Properties

Label 4006.51
Modulus $4006$
Conductor $2003$
Order $2002$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4006, base_ring=CyclotomicField(2002))
 
M = H._module
 
chi = DirichletCharacter(H, M([621]))
 
pari: [g,chi] = znchar(Mod(51,4006))
 

Basic properties

Modulus: \(4006\)
Conductor: \(2003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2002\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2003}(51,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4006.p

\(\chi_{4006}(5,\cdot)\) \(\chi_{4006}(7,\cdot)\) \(\chi_{4006}(15,\cdot)\) \(\chi_{4006}(29,\cdot)\) \(\chi_{4006}(31,\cdot)\) \(\chi_{4006}(33,\cdot)\) \(\chi_{4006}(37,\cdot)\) \(\chi_{4006}(41,\cdot)\) \(\chi_{4006}(43,\cdot)\) \(\chi_{4006}(51,\cdot)\) \(\chi_{4006}(61,\cdot)\) \(\chi_{4006}(63,\cdot)\) \(\chi_{4006}(83,\cdot)\) \(\chi_{4006}(93,\cdot)\) \(\chi_{4006}(97,\cdot)\) \(\chi_{4006}(103,\cdot)\) \(\chi_{4006}(109,\cdot)\) \(\chi_{4006}(123,\cdot)\) \(\chi_{4006}(125,\cdot)\) \(\chi_{4006}(127,\cdot)\) \(\chi_{4006}(129,\cdot)\) \(\chi_{4006}(131,\cdot)\) \(\chi_{4006}(133,\cdot)\) \(\chi_{4006}(135,\cdot)\) \(\chi_{4006}(137,\cdot)\) \(\chi_{4006}(139,\cdot)\) \(\chi_{4006}(143,\cdot)\) \(\chi_{4006}(151,\cdot)\) \(\chi_{4006}(153,\cdot)\) \(\chi_{4006}(157,\cdot)\) ...

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

\(5\) → \(e\left(\frac{621}{2002}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4006 }(51, a) \) \(-1\)\(1\)\(e\left(\frac{326}{1001}\right)\)\(e\left(\frac{621}{2002}\right)\)\(e\left(\frac{307}{2002}\right)\)\(e\left(\frac{652}{1001}\right)\)\(e\left(\frac{57}{286}\right)\)\(e\left(\frac{424}{1001}\right)\)\(e\left(\frac{1273}{2002}\right)\)\(e\left(\frac{55}{182}\right)\)\(e\left(\frac{883}{1001}\right)\)\(e\left(\frac{137}{286}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4006 }(51,a) \;\) at \(\;a = \) e.g. 2