Properties

Label 1157.271
Modulus $1157$
Conductor $1157$
Order $132$
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(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,48]))
 
pari: [g,chi] = znchar(Mod(271,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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.by

\(\chi_{1157}(2,\cdot)\) \(\chi_{1157}(32,\cdot)\) \(\chi_{1157}(45,\cdot)\) \(\chi_{1157}(67,\cdot)\) \(\chi_{1157}(93,\cdot)\) \(\chi_{1157}(97,\cdot)\) \(\chi_{1157}(128,\cdot)\) \(\chi_{1157}(167,\cdot)\) \(\chi_{1157}(180,\cdot)\) \(\chi_{1157}(210,\cdot)\) \(\chi_{1157}(223,\cdot)\) \(\chi_{1157}(245,\cdot)\) \(\chi_{1157}(271,\cdot)\) \(\chi_{1157}(275,\cdot)\) \(\chi_{1157}(306,\cdot)\) \(\chi_{1157}(331,\cdot)\) \(\chi_{1157}(345,\cdot)\) \(\chi_{1157}(358,\cdot)\) \(\chi_{1157}(388,\cdot)\) \(\chi_{1157}(401,\cdot)\) \(\chi_{1157}(423,\cdot)\) \(\chi_{1157}(449,\cdot)\) \(\chi_{1157}(453,\cdot)\) \(\chi_{1157}(461,\cdot)\) \(\chi_{1157}(509,\cdot)\) \(\chi_{1157}(566,\cdot)\) \(\chi_{1157}(579,\cdot)\) \(\chi_{1157}(631,\cdot)\) \(\chi_{1157}(639,\cdot)\) \(\chi_{1157}(687,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((535,92)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{4}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(271, a) \) \(-1\)\(1\)\(e\left(\frac{53}{132}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{13}{132}\right)\)\(e\left(\frac{115}{132}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{7}{66}\right)\)\(e\left(\frac{83}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(271,a) \;\) at \(\;a = \) e.g. 2