Properties

Label 2003.870
Modulus $2003$
Conductor $2003$
Order $154$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2003, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([97]))
 
pari: [g,chi] = znchar(Mod(870,2003))
 

Basic properties

Modulus: \(2003\)
Conductor: \(2003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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 2003.l

\(\chi_{2003}(50,\cdot)\) \(\chi_{2003}(65,\cdot)\) \(\chi_{2003}(99,\cdot)\) \(\chi_{2003}(111,\cdot)\) \(\chi_{2003}(124,\cdot)\) \(\chi_{2003}(199,\cdot)\) \(\chi_{2003}(207,\cdot)\) \(\chi_{2003}(210,\cdot)\) \(\chi_{2003}(214,\cdot)\) \(\chi_{2003}(234,\cdot)\) \(\chi_{2003}(245,\cdot)\) \(\chi_{2003}(257,\cdot)\) \(\chi_{2003}(273,\cdot)\) \(\chi_{2003}(329,\cdot)\) \(\chi_{2003}(371,\cdot)\) \(\chi_{2003}(397,\cdot)\) \(\chi_{2003}(459,\cdot)\) \(\chi_{2003}(566,\cdot)\) \(\chi_{2003}(628,\cdot)\) \(\chi_{2003}(648,\cdot)\) \(\chi_{2003}(726,\cdot)\) \(\chi_{2003}(756,\cdot)\) \(\chi_{2003}(757,\cdot)\) \(\chi_{2003}(797,\cdot)\) \(\chi_{2003}(814,\cdot)\) \(\chi_{2003}(847,\cdot)\) \(\chi_{2003}(870,\cdot)\) \(\chi_{2003}(935,\cdot)\) \(\chi_{2003}(1015,\cdot)\) \(\chi_{2003}(1059,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{97}{154}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2003 }(870, a) \) \(-1\)\(1\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{72}{77}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{97}{154}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{73}{154}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{67}{77}\right)\)\(e\left(\frac{59}{77}\right)\)\(e\left(\frac{19}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2003 }(870,a) \;\) at \(\;a = \) e.g. 2