Properties

Label 1792.41
Modulus $1792$
Conductor $896$
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([0,31,16]))
 
pari: [g,chi] = znchar(Mod(41,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(896\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{896}(461,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bm

\(\chi_{1792}(41,\cdot)\) \(\chi_{1792}(153,\cdot)\) \(\chi_{1792}(265,\cdot)\) \(\chi_{1792}(377,\cdot)\) \(\chi_{1792}(489,\cdot)\) \(\chi_{1792}(601,\cdot)\) \(\chi_{1792}(713,\cdot)\) \(\chi_{1792}(825,\cdot)\) \(\chi_{1792}(937,\cdot)\) \(\chi_{1792}(1049,\cdot)\) \(\chi_{1792}(1161,\cdot)\) \(\chi_{1792}(1273,\cdot)\) \(\chi_{1792}(1385,\cdot)\) \(\chi_{1792}(1497,\cdot)\) \(\chi_{1792}(1609,\cdot)\) \(\chi_{1792}(1721,\cdot)\)

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: 32.0.104303243075213755167445035578915122359095224799654955003407693930037248.1

Values on generators

\((1023,1541,1025)\) → \((1,e\left(\frac{31}{32}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{15}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(41,a) \;\) at \(\;a = \) e.g. 2