Properties

Label 2850.1493
Modulus $2850$
Conductor $285$
Order $12$
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(2850, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,9,8]))
 
pari: [g,chi] = znchar(Mod(1493,2850))
 

Basic properties

Modulus: \(2850\)
Conductor: \(285\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{285}(68,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2850.bd

\(\chi_{2850}(1493,\cdot)\) \(\chi_{2850}(1607,\cdot)\) \(\chi_{2850}(1793,\cdot)\) \(\chi_{2850}(1907,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.24181674720486328125.1

Values on generators

\((1901,1027,1351)\) → \((-1,-i,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(-i\)\(-1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(-i\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)
value at e.g. 2