Properties

Label 8016.1061
Modulus $8016$
Conductor $8016$
Order $332$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(332))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,83,166,226]))
 
pari: [g,chi] = znchar(Mod(1061,8016))
 

Basic properties

Modulus: \(8016\)
Conductor: \(8016\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(332\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8016.bt

\(\chi_{8016}(5,\cdot)\) \(\chi_{8016}(53,\cdot)\) \(\chi_{8016}(101,\cdot)\) \(\chi_{8016}(125,\cdot)\) \(\chi_{8016}(149,\cdot)\) \(\chi_{8016}(197,\cdot)\) \(\chi_{8016}(245,\cdot)\) \(\chi_{8016}(269,\cdot)\) \(\chi_{8016}(389,\cdot)\) \(\chi_{8016}(413,\cdot)\) \(\chi_{8016}(437,\cdot)\) \(\chi_{8016}(485,\cdot)\) \(\chi_{8016}(581,\cdot)\) \(\chi_{8016}(605,\cdot)\) \(\chi_{8016}(773,\cdot)\) \(\chi_{8016}(797,\cdot)\) \(\chi_{8016}(821,\cdot)\) \(\chi_{8016}(845,\cdot)\) \(\chi_{8016}(869,\cdot)\) \(\chi_{8016}(917,\cdot)\) \(\chi_{8016}(941,\cdot)\) \(\chi_{8016}(1037,\cdot)\) \(\chi_{8016}(1061,\cdot)\) \(\chi_{8016}(1085,\cdot)\) \(\chi_{8016}(1133,\cdot)\) \(\chi_{8016}(1157,\cdot)\) \(\chi_{8016}(1229,\cdot)\) \(\chi_{8016}(1325,\cdot)\) \(\chi_{8016}(1349,\cdot)\) \(\chi_{8016}(1373,\cdot)\) ...

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: $\Q(\zeta_{332})$
Fixed field: Number field defined by a degree 332 polynomial (not computed)

Values on generators

\((3007,2005,5345,673)\) → \((1,i,-1,e\left(\frac{113}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 8016 }(1061, a) \) \(1\)\(1\)\(e\left(\frac{143}{332}\right)\)\(e\left(\frac{137}{166}\right)\)\(e\left(\frac{269}{332}\right)\)\(e\left(\frac{287}{332}\right)\)\(e\left(\frac{48}{83}\right)\)\(e\left(\frac{77}{332}\right)\)\(e\left(\frac{65}{166}\right)\)\(e\left(\frac{143}{166}\right)\)\(e\left(\frac{119}{332}\right)\)\(e\left(\frac{22}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8016 }(1061,a) \;\) at \(\;a = \) e.g. 2