Properties

Label 4029.605
Modulus $4029$
Conductor $4029$
Order $208$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(208))
 
M = H._module
 
chi = DirichletCharacter(H, M([104,39,112]))
 
pari: [g,chi] = znchar(Mod(605,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(4029\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(208\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.cs

\(\chi_{4029}(62,\cdot)\) \(\chi_{4029}(65,\cdot)\) \(\chi_{4029}(125,\cdot)\) \(\chi_{4029}(131,\cdot)\) \(\chi_{4029}(143,\cdot)\) \(\chi_{4029}(146,\cdot)\) \(\chi_{4029}(176,\cdot)\) \(\chi_{4029}(245,\cdot)\) \(\chi_{4029}(275,\cdot)\) \(\chi_{4029}(299,\cdot)\) \(\chi_{4029}(326,\cdot)\) \(\chi_{4029}(362,\cdot)\) \(\chi_{4029}(368,\cdot)\) \(\chi_{4029}(380,\cdot)\) \(\chi_{4029}(413,\cdot)\) \(\chi_{4029}(482,\cdot)\) \(\chi_{4029}(539,\cdot)\) \(\chi_{4029}(575,\cdot)\) \(\chi_{4029}(605,\cdot)\) \(\chi_{4029}(617,\cdot)\) \(\chi_{4029}(653,\cdot)\) \(\chi_{4029}(719,\cdot)\) \(\chi_{4029}(776,\cdot)\) \(\chi_{4029}(836,\cdot)\) \(\chi_{4029}(857,\cdot)\) \(\chi_{4029}(887,\cdot)\) \(\chi_{4029}(890,\cdot)\) \(\chi_{4029}(1010,\cdot)\) \(\chi_{4029}(1013,\cdot)\) \(\chi_{4029}(1049,\cdot)\) ...

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: $\Q(\zeta_{208})$
Fixed field: Number field defined by a degree 208 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{7}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(605, a) \) \(1\)\(1\)\(e\left(\frac{29}{104}\right)\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{171}{208}\right)\)\(e\left(\frac{125}{208}\right)\)\(e\left(\frac{87}{104}\right)\)\(e\left(\frac{21}{208}\right)\)\(e\left(\frac{89}{208}\right)\)\(e\left(\frac{3}{52}\right)\)\(e\left(\frac{183}{208}\right)\)\(e\left(\frac{3}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(605,a) \;\) at \(\;a = \) e.g. 2