Properties

Label 1792.1525
Modulus $1792$
Conductor $1792$
Order $64$
Real no
Primitive yes
Minimal yes
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(64))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,21,32]))
 
pari: [g,chi] = znchar(Mod(1525,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(1792\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bt

\(\chi_{1792}(13,\cdot)\) \(\chi_{1792}(69,\cdot)\) \(\chi_{1792}(125,\cdot)\) \(\chi_{1792}(181,\cdot)\) \(\chi_{1792}(237,\cdot)\) \(\chi_{1792}(293,\cdot)\) \(\chi_{1792}(349,\cdot)\) \(\chi_{1792}(405,\cdot)\) \(\chi_{1792}(461,\cdot)\) \(\chi_{1792}(517,\cdot)\) \(\chi_{1792}(573,\cdot)\) \(\chi_{1792}(629,\cdot)\) \(\chi_{1792}(685,\cdot)\) \(\chi_{1792}(741,\cdot)\) \(\chi_{1792}(797,\cdot)\) \(\chi_{1792}(853,\cdot)\) \(\chi_{1792}(909,\cdot)\) \(\chi_{1792}(965,\cdot)\) \(\chi_{1792}(1021,\cdot)\) \(\chi_{1792}(1077,\cdot)\) \(\chi_{1792}(1133,\cdot)\) \(\chi_{1792}(1189,\cdot)\) \(\chi_{1792}(1245,\cdot)\) \(\chi_{1792}(1301,\cdot)\) \(\chi_{1792}(1357,\cdot)\) \(\chi_{1792}(1413,\cdot)\) \(\chi_{1792}(1469,\cdot)\) \(\chi_{1792}(1525,\cdot)\) \(\chi_{1792}(1581,\cdot)\) \(\chi_{1792}(1637,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((1023,1541,1025)\) → \((1,e\left(\frac{21}{64}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{63}{64}\right)\)\(e\left(\frac{53}{64}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{59}{64}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{3}{64}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{21}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(1525,a) \;\) at \(\;a = \) e.g. 2