Properties

Label 6001.3180
Modulus $6001$
Conductor $353$
Order $352$
Real no
Primitive no
Minimal yes
Parity odd

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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([0,1]))
 
pari: [g,chi] = znchar(Mod(3180,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(353\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(352\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{353}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6001.eh

\(\chi_{6001}(52,\cdot)\) \(\chi_{6001}(69,\cdot)\) \(\chi_{6001}(103,\cdot)\) \(\chi_{6001}(154,\cdot)\) \(\chi_{6001}(205,\cdot)\) \(\chi_{6001}(239,\cdot)\) \(\chi_{6001}(273,\cdot)\) \(\chi_{6001}(290,\cdot)\) \(\chi_{6001}(341,\cdot)\) \(\chi_{6001}(358,\cdot)\) \(\chi_{6001}(409,\cdot)\) \(\chi_{6001}(443,\cdot)\) \(\chi_{6001}(460,\cdot)\) \(\chi_{6001}(477,\cdot)\) \(\chi_{6001}(494,\cdot)\) \(\chi_{6001}(511,\cdot)\) \(\chi_{6001}(528,\cdot)\) \(\chi_{6001}(545,\cdot)\) \(\chi_{6001}(596,\cdot)\) \(\chi_{6001}(732,\cdot)\) \(\chi_{6001}(783,\cdot)\) \(\chi_{6001}(851,\cdot)\) \(\chi_{6001}(885,\cdot)\) \(\chi_{6001}(936,\cdot)\) \(\chi_{6001}(970,\cdot)\) \(\chi_{6001}(1004,\cdot)\) \(\chi_{6001}(1072,\cdot)\) \(\chi_{6001}(1174,\cdot)\) \(\chi_{6001}(1191,\cdot)\) \(\chi_{6001}(1208,\cdot)\) ...

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

\((2825,3180)\) → \((1,e\left(\frac{1}{352}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6001 }(3180, a) \) \(-1\)\(1\)\(e\left(\frac{41}{88}\right)\)\(e\left(\frac{1}{352}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{309}{352}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{35}{88}\right)\)\(e\left(\frac{1}{176}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{15}{88}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6001 }(3180,a) \;\) at \(\;a = \) e.g. 2