Properties

Label 1157.18
Modulus $1157$
Conductor $1157$
Order $44$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1157, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,9]))
 
pari: [g,chi] = znchar(Mod(18,1157))
 

Basic properties

Modulus: \(1157\)
Conductor: \(1157\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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.bk

\(\chi_{1157}(18,\cdot)\) \(\chi_{1157}(47,\cdot)\) \(\chi_{1157}(99,\cdot)\) \(\chi_{1157}(161,\cdot)\) \(\chi_{1157}(187,\cdot)\) \(\chi_{1157}(307,\cdot)\) \(\chi_{1157}(320,\cdot)\) \(\chi_{1157}(424,\cdot)\) \(\chi_{1157}(450,\cdot)\) \(\chi_{1157}(554,\cdot)\) \(\chi_{1157}(603,\cdot)\) \(\chi_{1157}(707,\cdot)\) \(\chi_{1157}(733,\cdot)\) \(\chi_{1157}(837,\cdot)\) \(\chi_{1157}(850,\cdot)\) \(\chi_{1157}(970,\cdot)\) \(\chi_{1157}(996,\cdot)\) \(\chi_{1157}(1058,\cdot)\) \(\chi_{1157}(1110,\cdot)\) \(\chi_{1157}(1139,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((535,92)\) → \((-i,e\left(\frac{9}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1157 }(18, a) \) \(-1\)\(1\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{19}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1157 }(18,a) \;\) at \(\;a = \) e.g. 2