Properties

Label 1157.323
Modulus $1157$
Conductor $1157$
Order $264$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1157, base_ring=CyclotomicField(264))
 
M = H._module
 
chi = DirichletCharacter(H, M([154,123]))
 
pari: [g,chi] = znchar(Mod(323,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(264\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1157.cc

\(\chi_{1157}(7,\cdot)\) \(\chi_{1157}(15,\cdot)\) \(\chi_{1157}(19,\cdot)\) \(\chi_{1157}(24,\cdot)\) \(\chi_{1157}(33,\cdot)\) \(\chi_{1157}(46,\cdot)\) \(\chi_{1157}(54,\cdot)\) \(\chi_{1157}(58,\cdot)\) \(\chi_{1157}(119,\cdot)\) \(\chi_{1157}(137,\cdot)\) \(\chi_{1157}(149,\cdot)\) \(\chi_{1157}(150,\cdot)\) \(\chi_{1157}(154,\cdot)\) \(\chi_{1157}(197,\cdot)\) \(\chi_{1157}(202,\cdot)\) \(\chi_{1157}(206,\cdot)\) \(\chi_{1157}(219,\cdot)\) \(\chi_{1157}(236,\cdot)\) \(\chi_{1157}(240,\cdot)\) \(\chi_{1157}(305,\cdot)\) \(\chi_{1157}(310,\cdot)\) \(\chi_{1157}(318,\cdot)\) \(\chi_{1157}(323,\cdot)\) \(\chi_{1157}(327,\cdot)\) \(\chi_{1157}(332,\cdot)\) \(\chi_{1157}(349,\cdot)\) \(\chi_{1157}(362,\cdot)\) \(\chi_{1157}(383,\cdot)\) \(\chi_{1157}(384,\cdot)\) \(\chi_{1157}(397,\cdot)\) ...

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: $\Q(\zeta_{264})$
Fixed field: Number field defined by a degree 264 polynomial (not computed)

Values on generators

\((535,92)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{41}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(323, a) \) \(1\)\(1\)\(e\left(\frac{5}{132}\right)\)\(e\left(\frac{211}{264}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{221}{264}\right)\)\(e\left(\frac{41}{264}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{79}{132}\right)\)\(e\left(\frac{119}{132}\right)\)\(e\left(\frac{29}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(323,a) \;\) at \(\;a = \) e.g. 2