Properties

Label 4034.27
Modulus $4034$
Conductor $2017$
Order $336$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4034, base_ring=CyclotomicField(336))
 
M = H._module
 
chi = DirichletCharacter(H, M([179]))
 
pari: [g,chi] = znchar(Mod(27,4034))
 

Basic properties

Modulus: \(4034\)
Conductor: \(2017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(336\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2017}(27,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4034.bf

\(\chi_{4034}(27,\cdot)\) \(\chi_{4034}(29,\cdot)\) \(\chi_{4034}(31,\cdot)\) \(\chi_{4034}(33,\cdot)\) \(\chi_{4034}(63,\cdot)\) \(\chi_{4034}(69,\cdot)\) \(\chi_{4034}(95,\cdot)\) \(\chi_{4034}(103,\cdot)\) \(\chi_{4034}(147,\cdot)\) \(\chi_{4034}(219,\cdot)\) \(\chi_{4034}(235,\cdot)\) \(\chi_{4034}(289,\cdot)\) \(\chi_{4034}(335,\cdot)\) \(\chi_{4034}(343,\cdot)\) \(\chi_{4034}(383,\cdot)\) \(\chi_{4034}(489,\cdot)\) \(\chi_{4034}(511,\cdot)\) \(\chi_{4034}(541,\cdot)\) \(\chi_{4034}(617,\cdot)\) \(\chi_{4034}(677,\cdot)\) \(\chi_{4034}(747,\cdot)\) \(\chi_{4034}(921,\cdot)\) \(\chi_{4034}(969,\cdot)\) \(\chi_{4034}(971,\cdot)\) \(\chi_{4034}(1009,\cdot)\) \(\chi_{4034}(1041,\cdot)\) \(\chi_{4034}(1123,\cdot)\) \(\chi_{4034}(1141,\cdot)\) \(\chi_{4034}(1165,\cdot)\) \(\chi_{4034}(1277,\cdot)\) ...

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: $\Q(\zeta_{336})$
Fixed field: Number field defined by a degree 336 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{179}{336}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4034 }(27, a) \) \(1\)\(1\)\(e\left(\frac{121}{168}\right)\)\(e\left(\frac{179}{336}\right)\)\(e\left(\frac{109}{168}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{85}{336}\right)\)\(e\left(\frac{59}{112}\right)\)\(e\left(\frac{19}{336}\right)\)\(e\left(\frac{31}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4034 }(27,a) \;\) at \(\;a = \) e.g. 2