Properties

Label 94.21
Modulus $94$
Conductor $47$
Order $23$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(94, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([6]))
 
pari: [g,chi] = znchar(Mod(21,94))
 

Basic properties

Modulus: \(94\)
Conductor: \(47\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{47}(21,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 94.c

\(\chi_{94}(3,\cdot)\) \(\chi_{94}(7,\cdot)\) \(\chi_{94}(9,\cdot)\) \(\chi_{94}(17,\cdot)\) \(\chi_{94}(21,\cdot)\) \(\chi_{94}(25,\cdot)\) \(\chi_{94}(27,\cdot)\) \(\chi_{94}(37,\cdot)\) \(\chi_{94}(49,\cdot)\) \(\chi_{94}(51,\cdot)\) \(\chi_{94}(53,\cdot)\) \(\chi_{94}(55,\cdot)\) \(\chi_{94}(59,\cdot)\) \(\chi_{94}(61,\cdot)\) \(\chi_{94}(63,\cdot)\) \(\chi_{94}(65,\cdot)\) \(\chi_{94}(71,\cdot)\) \(\chi_{94}(75,\cdot)\) \(\chi_{94}(79,\cdot)\) \(\chi_{94}(81,\cdot)\) \(\chi_{94}(83,\cdot)\) \(\chi_{94}(89,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 23 polynomial

Values on generators

\(5\) → \(e\left(\frac{3}{23}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 94 }(21, a) \) \(1\)\(1\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{17}{23}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{18}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 94 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 94 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 94 }(21,·),\chi_{ 94 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 94 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2