Properties

Label 1470.109
Modulus $1470$
Conductor $245$
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([0,21,40]))
 
pari: [g,chi] = znchar(Mod(109,1470))
 

Basic properties

Modulus: \(1470\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(109,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1470.bq

\(\chi_{1470}(109,\cdot)\) \(\chi_{1470}(289,\cdot)\) \(\chi_{1470}(319,\cdot)\) \(\chi_{1470}(499,\cdot)\) \(\chi_{1470}(529,\cdot)\) \(\chi_{1470}(709,\cdot)\) \(\chi_{1470}(739,\cdot)\) \(\chi_{1470}(919,\cdot)\) \(\chi_{1470}(1129,\cdot)\) \(\chi_{1470}(1159,\cdot)\) \(\chi_{1470}(1339,\cdot)\) \(\chi_{1470}(1369,\cdot)\)

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

\((491,1177,1081)\) → \((1,-1,e\left(\frac{20}{21}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{14}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1