Properties

Label 8026.119
Modulus $8026$
Conductor $4013$
Order $118$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8026, base_ring=CyclotomicField(118))
 
M = H._module
 
chi = DirichletCharacter(H, M([37]))
 
pari: [g,chi] = znchar(Mod(119,8026))
 

Basic properties

Modulus: \(8026\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(118\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4013}(119,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8026.h

\(\chi_{8026}(119,\cdot)\) \(\chi_{8026}(491,\cdot)\) \(\chi_{8026}(547,\cdot)\) \(\chi_{8026}(565,\cdot)\) \(\chi_{8026}(589,\cdot)\) \(\chi_{8026}(717,\cdot)\) \(\chi_{8026}(747,\cdot)\) \(\chi_{8026}(879,\cdot)\) \(\chi_{8026}(977,\cdot)\) \(\chi_{8026}(1075,\cdot)\) \(\chi_{8026}(1097,\cdot)\) \(\chi_{8026}(1131,\cdot)\) \(\chi_{8026}(1447,\cdot)\) \(\chi_{8026}(1519,\cdot)\) \(\chi_{8026}(1789,\cdot)\) \(\chi_{8026}(1815,\cdot)\) \(\chi_{8026}(1853,\cdot)\) \(\chi_{8026}(1891,\cdot)\) \(\chi_{8026}(2143,\cdot)\) \(\chi_{8026}(2413,\cdot)\) \(\chi_{8026}(2419,\cdot)\) \(\chi_{8026}(2535,\cdot)\) \(\chi_{8026}(2601,\cdot)\) \(\chi_{8026}(2963,\cdot)\) \(\chi_{8026}(3065,\cdot)\) \(\chi_{8026}(3323,\cdot)\) \(\chi_{8026}(3339,\cdot)\) \(\chi_{8026}(3495,\cdot)\) \(\chi_{8026}(3715,\cdot)\) \(\chi_{8026}(3811,\cdot)\) ...

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: $\Q(\zeta_{59})$
Fixed field: Number field defined by a degree 118 polynomial (not computed)

Values on generators

\(4015\) → \(e\left(\frac{37}{118}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8026 }(119, a) \) \(1\)\(1\)\(e\left(\frac{19}{118}\right)\)\(e\left(\frac{81}{118}\right)\)\(e\left(\frac{1}{59}\right)\)\(e\left(\frac{19}{59}\right)\)\(e\left(\frac{3}{59}\right)\)\(e\left(\frac{37}{59}\right)\)\(e\left(\frac{50}{59}\right)\)\(e\left(\frac{26}{59}\right)\)\(e\left(\frac{13}{59}\right)\)\(e\left(\frac{21}{118}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8026 }(119,a) \;\) at \(\;a = \) e.g. 2