Properties

Label 1157.765
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([77,117]))
 
pari: [g,chi] = znchar(Mod(765,1157))
 

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.bw

\(\chi_{1157}(20,\cdot)\) \(\chi_{1157}(84,\cdot)\) \(\chi_{1157}(110,\cdot)\) \(\chi_{1157}(158,\cdot)\) \(\chi_{1157}(214,\cdot)\) \(\chi_{1157}(227,\cdot)\) \(\chi_{1157}(258,\cdot)\) \(\chi_{1157}(262,\cdot)\) \(\chi_{1157}(284,\cdot)\) \(\chi_{1157}(288,\cdot)\) \(\chi_{1157}(392,\cdot)\) \(\chi_{1157}(405,\cdot)\) \(\chi_{1157}(427,\cdot)\) \(\chi_{1157}(435,\cdot)\) \(\chi_{1157}(436,\cdot)\) \(\chi_{1157}(462,\cdot)\) \(\chi_{1157}(487,\cdot)\) \(\chi_{1157}(492,\cdot)\) \(\chi_{1157}(544,\cdot)\) \(\chi_{1157}(552,\cdot)\) \(\chi_{1157}(605,\cdot)\) \(\chi_{1157}(613,\cdot)\) \(\chi_{1157}(665,\cdot)\) \(\chi_{1157}(670,\cdot)\) \(\chi_{1157}(695,\cdot)\) \(\chi_{1157}(721,\cdot)\) \(\chi_{1157}(722,\cdot)\) \(\chi_{1157}(730,\cdot)\) \(\chi_{1157}(752,\cdot)\) \(\chi_{1157}(765,\cdot)\) ...

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

\((535,92)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{39}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(765, a) \) \(-1\)\(1\)\(e\left(\frac{101}{132}\right)\)\(e\left(\frac{29}{132}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{71}{132}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(765,a) \;\) at \(\;a = \) e.g. 2