Properties

Label 8016.2009
Modulus $8016$
Conductor $4008$
Order $166$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,83,83,1]))
 
pari: [g,chi] = znchar(Mod(2009,8016))
 

Basic properties

Modulus: \(8016\)
Conductor: \(4008\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4008}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8016.bg

\(\chi_{8016}(41,\cdot)\) \(\chi_{8016}(377,\cdot)\) \(\chi_{8016}(425,\cdot)\) \(\chi_{8016}(473,\cdot)\) \(\chi_{8016}(521,\cdot)\) \(\chi_{8016}(569,\cdot)\) \(\chi_{8016}(665,\cdot)\) \(\chi_{8016}(713,\cdot)\) \(\chi_{8016}(905,\cdot)\) \(\chi_{8016}(953,\cdot)\) \(\chi_{8016}(1097,\cdot)\) \(\chi_{8016}(1145,\cdot)\) \(\chi_{8016}(1289,\cdot)\) \(\chi_{8016}(1481,\cdot)\) \(\chi_{8016}(1529,\cdot)\) \(\chi_{8016}(1577,\cdot)\) \(\chi_{8016}(1721,\cdot)\) \(\chi_{8016}(2009,\cdot)\) \(\chi_{8016}(2057,\cdot)\) \(\chi_{8016}(2105,\cdot)\) \(\chi_{8016}(2153,\cdot)\) \(\chi_{8016}(2201,\cdot)\) \(\chi_{8016}(2249,\cdot)\) \(\chi_{8016}(2393,\cdot)\) \(\chi_{8016}(2441,\cdot)\) \(\chi_{8016}(2489,\cdot)\) \(\chi_{8016}(2585,\cdot)\) \(\chi_{8016}(2777,\cdot)\) \(\chi_{8016}(2825,\cdot)\) \(\chi_{8016}(2873,\cdot)\) ...

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: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial (not computed)

Values on generators

\((3007,2005,5345,673)\) → \((1,-1,-1,e\left(\frac{1}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 8016 }(2009, a) \) \(1\)\(1\)\(e\left(\frac{1}{166}\right)\)\(e\left(\frac{59}{83}\right)\)\(e\left(\frac{14}{83}\right)\)\(e\left(\frac{10}{83}\right)\)\(e\left(\frac{68}{83}\right)\)\(e\left(\frac{141}{166}\right)\)\(e\left(\frac{8}{83}\right)\)\(e\left(\frac{1}{83}\right)\)\(e\left(\frac{75}{83}\right)\)\(e\left(\frac{45}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8016 }(2009,a) \;\) at \(\;a = \) e.g. 2