Properties

Label 6003.215
Modulus $6003$
Conductor $2001$
Order $44$
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([22,12,11]))
 
pari: [g,chi] = znchar(Mod(215,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2001}(215,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.cb

\(\chi_{6003}(215,\cdot)\) \(\chi_{6003}(278,\cdot)\) \(\chi_{6003}(476,\cdot)\) \(\chi_{6003}(800,\cdot)\) \(\chi_{6003}(998,\cdot)\) \(\chi_{6003}(1061,\cdot)\) \(\chi_{6003}(1520,\cdot)\) \(\chi_{6003}(1844,\cdot)\) \(\chi_{6003}(2042,\cdot)\) \(\chi_{6003}(2105,\cdot)\) \(\chi_{6003}(2303,\cdot)\) \(\chi_{6003}(2888,\cdot)\) \(\chi_{6003}(3086,\cdot)\) \(\chi_{6003}(3347,\cdot)\) \(\chi_{6003}(3410,\cdot)\) \(\chi_{6003}(4130,\cdot)\) \(\chi_{6003}(4652,\cdot)\) \(\chi_{6003}(4976,\cdot)\) \(\chi_{6003}(5237,\cdot)\) \(\chi_{6003}(5759,\cdot)\)

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

\((668,3133,4555)\) → \((-1,e\left(\frac{3}{11}\right),i)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{2}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial