Properties

Label 1792.9
Modulus $1792$
Conductor $896$
Order $96$
Real no
Primitive no
Minimal no
Parity even

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([0,9,32]))
 
pari: [g,chi] = znchar(Mod(9,1792))
 

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 \(\chi_{896}(765,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.by

\(\chi_{1792}(9,\cdot)\) \(\chi_{1792}(25,\cdot)\) \(\chi_{1792}(121,\cdot)\) \(\chi_{1792}(137,\cdot)\) \(\chi_{1792}(233,\cdot)\) \(\chi_{1792}(249,\cdot)\) \(\chi_{1792}(345,\cdot)\) \(\chi_{1792}(361,\cdot)\) \(\chi_{1792}(457,\cdot)\) \(\chi_{1792}(473,\cdot)\) \(\chi_{1792}(569,\cdot)\) \(\chi_{1792}(585,\cdot)\) \(\chi_{1792}(681,\cdot)\) \(\chi_{1792}(697,\cdot)\) \(\chi_{1792}(793,\cdot)\) \(\chi_{1792}(809,\cdot)\) \(\chi_{1792}(905,\cdot)\) \(\chi_{1792}(921,\cdot)\) \(\chi_{1792}(1017,\cdot)\) \(\chi_{1792}(1033,\cdot)\) \(\chi_{1792}(1129,\cdot)\) \(\chi_{1792}(1145,\cdot)\) \(\chi_{1792}(1241,\cdot)\) \(\chi_{1792}(1257,\cdot)\) \(\chi_{1792}(1353,\cdot)\) \(\chi_{1792}(1369,\cdot)\) \(\chi_{1792}(1465,\cdot)\) \(\chi_{1792}(1481,\cdot)\) \(\chi_{1792}(1577,\cdot)\) \(\chi_{1792}(1593,\cdot)\) ...

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

\((1023,1541,1025)\) → \((1,e\left(\frac{3}{32}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{59}{96}\right)\)\(e\left(\frac{73}{96}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{29}{96}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{79}{96}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{25}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(9,a) \;\) at \(\;a = \) e.g. 2