Properties

Label 6041.13
Modulus $6041$
Conductor $6041$
Order $862$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6041, base_ring=CyclotomicField(862))
 
M = H._module
 
chi = DirichletCharacter(H, M([431,99]))
 
pari: [g,chi] = znchar(Mod(13,6041))
 

Basic properties

Modulus: \(6041\)
Conductor: \(6041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(862\)
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 6041.k

\(\chi_{6041}(13,\cdot)\) \(\chi_{6041}(20,\cdot)\) \(\chi_{6041}(69,\cdot)\) \(\chi_{6041}(83,\cdot)\) \(\chi_{6041}(90,\cdot)\) \(\chi_{6041}(97,\cdot)\) \(\chi_{6041}(104,\cdot)\) \(\chi_{6041}(125,\cdot)\) \(\chi_{6041}(132,\cdot)\) \(\chi_{6041}(139,\cdot)\) \(\chi_{6041}(146,\cdot)\) \(\chi_{6041}(160,\cdot)\) \(\chi_{6041}(167,\cdot)\) \(\chi_{6041}(188,\cdot)\) \(\chi_{6041}(202,\cdot)\) \(\chi_{6041}(209,\cdot)\) \(\chi_{6041}(237,\cdot)\) \(\chi_{6041}(251,\cdot)\) \(\chi_{6041}(265,\cdot)\) \(\chi_{6041}(314,\cdot)\) \(\chi_{6041}(349,\cdot)\) \(\chi_{6041}(356,\cdot)\) \(\chi_{6041}(370,\cdot)\) \(\chi_{6041}(377,\cdot)\) \(\chi_{6041}(391,\cdot)\) \(\chi_{6041}(405,\cdot)\) \(\chi_{6041}(419,\cdot)\) \(\chi_{6041}(454,\cdot)\) \(\chi_{6041}(468,\cdot)\) \(\chi_{6041}(482,\cdot)\) ...

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: $\Q(\zeta_{431})$
Fixed field: Number field defined by a degree 862 polynomial (not computed)

Values on generators

\((864,4320)\) → \((-1,e\left(\frac{99}{862}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 6041 }(13, a) \) \(1\)\(1\)\(e\left(\frac{427}{431}\right)\)\(e\left(\frac{157}{862}\right)\)\(e\left(\frac{423}{431}\right)\)\(e\left(\frac{265}{431}\right)\)\(e\left(\frac{149}{862}\right)\)\(e\left(\frac{419}{431}\right)\)\(e\left(\frac{157}{431}\right)\)\(e\left(\frac{261}{431}\right)\)\(e\left(\frac{493}{862}\right)\)\(e\left(\frac{141}{862}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6041 }(13,a) \;\) at \(\;a = \) e.g. 2