Properties

Label 8018.123
Modulus $8018$
Conductor $4009$
Order $63$
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(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([56,36]))
 
pari: [g,chi] = znchar(Mod(123,8018))
 

Basic properties

Modulus: \(8018\)
Conductor: \(4009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(63\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4009}(123,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8018.cz

\(\chi_{8018}(123,\cdot)\) \(\chi_{8018}(199,\cdot)\) \(\chi_{8018}(359,\cdot)\) \(\chi_{8018}(593,\cdot)\) \(\chi_{8018}(777,\cdot)\) \(\chi_{8018}(967,\cdot)\) \(\chi_{8018}(1043,\cdot)\) \(\chi_{8018}(1203,\cdot)\) \(\chi_{8018}(1621,\cdot)\) \(\chi_{8018}(1811,\cdot)\) \(\chi_{8018}(1859,\cdot)\) \(\chi_{8018}(1887,\cdot)\) \(\chi_{8018}(2379,\cdot)\) \(\chi_{8018}(2703,\cdot)\) \(\chi_{8018}(3125,\cdot)\) \(\chi_{8018}(3645,\cdot)\) \(\chi_{8018}(3969,\cdot)\) \(\chi_{8018}(4489,\cdot)\) \(\chi_{8018}(4813,\cdot)\) \(\chi_{8018}(4911,\cdot)\) \(\chi_{8018}(5001,\cdot)\) \(\chi_{8018}(5419,\cdot)\) \(\chi_{8018}(5609,\cdot)\) \(\chi_{8018}(5685,\cdot)\) \(\chi_{8018}(5755,\cdot)\) \(\chi_{8018}(6267,\cdot)\) \(\chi_{8018}(6599,\cdot)\) \(\chi_{8018}(6685,\cdot)\) \(\chi_{8018}(6875,\cdot)\) \(\chi_{8018}(6951,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 63 polynomial

Values on generators

\((2111,1901)\) → \((e\left(\frac{4}{9}\right),e\left(\frac{2}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 8018 }(123, a) \) \(1\)\(1\)\(e\left(\frac{4}{63}\right)\)\(e\left(\frac{52}{63}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{8}{63}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{23}{63}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{19}{63}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8018 }(123,a) \;\) at \(\;a = \) e.g. 2