Properties

Label 6001.821
Modulus $6001$
Conductor $6001$
Order $352$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6001, base_ring=CyclotomicField(352))
 
M = H._module
 
chi = DirichletCharacter(H, M([110,315]))
 
pari: [g,chi] = znchar(Mod(821,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(352\)
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 6001.es

\(\chi_{6001}(14,\cdot)\) \(\chi_{6001}(40,\cdot)\) \(\chi_{6001}(71,\cdot)\) \(\chi_{6001}(90,\cdot)\) \(\chi_{6001}(141,\cdot)\) \(\chi_{6001}(148,\cdot)\) \(\chi_{6001}(173,\cdot)\) \(\chi_{6001}(190,\cdot)\) \(\chi_{6001}(199,\cdot)\) \(\chi_{6001}(250,\cdot)\) \(\chi_{6001}(333,\cdot)\) \(\chi_{6001}(384,\cdot)\) \(\chi_{6001}(422,\cdot)\) \(\chi_{6001}(449,\cdot)\) \(\chi_{6001}(581,\cdot)\) \(\chi_{6001}(632,\cdot)\) \(\chi_{6001}(673,\cdot)\) \(\chi_{6001}(703,\cdot)\) \(\chi_{6001}(754,\cdot)\) \(\chi_{6001}(821,\cdot)\) \(\chi_{6001}(840,\cdot)\) \(\chi_{6001}(864,\cdot)\) \(\chi_{6001}(925,\cdot)\) \(\chi_{6001}(947,\cdot)\) \(\chi_{6001}(1032,\cdot)\) \(\chi_{6001}(1234,\cdot)\) \(\chi_{6001}(1252,\cdot)\) \(\chi_{6001}(1270,\cdot)\) \(\chi_{6001}(1286,\cdot)\) \(\chi_{6001}(1297,\cdot)\) ...

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: $\Q(\zeta_{352})$
Fixed field: Number field defined by a degree 352 polynomial (not computed)

Values on generators

\((2825,3180)\) → \((e\left(\frac{5}{16}\right),e\left(\frac{315}{352}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6001 }(821, a) \) \(1\)\(1\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{73}{352}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{29}{352}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{73}{176}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{155}{176}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6001 }(821,a) \;\) at \(\;a = \) e.g. 2