Properties

Label 2304.19
Modulus $2304$
Conductor $256$
Order $64$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2304, base_ring=CyclotomicField(64))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([32,23,0]))
 
pari: [g,chi] = znchar(Mod(19,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(256\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{256}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.bv

\(\chi_{2304}(19,\cdot)\) \(\chi_{2304}(91,\cdot)\) \(\chi_{2304}(163,\cdot)\) \(\chi_{2304}(235,\cdot)\) \(\chi_{2304}(307,\cdot)\) \(\chi_{2304}(379,\cdot)\) \(\chi_{2304}(451,\cdot)\) \(\chi_{2304}(523,\cdot)\) \(\chi_{2304}(595,\cdot)\) \(\chi_{2304}(667,\cdot)\) \(\chi_{2304}(739,\cdot)\) \(\chi_{2304}(811,\cdot)\) \(\chi_{2304}(883,\cdot)\) \(\chi_{2304}(955,\cdot)\) \(\chi_{2304}(1027,\cdot)\) \(\chi_{2304}(1099,\cdot)\) \(\chi_{2304}(1171,\cdot)\) \(\chi_{2304}(1243,\cdot)\) \(\chi_{2304}(1315,\cdot)\) \(\chi_{2304}(1387,\cdot)\) \(\chi_{2304}(1459,\cdot)\) \(\chi_{2304}(1531,\cdot)\) \(\chi_{2304}(1603,\cdot)\) \(\chi_{2304}(1675,\cdot)\) \(\chi_{2304}(1747,\cdot)\) \(\chi_{2304}(1819,\cdot)\) \(\chi_{2304}(1891,\cdot)\) \(\chi_{2304}(1963,\cdot)\) \(\chi_{2304}(2035,\cdot)\) \(\chi_{2304}(2107,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((1279,2053,1793)\) → \((-1,e\left(\frac{23}{64}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{3}{64}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{49}{64}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{13}{64}\right)\)\(e\left(\frac{3}{8}\right)\)
value at e.g. 2