Properties

Label 8016.2737
Modulus $8016$
Conductor $167$
Order $83$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,104]))
 
pari: [g,chi] = znchar(Mod(2737,8016))
 

Basic properties

Modulus: \(8016\)
Conductor: \(167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(83\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{167}(65,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8016.y

\(\chi_{8016}(49,\cdot)\) \(\chi_{8016}(97,\cdot)\) \(\chi_{8016}(289,\cdot)\) \(\chi_{8016}(337,\cdot)\) \(\chi_{8016}(433,\cdot)\) \(\chi_{8016}(481,\cdot)\) \(\chi_{8016}(529,\cdot)\) \(\chi_{8016}(577,\cdot)\) \(\chi_{8016}(625,\cdot)\) \(\chi_{8016}(961,\cdot)\) \(\chi_{8016}(1009,\cdot)\) \(\chi_{8016}(1201,\cdot)\) \(\chi_{8016}(1297,\cdot)\) \(\chi_{8016}(1345,\cdot)\) \(\chi_{8016}(1393,\cdot)\) \(\chi_{8016}(1633,\cdot)\) \(\chi_{8016}(1681,\cdot)\) \(\chi_{8016}(1777,\cdot)\) \(\chi_{8016}(1873,\cdot)\) \(\chi_{8016}(1921,\cdot)\) \(\chi_{8016}(1969,\cdot)\) \(\chi_{8016}(2065,\cdot)\) \(\chi_{8016}(2161,\cdot)\) \(\chi_{8016}(2209,\cdot)\) \(\chi_{8016}(2401,\cdot)\) \(\chi_{8016}(2593,\cdot)\) \(\chi_{8016}(2737,\cdot)\) \(\chi_{8016}(2881,\cdot)\) \(\chi_{8016}(3025,\cdot)\) \(\chi_{8016}(3121,\cdot)\) ...

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 83 polynomial

Values on generators

\((3007,2005,5345,673)\) → \((1,1,1,e\left(\frac{52}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 8016 }(2737, a) \) \(1\)\(1\)\(e\left(\frac{52}{83}\right)\)\(e\left(\frac{77}{83}\right)\)\(e\left(\frac{45}{83}\right)\)\(e\left(\frac{44}{83}\right)\)\(e\left(\frac{17}{83}\right)\)\(e\left(\frac{28}{83}\right)\)\(e\left(\frac{2}{83}\right)\)\(e\left(\frac{21}{83}\right)\)\(e\left(\frac{81}{83}\right)\)\(e\left(\frac{32}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8016 }(2737,a) \;\) at \(\;a = \) e.g. 2