Properties

Label 6003.82
Modulus $6003$
Conductor $667$
Order $77$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(6003)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,49,22]))
 
pari: [g,chi] = znchar(Mod(82,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(77\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(82,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.cm

\(\chi_{6003}(82,\cdot)\) \(\chi_{6003}(190,\cdot)\) \(\chi_{6003}(397,\cdot)\) \(\chi_{6003}(487,\cdot)\) \(\chi_{6003}(604,\cdot)\) \(\chi_{6003}(703,\cdot)\) \(\chi_{6003}(721,\cdot)\) \(\chi_{6003}(748,\cdot)\) \(\chi_{6003}(982,\cdot)\) \(\chi_{6003}(1225,\cdot)\) \(\chi_{6003}(1504,\cdot)\) \(\chi_{6003}(1531,\cdot)\) \(\chi_{6003}(1756,\cdot)\) \(\chi_{6003}(1963,\cdot)\) \(\chi_{6003}(2017,\cdot)\) \(\chi_{6003}(2026,\cdot)\) \(\chi_{6003}(2053,\cdot)\) \(\chi_{6003}(2170,\cdot)\) \(\chi_{6003}(2224,\cdot)\) \(\chi_{6003}(2431,\cdot)\) \(\chi_{6003}(2539,\cdot)\) \(\chi_{6003}(2548,\cdot)\) \(\chi_{6003}(2746,\cdot)\) \(\chi_{6003}(2791,\cdot)\) \(\chi_{6003}(2809,\cdot)\) \(\chi_{6003}(2953,\cdot)\) \(\chi_{6003}(3052,\cdot)\) \(\chi_{6003}(3061,\cdot)\) \(\chi_{6003}(3268,\cdot)\) \(\chi_{6003}(3475,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((668,3133,4555)\) → \((1,e\left(\frac{7}{11}\right),e\left(\frac{2}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{43}{77}\right)\)\(e\left(\frac{9}{77}\right)\)\(e\left(\frac{71}{77}\right)\)\(e\left(\frac{40}{77}\right)\)\(e\left(\frac{52}{77}\right)\)\(e\left(\frac{37}{77}\right)\)\(e\left(\frac{67}{77}\right)\)\(e\left(\frac{4}{77}\right)\)\(e\left(\frac{6}{77}\right)\)\(e\left(\frac{18}{77}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 77 polynomial