Properties

Label 8041.1125
Modulus $8041$
Conductor $8041$
Order $240$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(240))
 
M = H._module
 
chi = DirichletCharacter(H, M([192,15,200]))
 
pari: [g,chi] = znchar(Mod(1125,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(240\)
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 8041.fc

\(\chi_{8041}(37,\cdot)\) \(\chi_{8041}(80,\cdot)\) \(\chi_{8041}(295,\cdot)\) \(\chi_{8041}(394,\cdot)\) \(\chi_{8041}(566,\cdot)\) \(\chi_{8041}(609,\cdot)\) \(\chi_{8041}(652,\cdot)\) \(\chi_{8041}(768,\cdot)\) \(\chi_{8041}(940,\cdot)\) \(\chi_{8041}(983,\cdot)\) \(\chi_{8041}(1026,\cdot)\) \(\chi_{8041}(1082,\cdot)\) \(\chi_{8041}(1125,\cdot)\) \(\chi_{8041}(1340,\cdot)\) \(\chi_{8041}(1456,\cdot)\) \(\chi_{8041}(1499,\cdot)\) \(\chi_{8041}(1714,\cdot)\) \(\chi_{8041}(1813,\cdot)\) \(\chi_{8041}(2028,\cdot)\) \(\chi_{8041}(2071,\cdot)\) \(\chi_{8041}(2187,\cdot)\) \(\chi_{8041}(2402,\cdot)\) \(\chi_{8041}(2445,\cdot)\) \(\chi_{8041}(2458,\cdot)\) \(\chi_{8041}(2544,\cdot)\) \(\chi_{8041}(2759,\cdot)\) \(\chi_{8041}(2832,\cdot)\) \(\chi_{8041}(2918,\cdot)\) \(\chi_{8041}(2931,\cdot)\) \(\chi_{8041}(3133,\cdot)\) ...

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: $\Q(\zeta_{240})$
Fixed field: Number field defined by a degree 240 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{1}{16}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(1125, a) \) \(1\)\(1\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{71}{240}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{83}{240}\right)\)\(e\left(\frac{113}{240}\right)\)\(e\left(\frac{109}{240}\right)\)\(e\left(\frac{21}{40}\right)\)\(e\left(\frac{71}{120}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{31}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(1125,a) \;\) at \(\;a = \) e.g. 2