Properties

Label 2004.1487
Modulus $2004$
Conductor $2004$
Order $166$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2004, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([83,83,77]))
 
pari: [g,chi] = znchar(Mod(1487,2004))
 

Basic properties

Modulus: \(2004\)
Conductor: \(2004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
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 2004.k

\(\chi_{2004}(23,\cdot)\) \(\chi_{2004}(35,\cdot)\) \(\chi_{2004}(59,\cdot)\) \(\chi_{2004}(71,\cdot)\) \(\chi_{2004}(83,\cdot)\) \(\chi_{2004}(95,\cdot)\) \(\chi_{2004}(119,\cdot)\) \(\chi_{2004}(131,\cdot)\) \(\chi_{2004}(143,\cdot)\) \(\chi_{2004}(155,\cdot)\) \(\chi_{2004}(227,\cdot)\) \(\chi_{2004}(287,\cdot)\) \(\chi_{2004}(323,\cdot)\) \(\chi_{2004}(347,\cdot)\) \(\chi_{2004}(371,\cdot)\) \(\chi_{2004}(407,\cdot)\) \(\chi_{2004}(443,\cdot)\) \(\chi_{2004}(479,\cdot)\) \(\chi_{2004}(527,\cdot)\) \(\chi_{2004}(575,\cdot)\) \(\chi_{2004}(587,\cdot)\) \(\chi_{2004}(611,\cdot)\) \(\chi_{2004}(635,\cdot)\) \(\chi_{2004}(647,\cdot)\) \(\chi_{2004}(659,\cdot)\) \(\chi_{2004}(683,\cdot)\) \(\chi_{2004}(707,\cdot)\) \(\chi_{2004}(719,\cdot)\) \(\chi_{2004}(779,\cdot)\) \(\chi_{2004}(791,\cdot)\) ...

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: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial (not computed)

Values on generators

\((1003,1337,673)\) → \((-1,-1,e\left(\frac{77}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 2004 }(1487, a) \) \(-1\)\(1\)\(e\left(\frac{80}{83}\right)\)\(e\left(\frac{39}{166}\right)\)\(e\left(\frac{82}{83}\right)\)\(e\left(\frac{129}{166}\right)\)\(e\left(\frac{7}{83}\right)\)\(e\left(\frac{67}{166}\right)\)\(e\left(\frac{153}{166}\right)\)\(e\left(\frac{77}{83}\right)\)\(e\left(\frac{13}{166}\right)\)\(e\left(\frac{41}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2004 }(1487,a) \;\) at \(\;a = \) e.g. 2