Properties

Label 1157.880
Modulus $1157$
Conductor $1157$
Order $132$
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(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([88,63]))
 
pari: [g,chi] = znchar(Mod(880,1157))
 

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

Galois orbit 1157.bz

\(\chi_{1157}(9,\cdot)\) \(\chi_{1157}(42,\cdot)\) \(\chi_{1157}(68,\cdot)\) \(\chi_{1157}(94,\cdot)\) \(\chi_{1157}(107,\cdot)\) \(\chi_{1157}(198,\cdot)\) \(\chi_{1157}(250,\cdot)\) \(\chi_{1157}(276,\cdot)\) \(\chi_{1157}(347,\cdot)\) \(\chi_{1157}(373,\cdot)\) \(\chi_{1157}(425,\cdot)\) \(\chi_{1157}(516,\cdot)\) \(\chi_{1157}(529,\cdot)\) \(\chi_{1157}(555,\cdot)\) \(\chi_{1157}(581,\cdot)\) \(\chi_{1157}(614,\cdot)\) \(\chi_{1157}(633,\cdot)\) \(\chi_{1157}(640,\cdot)\) \(\chi_{1157}(659,\cdot)\) \(\chi_{1157}(672,\cdot)\) \(\chi_{1157}(692,\cdot)\) \(\chi_{1157}(783,\cdot)\) \(\chi_{1157}(796,\cdot)\) \(\chi_{1157}(822,\cdot)\) \(\chi_{1157}(841,\cdot)\) \(\chi_{1157}(848,\cdot)\) \(\chi_{1157}(854,\cdot)\) \(\chi_{1157}(880,\cdot)\) \(\chi_{1157}(900,\cdot)\) \(\chi_{1157}(926,\cdot)\) ...

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

\((535,92)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{21}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(880, a) \) \(1\)\(1\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{19}{132}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{59}{132}\right)\)\(e\left(\frac{131}{132}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{25}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(880,a) \;\) at \(\;a = \) e.g. 2