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Properties

Label 6815.3191
Modulus $6815$
Conductor $47$
Order $23$
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(6815, base_ring=CyclotomicField(46)) M = H._module chi = DirichletCharacter(H, M([0,0,24]))
 
Copy content gp:[g,chi] = znchar(Mod(3191, 6815))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6815.3191");
 

Basic properties

Modulus: \(6815\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(47\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(23\)
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 \(\chi_{47}(42,\cdot)\)
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 6815.bc

\(\chi_{6815}(291,\cdot)\) \(\chi_{6815}(581,\cdot)\) \(\chi_{6815}(726,\cdot)\) \(\chi_{6815}(871,\cdot)\) \(\chi_{6815}(1306,\cdot)\) \(\chi_{6815}(1741,\cdot)\) \(\chi_{6815}(1886,\cdot)\) \(\chi_{6815}(2176,\cdot)\) \(\chi_{6815}(2321,\cdot)\) \(\chi_{6815}(2901,\cdot)\) \(\chi_{6815}(3191,\cdot)\) \(\chi_{6815}(3481,\cdot)\) \(\chi_{6815}(3626,\cdot)\) \(\chi_{6815}(4351,\cdot)\) \(\chi_{6815}(5221,\cdot)\) \(\chi_{6815}(5366,\cdot)\) \(\chi_{6815}(5511,\cdot)\) \(\chi_{6815}(5656,\cdot)\) \(\chi_{6815}(5946,\cdot)\) \(\chi_{6815}(6091,\cdot)\) \(\chi_{6815}(6236,\cdot)\) \(\chi_{6815}(6381,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 23 polynomial

Values on generators

\((2727,2351,146)\) → \((1,1,e\left(\frac{12}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6815 }(3191, a) \) \(1\)\(1\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{17}{23}\right)\)
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)
 
\( \chi_{ 6815 }(3191,a) \;\) at \(\;a = \) e.g. 2