Properties

Label 1792.57
Modulus $1792$
Conductor $128$
Order $32$
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(32))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,29,0]))
 
pari: [g,chi] = znchar(Mod(57,1792))
 

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 \(\chi_{128}(85,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bn

\(\chi_{1792}(57,\cdot)\) \(\chi_{1792}(169,\cdot)\) \(\chi_{1792}(281,\cdot)\) \(\chi_{1792}(393,\cdot)\) \(\chi_{1792}(505,\cdot)\) \(\chi_{1792}(617,\cdot)\) \(\chi_{1792}(729,\cdot)\) \(\chi_{1792}(841,\cdot)\) \(\chi_{1792}(953,\cdot)\) \(\chi_{1792}(1065,\cdot)\) \(\chi_{1792}(1177,\cdot)\) \(\chi_{1792}(1289,\cdot)\) \(\chi_{1792}(1401,\cdot)\) \(\chi_{1792}(1513,\cdot)\) \(\chi_{1792}(1625,\cdot)\) \(\chi_{1792}(1737,\cdot)\)

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: \(\Q(\zeta_{128})^+\)

Values on generators

\((1023,1541,1025)\) → \((1,e\left(\frac{29}{32}\right),1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{13}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(57,a) \;\) at \(\;a = \) e.g. 2