Properties

Label 1470.73
Modulus $1470$
Conductor $245$
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([0,63,74]))
 
pari: [g,chi] = znchar(Mod(73,1470))
 

Basic properties

Modulus: \(1470\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(73,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1470.bu

\(\chi_{1470}(73,\cdot)\) \(\chi_{1470}(103,\cdot)\) \(\chi_{1470}(157,\cdot)\) \(\chi_{1470}(187,\cdot)\) \(\chi_{1470}(283,\cdot)\) \(\chi_{1470}(367,\cdot)\) \(\chi_{1470}(397,\cdot)\) \(\chi_{1470}(493,\cdot)\) \(\chi_{1470}(523,\cdot)\) \(\chi_{1470}(577,\cdot)\) \(\chi_{1470}(703,\cdot)\) \(\chi_{1470}(733,\cdot)\) \(\chi_{1470}(787,\cdot)\) \(\chi_{1470}(817,\cdot)\) \(\chi_{1470}(943,\cdot)\) \(\chi_{1470}(997,\cdot)\) \(\chi_{1470}(1027,\cdot)\) \(\chi_{1470}(1123,\cdot)\) \(\chi_{1470}(1153,\cdot)\) \(\chi_{1470}(1237,\cdot)\) \(\chi_{1470}(1333,\cdot)\) \(\chi_{1470}(1363,\cdot)\) \(\chi_{1470}(1417,\cdot)\) \(\chi_{1470}(1447,\cdot)\)

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

\((491,1177,1081)\) → \((1,-i,e\left(\frac{37}{42}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{65}{84}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{15}{28}\right)\)
value at e.g. 2

Related number fields

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