Properties

Label 6003.2026
Modulus $6003$
Conductor $667$
Order $77$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14,88]))
 
pari: [g,chi] = znchar(Mod(2026,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(77\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.cm

\(\chi_{6003}(82,\cdot)\) \(\chi_{6003}(190,\cdot)\) \(\chi_{6003}(397,\cdot)\) \(\chi_{6003}(487,\cdot)\) \(\chi_{6003}(604,\cdot)\) \(\chi_{6003}(703,\cdot)\) \(\chi_{6003}(721,\cdot)\) \(\chi_{6003}(748,\cdot)\) \(\chi_{6003}(982,\cdot)\) \(\chi_{6003}(1225,\cdot)\) \(\chi_{6003}(1504,\cdot)\) \(\chi_{6003}(1531,\cdot)\) \(\chi_{6003}(1756,\cdot)\) \(\chi_{6003}(1963,\cdot)\) \(\chi_{6003}(2017,\cdot)\) \(\chi_{6003}(2026,\cdot)\) \(\chi_{6003}(2053,\cdot)\) \(\chi_{6003}(2170,\cdot)\) \(\chi_{6003}(2224,\cdot)\) \(\chi_{6003}(2431,\cdot)\) \(\chi_{6003}(2539,\cdot)\) \(\chi_{6003}(2548,\cdot)\) \(\chi_{6003}(2746,\cdot)\) \(\chi_{6003}(2791,\cdot)\) \(\chi_{6003}(2809,\cdot)\) \(\chi_{6003}(2953,\cdot)\) \(\chi_{6003}(3052,\cdot)\) \(\chi_{6003}(3061,\cdot)\) \(\chi_{6003}(3268,\cdot)\) \(\chi_{6003}(3475,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 77 polynomial

Values on generators

\((668,3133,4555)\) → \((1,e\left(\frac{1}{11}\right),e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(2026, a) \) \(1\)\(1\)\(e\left(\frac{58}{77}\right)\)\(e\left(\frac{39}{77}\right)\)\(e\left(\frac{51}{77}\right)\)\(e\left(\frac{45}{77}\right)\)\(e\left(\frac{20}{77}\right)\)\(e\left(\frac{32}{77}\right)\)\(e\left(\frac{8}{77}\right)\)\(e\left(\frac{43}{77}\right)\)\(e\left(\frac{26}{77}\right)\)\(e\left(\frac{1}{77}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(2026,a) \;\) at \(\;a = \) e.g. 2