Properties

Label 1470.59
Modulus $1470$
Conductor $735$
Order $42$
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(1470)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,21,13]))
 
pari: [g,chi] = znchar(Mod(59,1470))
 

Basic properties

Modulus: \(1470\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{735}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1470.bm

\(\chi_{1470}(59,\cdot)\) \(\chi_{1470}(89,\cdot)\) \(\chi_{1470}(269,\cdot)\) \(\chi_{1470}(299,\cdot)\) \(\chi_{1470}(479,\cdot)\) \(\chi_{1470}(689,\cdot)\) \(\chi_{1470}(719,\cdot)\) \(\chi_{1470}(899,\cdot)\) \(\chi_{1470}(929,\cdot)\) \(\chi_{1470}(1139,\cdot)\) \(\chi_{1470}(1319,\cdot)\) \(\chi_{1470}(1349,\cdot)\)

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

\((491,1177,1081)\) → \((-1,-1,e\left(\frac{13}{42}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{14}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.42.589475176645907082922286550311127085690444572711075874815443834048428072939395904541015625.1