Properties

Label 1077.16
Modulus $1077$
Conductor $359$
Order $179$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1077, base_ring=CyclotomicField(358))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,182]))
 
pari: [g,chi] = znchar(Mod(16,1077))
 

Basic properties

Modulus: \(1077\)
Conductor: \(359\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(179\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{359}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1077.e

\(\chi_{1077}(4,\cdot)\) \(\chi_{1077}(10,\cdot)\) \(\chi_{1077}(16,\cdot)\) \(\chi_{1077}(22,\cdot)\) \(\chi_{1077}(25,\cdot)\) \(\chi_{1077}(34,\cdot)\) \(\chi_{1077}(37,\cdot)\) \(\chi_{1077}(40,\cdot)\) \(\chi_{1077}(46,\cdot)\) \(\chi_{1077}(49,\cdot)\) \(\chi_{1077}(55,\cdot)\) \(\chi_{1077}(64,\cdot)\) \(\chi_{1077}(73,\cdot)\) \(\chi_{1077}(79,\cdot)\) \(\chi_{1077}(82,\cdot)\) \(\chi_{1077}(85,\cdot)\) \(\chi_{1077}(88,\cdot)\) \(\chi_{1077}(91,\cdot)\) \(\chi_{1077}(94,\cdot)\) \(\chi_{1077}(100,\cdot)\) \(\chi_{1077}(115,\cdot)\) \(\chi_{1077}(121,\cdot)\) \(\chi_{1077}(127,\cdot)\) \(\chi_{1077}(133,\cdot)\) \(\chi_{1077}(136,\cdot)\) \(\chi_{1077}(148,\cdot)\) \(\chi_{1077}(151,\cdot)\) \(\chi_{1077}(160,\cdot)\) \(\chi_{1077}(169,\cdot)\) \(\chi_{1077}(181,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{179})$
Fixed field: Number field defined by a degree 179 polynomial (not computed)

Values on generators

\((719,7)\) → \((1,e\left(\frac{91}{179}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1077 }(16, a) \) \(1\)\(1\)\(e\left(\frac{113}{179}\right)\)\(e\left(\frac{47}{179}\right)\)\(e\left(\frac{177}{179}\right)\)\(e\left(\frac{91}{179}\right)\)\(e\left(\frac{160}{179}\right)\)\(e\left(\frac{111}{179}\right)\)\(e\left(\frac{63}{179}\right)\)\(e\left(\frac{39}{179}\right)\)\(e\left(\frac{25}{179}\right)\)\(e\left(\frac{94}{179}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1077 }(16,a) \;\) at \(\;a = \) e.g. 2