Properties

Label 8018.379
Modulus $8018$
Conductor $4009$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8018, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,37]))
 
pari: [g,chi] = znchar(Mod(379,8018))
 

Basic properties

Modulus: \(8018\)
Conductor: \(4009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4009}(379,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8018.cq

\(\chi_{8018}(379,\cdot)\) \(\chi_{8018}(455,\cdot)\) \(\chi_{8018}(683,\cdot)\) \(\chi_{8018}(1443,\cdot)\) \(\chi_{8018}(1937,\cdot)\) \(\chi_{8018}(2431,\cdot)\) \(\chi_{8018}(3191,\cdot)\) \(\chi_{8018}(3533,\cdot)\) \(\chi_{8018}(4103,\cdot)\) \(\chi_{8018}(4673,\cdot)\) \(\chi_{8018}(6573,\cdot)\) \(\chi_{8018}(7523,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((2111,1901)\) → \((-1,e\left(\frac{37}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 8018 }(379, a) \) \(1\)\(1\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8018 }(379,a) \;\) at \(\;a = \) e.g. 2