Properties

Label 8049.73
Modulus $8049$
Conductor $2683$
Order $1341$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8049, base_ring=CyclotomicField(2682))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1952]))
 
pari: [g,chi] = znchar(Mod(73,8049))
 

Basic properties

Modulus: \(8049\)
Conductor: \(2683\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1341\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2683}(73,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8049.u

\(\chi_{8049}(4,\cdot)\) \(\chi_{8049}(16,\cdot)\) \(\chi_{8049}(25,\cdot)\) \(\chi_{8049}(34,\cdot)\) \(\chi_{8049}(40,\cdot)\) \(\chi_{8049}(58,\cdot)\) \(\chi_{8049}(67,\cdot)\) \(\chi_{8049}(73,\cdot)\) \(\chi_{8049}(85,\cdot)\) \(\chi_{8049}(94,\cdot)\) \(\chi_{8049}(103,\cdot)\) \(\chi_{8049}(106,\cdot)\) \(\chi_{8049}(118,\cdot)\) \(\chi_{8049}(121,\cdot)\) \(\chi_{8049}(124,\cdot)\) \(\chi_{8049}(145,\cdot)\) \(\chi_{8049}(154,\cdot)\) \(\chi_{8049}(157,\cdot)\) \(\chi_{8049}(160,\cdot)\) \(\chi_{8049}(163,\cdot)\) \(\chi_{8049}(166,\cdot)\) \(\chi_{8049}(196,\cdot)\) \(\chi_{8049}(202,\cdot)\) \(\chi_{8049}(241,\cdot)\) \(\chi_{8049}(244,\cdot)\) \(\chi_{8049}(250,\cdot)\) \(\chi_{8049}(253,\cdot)\) \(\chi_{8049}(256,\cdot)\) \(\chi_{8049}(259,\cdot)\) \(\chi_{8049}(262,\cdot)\) ...

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: $\Q(\zeta_{1341})$
Fixed field: Number field defined by a degree 1341 polynomial (not computed)

Values on generators

\((2684,5368)\) → \((1,e\left(\frac{976}{1341}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 8049 }(73, a) \) \(1\)\(1\)\(e\left(\frac{976}{1341}\right)\)\(e\left(\frac{611}{1341}\right)\)\(e\left(\frac{116}{1341}\right)\)\(e\left(\frac{140}{447}\right)\)\(e\left(\frac{82}{447}\right)\)\(e\left(\frac{364}{447}\right)\)\(e\left(\frac{565}{1341}\right)\)\(e\left(\frac{6}{149}\right)\)\(e\left(\frac{55}{1341}\right)\)\(e\left(\frac{1222}{1341}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8049 }(73,a) \;\) at \(\;a = \) e.g. 2