Properties

Label 1157.209
Modulus $1157$
Conductor $89$
Order $88$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1157, base_ring=CyclotomicField(88))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,31]))
 
pari: [g,chi] = znchar(Mod(209,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(89\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{89}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1157.bq

\(\chi_{1157}(14,\cdot)\) \(\chi_{1157}(27,\cdot)\) \(\chi_{1157}(66,\cdot)\) \(\chi_{1157}(92,\cdot)\) \(\chi_{1157}(118,\cdot)\) \(\chi_{1157}(209,\cdot)\) \(\chi_{1157}(248,\cdot)\) \(\chi_{1157}(261,\cdot)\) \(\chi_{1157}(274,\cdot)\) \(\chi_{1157}(300,\cdot)\) \(\chi_{1157}(313,\cdot)\) \(\chi_{1157}(326,\cdot)\) \(\chi_{1157}(391,\cdot)\) \(\chi_{1157}(404,\cdot)\) \(\chi_{1157}(417,\cdot)\) \(\chi_{1157}(430,\cdot)\) \(\chi_{1157}(469,\cdot)\) \(\chi_{1157}(508,\cdot)\) \(\chi_{1157}(521,\cdot)\) \(\chi_{1157}(547,\cdot)\) \(\chi_{1157}(560,\cdot)\) \(\chi_{1157}(599,\cdot)\) \(\chi_{1157}(638,\cdot)\) \(\chi_{1157}(651,\cdot)\) \(\chi_{1157}(664,\cdot)\) \(\chi_{1157}(677,\cdot)\) \(\chi_{1157}(742,\cdot)\) \(\chi_{1157}(755,\cdot)\) \(\chi_{1157}(768,\cdot)\) \(\chi_{1157}(794,\cdot)\) ...

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: $\Q(\zeta_{88})$
Fixed field: Number field defined by a degree 88 polynomial

Values on generators

\((535,92)\) → \((1,e\left(\frac{31}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(209, a) \) \(-1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{31}{88}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{87}{88}\right)\)\(e\left(\frac{47}{88}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{13}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(209,a) \;\) at \(\;a = \) e.g. 2