Properties

Label 1792.73
Modulus $1792$
Conductor $896$
Order $96$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1792, base_ring=CyclotomicField(96))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,81,16]))
 
pari: [g,chi] = znchar(Mod(73,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(896\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(96\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{896}(157,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bz

\(\chi_{1792}(73,\cdot)\) \(\chi_{1792}(89,\cdot)\) \(\chi_{1792}(185,\cdot)\) \(\chi_{1792}(201,\cdot)\) \(\chi_{1792}(297,\cdot)\) \(\chi_{1792}(313,\cdot)\) \(\chi_{1792}(409,\cdot)\) \(\chi_{1792}(425,\cdot)\) \(\chi_{1792}(521,\cdot)\) \(\chi_{1792}(537,\cdot)\) \(\chi_{1792}(633,\cdot)\) \(\chi_{1792}(649,\cdot)\) \(\chi_{1792}(745,\cdot)\) \(\chi_{1792}(761,\cdot)\) \(\chi_{1792}(857,\cdot)\) \(\chi_{1792}(873,\cdot)\) \(\chi_{1792}(969,\cdot)\) \(\chi_{1792}(985,\cdot)\) \(\chi_{1792}(1081,\cdot)\) \(\chi_{1792}(1097,\cdot)\) \(\chi_{1792}(1193,\cdot)\) \(\chi_{1792}(1209,\cdot)\) \(\chi_{1792}(1305,\cdot)\) \(\chi_{1792}(1321,\cdot)\) \(\chi_{1792}(1417,\cdot)\) \(\chi_{1792}(1433,\cdot)\) \(\chi_{1792}(1529,\cdot)\) \(\chi_{1792}(1545,\cdot)\) \(\chi_{1792}(1641,\cdot)\) \(\chi_{1792}(1657,\cdot)\) ...

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

\((1023,1541,1025)\) → \((1,e\left(\frac{27}{32}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{67}{96}\right)\)\(e\left(\frac{65}{96}\right)\)\(e\left(\frac{19}{48}\right)\)\(e\left(\frac{37}{96}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{23}{96}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{17}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(73,a) \;\) at \(\;a = \) e.g. 2