Properties

Label 8041.1695
Modulus $8041$
Conductor $731$
Order $336$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(336))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,273,232]))
 
pari: [g,chi] = znchar(Mod(1695,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(336\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{731}(233,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.fk

\(\chi_{8041}(12,\cdot)\) \(\chi_{8041}(177,\cdot)\) \(\chi_{8041}(243,\cdot)\) \(\chi_{8041}(320,\cdot)\) \(\chi_{8041}(364,\cdot)\) \(\chi_{8041}(507,\cdot)\) \(\chi_{8041}(760,\cdot)\) \(\chi_{8041}(793,\cdot)\) \(\chi_{8041}(804,\cdot)\) \(\chi_{8041}(958,\cdot)\) \(\chi_{8041}(980,\cdot)\) \(\chi_{8041}(1431,\cdot)\) \(\chi_{8041}(1508,\cdot)\) \(\chi_{8041}(1574,\cdot)\) \(\chi_{8041}(1695,\cdot)\) \(\chi_{8041}(1706,\cdot)\) \(\chi_{8041}(1739,\cdot)\) \(\chi_{8041}(1882,\cdot)\) \(\chi_{8041}(1926,\cdot)\) \(\chi_{8041}(2047,\cdot)\) \(\chi_{8041}(2069,\cdot)\) \(\chi_{8041}(2135,\cdot)\) \(\chi_{8041}(2179,\cdot)\) \(\chi_{8041}(2256,\cdot)\) \(\chi_{8041}(2377,\cdot)\) \(\chi_{8041}(2454,\cdot)\) \(\chi_{8041}(2608,\cdot)\) \(\chi_{8041}(2641,\cdot)\) \(\chi_{8041}(2696,\cdot)\) \(\chi_{8041}(2828,\cdot)\) ...

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: $\Q(\zeta_{336})$
Fixed field: Number field defined by a degree 336 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(1695, a) \) \(1\)\(1\)\(e\left(\frac{1}{56}\right)\)\(e\left(\frac{169}{336}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{109}{336}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{1}{168}\right)\)\(e\left(\frac{115}{336}\right)\)\(e\left(\frac{181}{336}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(1695,a) \;\) at \(\;a = \) e.g. 2