Properties

Label 8016.1535
Modulus $8016$
Conductor $2004$
Order $166$
Real no
Primitive no
Minimal yes
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([83,0,83,34]))
 
pari: [g,chi] = znchar(Mod(1535,8016))
 

Basic properties

Modulus: \(8016\)
Conductor: \(2004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2004}(1535,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8016.bk

\(\chi_{8016}(47,\cdot)\) \(\chi_{8016}(191,\cdot)\) \(\chi_{8016}(239,\cdot)\) \(\chi_{8016}(383,\cdot)\) \(\chi_{8016}(431,\cdot)\) \(\chi_{8016}(623,\cdot)\) \(\chi_{8016}(671,\cdot)\) \(\chi_{8016}(767,\cdot)\) \(\chi_{8016}(815,\cdot)\) \(\chi_{8016}(863,\cdot)\) \(\chi_{8016}(911,\cdot)\) \(\chi_{8016}(959,\cdot)\) \(\chi_{8016}(1295,\cdot)\) \(\chi_{8016}(1343,\cdot)\) \(\chi_{8016}(1535,\cdot)\) \(\chi_{8016}(1631,\cdot)\) \(\chi_{8016}(1679,\cdot)\) \(\chi_{8016}(1727,\cdot)\) \(\chi_{8016}(1967,\cdot)\) \(\chi_{8016}(2015,\cdot)\) \(\chi_{8016}(2111,\cdot)\) \(\chi_{8016}(2207,\cdot)\) \(\chi_{8016}(2255,\cdot)\) \(\chi_{8016}(2303,\cdot)\) \(\chi_{8016}(2399,\cdot)\) \(\chi_{8016}(2495,\cdot)\) \(\chi_{8016}(2543,\cdot)\) \(\chi_{8016}(2735,\cdot)\) \(\chi_{8016}(2927,\cdot)\) \(\chi_{8016}(3071,\cdot)\) ...

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

\((3007,2005,5345,673)\) → \((-1,1,-1,e\left(\frac{17}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 8016 }(1535, a) \) \(1\)\(1\)\(e\left(\frac{117}{166}\right)\)\(e\left(\frac{111}{166}\right)\)\(e\left(\frac{61}{83}\right)\)\(e\left(\frac{8}{83}\right)\)\(e\left(\frac{59}{166}\right)\)\(e\left(\frac{63}{166}\right)\)\(e\left(\frac{23}{83}\right)\)\(e\left(\frac{34}{83}\right)\)\(e\left(\frac{37}{166}\right)\)\(e\left(\frac{155}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8016 }(1535,a) \;\) at \(\;a = \) e.g. 2