Properties

Label 1470.23
Modulus $1470$
Conductor $735$
Order $84$
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([42,63,76]))
 
pari: [g,chi] = znchar(Mod(23,1470))
 

Basic properties

Modulus: \(1470\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{735}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1470.bt

\(\chi_{1470}(23,\cdot)\) \(\chi_{1470}(53,\cdot)\) \(\chi_{1470}(107,\cdot)\) \(\chi_{1470}(137,\cdot)\) \(\chi_{1470}(233,\cdot)\) \(\chi_{1470}(317,\cdot)\) \(\chi_{1470}(347,\cdot)\) \(\chi_{1470}(443,\cdot)\) \(\chi_{1470}(473,\cdot)\) \(\chi_{1470}(527,\cdot)\) \(\chi_{1470}(653,\cdot)\) \(\chi_{1470}(683,\cdot)\) \(\chi_{1470}(737,\cdot)\) \(\chi_{1470}(767,\cdot)\) \(\chi_{1470}(893,\cdot)\) \(\chi_{1470}(947,\cdot)\) \(\chi_{1470}(977,\cdot)\) \(\chi_{1470}(1073,\cdot)\) \(\chi_{1470}(1103,\cdot)\) \(\chi_{1470}(1187,\cdot)\) \(\chi_{1470}(1283,\cdot)\) \(\chi_{1470}(1313,\cdot)\) \(\chi_{1470}(1367,\cdot)\) \(\chi_{1470}(1397,\cdot)\)

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

\((491,1177,1081)\) → \((-1,-i,e\left(\frac{19}{21}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{19}{28}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial