Properties

Label 4410.323
Modulus $4410$
Conductor $735$
Order $28$
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(4410)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,21,12]))
 
pari: [g,chi] = znchar(Mod(323,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{735}(323,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.cw

\(\chi_{4410}(323,\cdot)\) \(\chi_{4410}(827,\cdot)\) \(\chi_{4410}(953,\cdot)\) \(\chi_{4410}(1457,\cdot)\) \(\chi_{4410}(1583,\cdot)\) \(\chi_{4410}(2087,\cdot)\) \(\chi_{4410}(2213,\cdot)\) \(\chi_{4410}(2717,\cdot)\) \(\chi_{4410}(3347,\cdot)\) \(\chi_{4410}(3473,\cdot)\) \(\chi_{4410}(3977,\cdot)\) \(\chi_{4410}(4103,\cdot)\)

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

\((3431,2647,1081)\) → \((-1,-i,e\left(\frac{3}{7}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(-1\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{5}{7}\right)\)\(1\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{23}{28}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: 28.28.83709274472667248662717543234670163936409630856037139892578125.1