Properties

Label 1157.82
Modulus $1157$
Conductor $1157$
Order $264$
Real no
Primitive yes
Minimal yes
Parity odd

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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([44,111]))
 
pari: [g,chi] = znchar(Mod(82,1157))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1157.ca

\(\chi_{1157}(23,\cdot)\) \(\chi_{1157}(30,\cdot)\) \(\chi_{1157}(43,\cdot)\) \(\chi_{1157}(56,\cdot)\) \(\chi_{1157}(62,\cdot)\) \(\chi_{1157}(75,\cdot)\) \(\chi_{1157}(82,\cdot)\) \(\chi_{1157}(95,\cdot)\) \(\chi_{1157}(108,\cdot)\) \(\chi_{1157}(127,\cdot)\) \(\chi_{1157}(140,\cdot)\) \(\chi_{1157}(147,\cdot)\) \(\chi_{1157}(192,\cdot)\) \(\chi_{1157}(205,\cdot)\) \(\chi_{1157}(238,\cdot)\) \(\chi_{1157}(244,\cdot)\) \(\chi_{1157}(264,\cdot)\) \(\chi_{1157}(270,\cdot)\) \(\chi_{1157}(290,\cdot)\) \(\chi_{1157}(296,\cdot)\) \(\chi_{1157}(329,\cdot)\) \(\chi_{1157}(342,\cdot)\) \(\chi_{1157}(387,\cdot)\) \(\chi_{1157}(394,\cdot)\) \(\chi_{1157}(407,\cdot)\) \(\chi_{1157}(426,\cdot)\) \(\chi_{1157}(439,\cdot)\) \(\chi_{1157}(452,\cdot)\) \(\chi_{1157}(459,\cdot)\) \(\chi_{1157}(472,\cdot)\) ...

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

\((535,92)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{37}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(82, a) \) \(-1\)\(1\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{23}{264}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{259}{264}\right)\)\(e\left(\frac{235}{264}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{23}{132}\right)\)\(e\left(\frac{109}{132}\right)\)\(e\left(\frac{16}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(82,a) \;\) at \(\;a = \) e.g. 2