Properties

Label 1792.71
Modulus $1792$
Conductor $128$
Order $32$
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(32))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([16,13,0]))
 
pari: [g,chi] = znchar(Mod(71,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(128\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{128}(107,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bl

\(\chi_{1792}(71,\cdot)\) \(\chi_{1792}(183,\cdot)\) \(\chi_{1792}(295,\cdot)\) \(\chi_{1792}(407,\cdot)\) \(\chi_{1792}(519,\cdot)\) \(\chi_{1792}(631,\cdot)\) \(\chi_{1792}(743,\cdot)\) \(\chi_{1792}(855,\cdot)\) \(\chi_{1792}(967,\cdot)\) \(\chi_{1792}(1079,\cdot)\) \(\chi_{1792}(1191,\cdot)\) \(\chi_{1792}(1303,\cdot)\) \(\chi_{1792}(1415,\cdot)\) \(\chi_{1792}(1527,\cdot)\) \(\chi_{1792}(1639,\cdot)\) \(\chi_{1792}(1751,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.0.3138550867693340381917894711603833208051177722232017256448.1

Values on generators

\((1023,1541,1025)\) → \((-1,e\left(\frac{13}{32}\right),1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{13}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(71,a) \;\) at \(\;a = \) e.g. 2