Properties

Label 2304.7
Modulus $2304$
Conductor $1152$
Order $96$
Real no
Primitive no
Minimal no
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(96))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([48,15,64]))
 
pari: [g,chi] = znchar(Mod(7,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(1152\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(96\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1152}(331,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.by

\(\chi_{2304}(7,\cdot)\) \(\chi_{2304}(103,\cdot)\) \(\chi_{2304}(151,\cdot)\) \(\chi_{2304}(247,\cdot)\) \(\chi_{2304}(295,\cdot)\) \(\chi_{2304}(391,\cdot)\) \(\chi_{2304}(439,\cdot)\) \(\chi_{2304}(535,\cdot)\) \(\chi_{2304}(583,\cdot)\) \(\chi_{2304}(679,\cdot)\) \(\chi_{2304}(727,\cdot)\) \(\chi_{2304}(823,\cdot)\) \(\chi_{2304}(871,\cdot)\) \(\chi_{2304}(967,\cdot)\) \(\chi_{2304}(1015,\cdot)\) \(\chi_{2304}(1111,\cdot)\) \(\chi_{2304}(1159,\cdot)\) \(\chi_{2304}(1255,\cdot)\) \(\chi_{2304}(1303,\cdot)\) \(\chi_{2304}(1399,\cdot)\) \(\chi_{2304}(1447,\cdot)\) \(\chi_{2304}(1543,\cdot)\) \(\chi_{2304}(1591,\cdot)\) \(\chi_{2304}(1687,\cdot)\) \(\chi_{2304}(1735,\cdot)\) \(\chi_{2304}(1831,\cdot)\) \(\chi_{2304}(1879,\cdot)\) \(\chi_{2304}(1975,\cdot)\) \(\chi_{2304}(2023,\cdot)\) \(\chi_{2304}(2119,\cdot)\) ...

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: $\Q(\zeta_{96})$
Fixed field: Number field defined by a degree 96 polynomial

Values on generators

\((1279,2053,1793)\) → \((-1,e\left(\frac{5}{32}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{47}{96}\right)\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{43}{96}\right)\)\(e\left(\frac{65}{96}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{85}{96}\right)\)\(e\left(\frac{1}{12}\right)\)
value at e.g. 2