Properties

Label 8035.1163
Modulus $8035$
Conductor $8035$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8035, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,38]))
 
pari: [g,chi] = znchar(Mod(1163,8035))
 

Basic properties

Modulus: \(8035\)
Conductor: \(8035\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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 8035.l

\(\chi_{8035}(1163,\cdot)\) \(\chi_{8035}(1692,\cdot)\) \(\chi_{8035}(1858,\cdot)\) \(\chi_{8035}(1978,\cdot)\) \(\chi_{8035}(2132,\cdot)\) \(\chi_{8035}(2168,\cdot)\) \(\chi_{8035}(2212,\cdot)\) \(\chi_{8035}(2417,\cdot)\) \(\chi_{8035}(3993,\cdot)\) \(\chi_{8035}(4377,\cdot)\) \(\chi_{8035}(4493,\cdot)\) \(\chi_{8035}(5072,\cdot)\) \(\chi_{8035}(5192,\cdot)\) \(\chi_{8035}(5382,\cdot)\) \(\chi_{8035}(6513,\cdot)\) \(\chi_{8035}(6953,\cdot)\) \(\chi_{8035}(7033,\cdot)\) \(\chi_{8035}(7207,\cdot)\) \(\chi_{8035}(7238,\cdot)\) \(\chi_{8035}(7707,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((4822,4826)\) → \((-i,e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 8035 }(1163, a) \) \(1\)\(1\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{9}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8035 }(1163,a) \;\) at \(\;a = \) e.g. 2