Properties

Label 2304.5
Modulus $2304$
Conductor $2304$
Order $192$
Real no
Primitive yes
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(192))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,160]))
 
pari: [g,chi] = znchar(Mod(5,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(2304\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(192\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.cb

\(\chi_{2304}(5,\cdot)\) \(\chi_{2304}(29,\cdot)\) \(\chi_{2304}(77,\cdot)\) \(\chi_{2304}(101,\cdot)\) \(\chi_{2304}(149,\cdot)\) \(\chi_{2304}(173,\cdot)\) \(\chi_{2304}(221,\cdot)\) \(\chi_{2304}(245,\cdot)\) \(\chi_{2304}(293,\cdot)\) \(\chi_{2304}(317,\cdot)\) \(\chi_{2304}(365,\cdot)\) \(\chi_{2304}(389,\cdot)\) \(\chi_{2304}(437,\cdot)\) \(\chi_{2304}(461,\cdot)\) \(\chi_{2304}(509,\cdot)\) \(\chi_{2304}(533,\cdot)\) \(\chi_{2304}(581,\cdot)\) \(\chi_{2304}(605,\cdot)\) \(\chi_{2304}(653,\cdot)\) \(\chi_{2304}(677,\cdot)\) \(\chi_{2304}(725,\cdot)\) \(\chi_{2304}(749,\cdot)\) \(\chi_{2304}(797,\cdot)\) \(\chi_{2304}(821,\cdot)\) \(\chi_{2304}(869,\cdot)\) \(\chi_{2304}(893,\cdot)\) \(\chi_{2304}(941,\cdot)\) \(\chi_{2304}(965,\cdot)\) \(\chi_{2304}(1013,\cdot)\) \(\chi_{2304}(1037,\cdot)\) ...

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: $\Q(\zeta_{192})$
Fixed field: Number field defined by a degree 192 polynomial (not computed)

Values on generators

\((1279,2053,1793)\) → \((1,e\left(\frac{1}{64}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{35}{192}\right)\)\(e\left(\frac{47}{96}\right)\)\(e\left(\frac{31}{192}\right)\)\(e\left(\frac{77}{192}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{37}{96}\right)\)\(e\left(\frac{35}{96}\right)\)\(e\left(\frac{145}{192}\right)\)\(e\left(\frac{19}{24}\right)\)
value at e.g. 2