Properties

Label 1157.912
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([11,18]))
 
pari: [g,chi] = znchar(Mod(912,1157))
 

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.bv

\(\chi_{1157}(11,\cdot)\) \(\chi_{1157}(50,\cdot)\) \(\chi_{1157}(85,\cdot)\) \(\chi_{1157}(111,\cdot)\) \(\chi_{1157}(162,\cdot)\) \(\chi_{1157}(176,\cdot)\) \(\chi_{1157}(189,\cdot)\) \(\chi_{1157}(228,\cdot)\) \(\chi_{1157}(292,\cdot)\) \(\chi_{1157}(340,\cdot)\) \(\chi_{1157}(470,\cdot)\) \(\chi_{1157}(518,\cdot)\) \(\chi_{1157}(526,\cdot)\) \(\chi_{1157}(578,\cdot)\) \(\chi_{1157}(591,\cdot)\) \(\chi_{1157}(648,\cdot)\) \(\chi_{1157}(696,\cdot)\) \(\chi_{1157}(704,\cdot)\) \(\chi_{1157}(708,\cdot)\) \(\chi_{1157}(734,\cdot)\) \(\chi_{1157}(756,\cdot)\) \(\chi_{1157}(769,\cdot)\) \(\chi_{1157}(799,\cdot)\) \(\chi_{1157}(812,\cdot)\) \(\chi_{1157}(826,\cdot)\) \(\chi_{1157}(851,\cdot)\) \(\chi_{1157}(882,\cdot)\) \(\chi_{1157}(886,\cdot)\) \(\chi_{1157}(912,\cdot)\) \(\chi_{1157}(934,\cdot)\) ...

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

\((535,92)\) → \((e\left(\frac{1}{12}\right),e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(912, a) \) \(-1\)\(1\)\(e\left(\frac{35}{132}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{97}{132}\right)\)\(e\left(\frac{127}{132}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{5}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(912,a) \;\) at \(\;a = \) e.g. 2