Properties

Label 1152.7
Modulus $1152$
Conductor $576$
Order $48$
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(1152, base_ring=CyclotomicField(48))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([24,15,32]))
 
pari: [g,chi] = znchar(Mod(7,1152))
 

Basic properties

Modulus: \(1152\)
Conductor: \(576\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{576}(331,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1152.bo

\(\chi_{1152}(7,\cdot)\) \(\chi_{1152}(103,\cdot)\) \(\chi_{1152}(151,\cdot)\) \(\chi_{1152}(247,\cdot)\) \(\chi_{1152}(295,\cdot)\) \(\chi_{1152}(391,\cdot)\) \(\chi_{1152}(439,\cdot)\) \(\chi_{1152}(535,\cdot)\) \(\chi_{1152}(583,\cdot)\) \(\chi_{1152}(679,\cdot)\) \(\chi_{1152}(727,\cdot)\) \(\chi_{1152}(823,\cdot)\) \(\chi_{1152}(871,\cdot)\) \(\chi_{1152}(967,\cdot)\) \(\chi_{1152}(1015,\cdot)\) \(\chi_{1152}(1111,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((127,901,641)\) → \((-1,e\left(\frac{5}{16}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{1}{48}\right)\)\(-i\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2