Properties

Label 1157.866
Modulus $1157$
Conductor $1157$
Order $88$
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(88))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,5]))
 
pari: [g,chi] = znchar(Mod(866,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
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.bs

\(\chi_{1157}(31,\cdot)\) \(\chi_{1157}(70,\cdot)\) \(\chi_{1157}(86,\cdot)\) \(\chi_{1157}(112,\cdot)\) \(\chi_{1157}(148,\cdot)\) \(\chi_{1157}(164,\cdot)\) \(\chi_{1157}(213,\cdot)\) \(\chi_{1157}(252,\cdot)\) \(\chi_{1157}(281,\cdot)\) \(\chi_{1157}(330,\cdot)\) \(\chi_{1157}(333,\cdot)\) \(\chi_{1157}(343,\cdot)\) \(\chi_{1157}(359,\cdot)\) \(\chi_{1157}(369,\cdot)\) \(\chi_{1157}(382,\cdot)\) \(\chi_{1157}(460,\cdot)\) \(\chi_{1157}(499,\cdot)\) \(\chi_{1157}(528,\cdot)\) \(\chi_{1157}(541,\cdot)\) \(\chi_{1157}(564,\cdot)\) \(\chi_{1157}(567,\cdot)\) \(\chi_{1157}(580,\cdot)\) \(\chi_{1157}(642,\cdot)\) \(\chi_{1157}(671,\cdot)\) \(\chi_{1157}(681,\cdot)\) \(\chi_{1157}(684,\cdot)\) \(\chi_{1157}(736,\cdot)\) \(\chi_{1157}(772,\cdot)\) \(\chi_{1157}(863,\cdot)\) \(\chi_{1157}(866,\cdot)\) ...

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

\((535,92)\) → \((i,e\left(\frac{5}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(866, a) \) \(1\)\(1\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{5}{88}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{19}{88}\right)\)\(e\left(\frac{31}{88}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{23}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(866,a) \;\) at \(\;a = \) e.g. 2