Properties

Label 6003.10
Modulus $6003$
Conductor $667$
Order $308$
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([0,42,253]))
 
pari: [g,chi] = znchar(Mod(10,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(10,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.df

\(\chi_{6003}(10,\cdot)\) \(\chi_{6003}(19,\cdot)\) \(\chi_{6003}(37,\cdot)\) \(\chi_{6003}(172,\cdot)\) \(\chi_{6003}(217,\cdot)\) \(\chi_{6003}(235,\cdot)\) \(\chi_{6003}(316,\cdot)\) \(\chi_{6003}(379,\cdot)\) \(\chi_{6003}(388,\cdot)\) \(\chi_{6003}(424,\cdot)\) \(\chi_{6003}(433,\cdot)\) \(\chi_{6003}(559,\cdot)\) \(\chi_{6003}(595,\cdot)\) \(\chi_{6003}(640,\cdot)\) \(\chi_{6003}(649,\cdot)\) \(\chi_{6003}(757,\cdot)\) \(\chi_{6003}(793,\cdot)\) \(\chi_{6003}(802,\cdot)\) \(\chi_{6003}(820,\cdot)\) \(\chi_{6003}(838,\cdot)\) \(\chi_{6003}(856,\cdot)\) \(\chi_{6003}(1000,\cdot)\) \(\chi_{6003}(1054,\cdot)\) \(\chi_{6003}(1063,\cdot)\) \(\chi_{6003}(1171,\cdot)\) \(\chi_{6003}(1207,\cdot)\) \(\chi_{6003}(1216,\cdot)\) \(\chi_{6003}(1261,\cdot)\) \(\chi_{6003}(1279,\cdot)\) \(\chi_{6003}(1378,\cdot)\) ...

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

\((668,3133,4555)\) → \((1,e\left(\frac{3}{22}\right),e\left(\frac{23}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{29}{308}\right)\)\(e\left(\frac{29}{154}\right)\)\(e\left(\frac{16}{77}\right)\)\(e\left(\frac{69}{154}\right)\)\(e\left(\frac{87}{308}\right)\)\(e\left(\frac{93}{308}\right)\)\(e\left(\frac{235}{308}\right)\)\(e\left(\frac{107}{154}\right)\)\(e\left(\frac{167}{308}\right)\)\(e\left(\frac{29}{77}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{308})$
Fixed field: Number field defined by a degree 308 polynomial