Properties

Label 1148.561
Modulus $1148$
Conductor $41$
Order $40$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,11]))
 
pari: [g,chi] = znchar(Mod(561,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(28,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1148.cd

\(\chi_{1148}(29,\cdot)\) \(\chi_{1148}(253,\cdot)\) \(\chi_{1148}(281,\cdot)\) \(\chi_{1148}(309,\cdot)\) \(\chi_{1148}(393,\cdot)\) \(\chi_{1148}(421,\cdot)\) \(\chi_{1148}(477,\cdot)\) \(\chi_{1148}(505,\cdot)\) \(\chi_{1148}(561,\cdot)\) \(\chi_{1148}(589,\cdot)\) \(\chi_{1148}(645,\cdot)\) \(\chi_{1148}(673,\cdot)\) \(\chi_{1148}(757,\cdot)\) \(\chi_{1148}(785,\cdot)\) \(\chi_{1148}(813,\cdot)\) \(\chi_{1148}(1037,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((575,493,785)\) → \((1,1,e\left(\frac{11}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1148 }(561, a) \) \(-1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{20}\right)\)\(i\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{21}{40}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1148 }(561,a) \;\) at \(\;a = \) e.g. 2