Properties

Label 1157.191
Modulus $1157$
Conductor $1157$
Order $264$
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(264))
 
M = H._module
 
chi = DirichletCharacter(H, M([176,69]))
 
pari: [g,chi] = znchar(Mod(191,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(264\)
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.cd

\(\chi_{1157}(3,\cdot)\) \(\chi_{1157}(29,\cdot)\) \(\chi_{1157}(35,\cdot)\) \(\chi_{1157}(48,\cdot)\) \(\chi_{1157}(61,\cdot)\) \(\chi_{1157}(74,\cdot)\) \(\chi_{1157}(113,\cdot)\) \(\chi_{1157}(120,\cdot)\) \(\chi_{1157}(152,\cdot)\) \(\chi_{1157}(159,\cdot)\) \(\chi_{1157}(165,\cdot)\) \(\chi_{1157}(172,\cdot)\) \(\chi_{1157}(185,\cdot)\) \(\chi_{1157}(191,\cdot)\) \(\chi_{1157}(204,\cdot)\) \(\chi_{1157}(211,\cdot)\) \(\chi_{1157}(224,\cdot)\) \(\chi_{1157}(237,\cdot)\) \(\chi_{1157}(243,\cdot)\) \(\chi_{1157}(282,\cdot)\) \(\chi_{1157}(295,\cdot)\) \(\chi_{1157}(302,\cdot)\) \(\chi_{1157}(308,\cdot)\) \(\chi_{1157}(315,\cdot)\) \(\chi_{1157}(321,\cdot)\) \(\chi_{1157}(328,\cdot)\) \(\chi_{1157}(341,\cdot)\) \(\chi_{1157}(380,\cdot)\) \(\chi_{1157}(386,\cdot)\) \(\chi_{1157}(399,\cdot)\) ...

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: $\Q(\zeta_{264})$
Fixed field: Number field defined by a degree 264 polynomial (not computed)

Values on generators

\((535,92)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{23}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(191, a) \) \(-1\)\(1\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{245}{264}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{205}{264}\right)\)\(e\left(\frac{133}{264}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{113}{132}\right)\)\(e\left(\frac{19}{132}\right)\)\(e\left(\frac{41}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(191,a) \;\) at \(\;a = \) e.g. 2