Properties

Label 8018.249
Modulus $8018$
Conductor $4009$
Order $126$
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([7,93]))
 
pari: [g,chi] = znchar(Mod(249,8018))
 

Basic properties

Modulus: \(8018\)
Conductor: \(4009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4009}(249,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8018.dz

\(\chi_{8018}(249,\cdot)\) \(\chi_{8018}(659,\cdot)\) \(\chi_{8018}(801,\cdot)\) \(\chi_{8018}(1001,\cdot)\) \(\chi_{8018}(1021,\cdot)\) \(\chi_{8018}(1515,\cdot)\) \(\chi_{8018}(1587,\cdot)\) \(\chi_{8018}(1865,\cdot)\) \(\chi_{8018}(2141,\cdot)\) \(\chi_{8018}(2143,\cdot)\) \(\chi_{8018}(2347,\cdot)\) \(\chi_{8018}(2359,\cdot)\) \(\chi_{8018}(2371,\cdot)\) \(\chi_{8018}(2415,\cdot)\) \(\chi_{8018}(2689,\cdot)\) \(\chi_{8018}(3681,\cdot)\) \(\chi_{8018}(3829,\cdot)\) \(\chi_{8018}(4119,\cdot)\) \(\chi_{8018}(4525,\cdot)\) \(\chi_{8018}(4879,\cdot)\) \(\chi_{8018}(4885,\cdot)\) \(\chi_{8018}(5221,\cdot)\) \(\chi_{8018}(5835,\cdot)\) \(\chi_{8018}(5941,\cdot)\) \(\chi_{8018}(6151,\cdot)\) \(\chi_{8018}(6169,\cdot)\) \(\chi_{8018}(6361,\cdot)\) \(\chi_{8018}(6709,\cdot)\) \(\chi_{8018}(6995,\cdot)\) \(\chi_{8018}(7101,\cdot)\) ...

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 126 polynomial (not computed)

Values on generators

\((2111,1901)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{31}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 8018 }(249, a) \) \(1\)\(1\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{20}{63}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{58}{63}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{71}{126}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{55}{126}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{11}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8018 }(249,a) \;\) at \(\;a = \) e.g. 2