(\frac{241}{3403}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 68890 }(501, a) \) \(1\)\(1\)\(e\left(\frac{2756}{3403}\right)\)\(e\left(\frac{2420}{3403}\right)\)\(e\left(\frac{2109}{3403}\right)\)\(e\left(\frac{3037}{3403}\right)\)\(e\left(\frac{3264}{3403}\right)\)\(e\left(\frac{3164}{3403}\right)\)\(e\left(\frac{2102}{3403}\right)\)\(e\left(\frac{1773}{3403}\right)\)\(e\left(\frac{3390}{3403}\right)\)\(e\left(\frac{1462}{3403}\right)\)
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)
 
\( \chi_{ 68890 }(501,a) \;\) at \(\;a = \) e.g. 2