Properties

Label 735.188
Modulus $735$
Conductor $735$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,21,10]))
 
pari: [g,chi] = znchar(Mod(188,735))
 

Basic properties

Modulus: \(735\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 735.bk

\(\chi_{735}(62,\cdot)\) \(\chi_{735}(83,\cdot)\) \(\chi_{735}(167,\cdot)\) \(\chi_{735}(188,\cdot)\) \(\chi_{735}(272,\cdot)\) \(\chi_{735}(377,\cdot)\) \(\chi_{735}(398,\cdot)\) \(\chi_{735}(482,\cdot)\) \(\chi_{735}(503,\cdot)\) \(\chi_{735}(608,\cdot)\) \(\chi_{735}(692,\cdot)\) \(\chi_{735}(713,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.0.4101754449160695184473159618498838032884071911945819854736328125.1

Values on generators

\((491,442,346)\) → \((-1,-i,e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 735 }(188, a) \) \(-1\)\(1\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{28}\right)\)\(1\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{9}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 735 }(188,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 735 }(188,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 735 }(188,·),\chi_{ 735 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 735 }(188,·)) \;\) at \(\; a,b = \) e.g. 1,2