Properties

Label 1470.121
Modulus $1470$
Conductor $49$
Order $21$
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([0,0,19]))
 
pari: [g,chi] = znchar(Mod(121,1470))
 

Basic properties

Modulus: \(1470\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1470.bg

\(\chi_{1470}(121,\cdot)\) \(\chi_{1470}(151,\cdot)\) \(\chi_{1470}(331,\cdot)\) \(\chi_{1470}(541,\cdot)\) \(\chi_{1470}(571,\cdot)\) \(\chi_{1470}(751,\cdot)\) \(\chi_{1470}(781,\cdot)\) \(\chi_{1470}(991,\cdot)\) \(\chi_{1470}(1171,\cdot)\) \(\chi_{1470}(1201,\cdot)\) \(\chi_{1470}(1381,\cdot)\) \(\chi_{1470}(1411,\cdot)\)

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

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

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})^+\)