Properties

Label 2003.70
Modulus $2003$
Conductor $2003$
Order $2002$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2003, base_ring=CyclotomicField(2002))
 
M = H._module
 
chi = DirichletCharacter(H, M([1461]))
 
pari: [g,chi] = znchar(Mod(70,2003))
 

Basic properties

Modulus: \(2003\)
Conductor: \(2003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2002\)
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.p

\(\chi_{2003}(5,\cdot)\) \(\chi_{2003}(7,\cdot)\) \(\chi_{2003}(15,\cdot)\) \(\chi_{2003}(18,\cdot)\) \(\chi_{2003}(20,\cdot)\) \(\chi_{2003}(24,\cdot)\) \(\chi_{2003}(26,\cdot)\) \(\chi_{2003}(28,\cdot)\) \(\chi_{2003}(29,\cdot)\) \(\chi_{2003}(31,\cdot)\) \(\chi_{2003}(33,\cdot)\) \(\chi_{2003}(37,\cdot)\) \(\chi_{2003}(38,\cdot)\) \(\chi_{2003}(41,\cdot)\) \(\chi_{2003}(43,\cdot)\) \(\chi_{2003}(51,\cdot)\) \(\chi_{2003}(54,\cdot)\) \(\chi_{2003}(60,\cdot)\) \(\chi_{2003}(61,\cdot)\) \(\chi_{2003}(63,\cdot)\) \(\chi_{2003}(68,\cdot)\) \(\chi_{2003}(70,\cdot)\) \(\chi_{2003}(72,\cdot)\) \(\chi_{2003}(78,\cdot)\) \(\chi_{2003}(80,\cdot)\) \(\chi_{2003}(83,\cdot)\) \(\chi_{2003}(93,\cdot)\) \(\chi_{2003}(94,\cdot)\) \(\chi_{2003}(96,\cdot)\) \(\chi_{2003}(97,\cdot)\) ...

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: $\Q(\zeta_{1001})$
Fixed field: Number field defined by a degree 2002 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{1461}{2002}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2003 }(70, a) \) \(-1\)\(1\)\(e\left(\frac{201}{286}\right)\)\(e\left(\frac{501}{1001}\right)\)\(e\left(\frac{58}{143}\right)\)\(e\left(\frac{1461}{2002}\right)\)\(e\left(\frac{37}{182}\right)\)\(e\left(\frac{1525}{2002}\right)\)\(e\left(\frac{31}{286}\right)\)\(e\left(\frac{1}{1001}\right)\)\(e\left(\frac{433}{1001}\right)\)\(e\left(\frac{217}{286}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2003 }(70,a) \;\) at \(\;a = \) e.g. 2