Properties

Label 1157.868
Modulus $1157$
Conductor $1157$
Order $66$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,42]))
 
pari: [g,chi] = znchar(Mod(868,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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.bo

\(\chi_{1157}(4,\cdot)\) \(\chi_{1157}(121,\cdot)\) \(\chi_{1157}(134,\cdot)\) \(\chi_{1157}(153,\cdot)\) \(\chi_{1157}(186,\cdot)\) \(\chi_{1157}(283,\cdot)\) \(\chi_{1157}(420,\cdot)\) \(\chi_{1157}(550,\cdot)\) \(\chi_{1157}(751,\cdot)\) \(\chi_{1157}(790,\cdot)\) \(\chi_{1157}(803,\cdot)\) \(\chi_{1157}(868,\cdot)\) \(\chi_{1157}(894,\cdot)\) \(\chi_{1157}(1011,\cdot)\) \(\chi_{1157}(1018,\cdot)\) \(\chi_{1157}(1024,\cdot)\) \(\chi_{1157}(1057,\cdot)\) \(\chi_{1157}(1070,\cdot)\) \(\chi_{1157}(1076,\cdot)\) \(\chi_{1157}(1135,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((535,92)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(868, a) \) \(1\)\(1\)\(e\left(\frac{1}{66}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{19}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(868,a) \;\) at \(\;a = \) e.g. 2