Properties

Label 223.28
Modulus $223$
Conductor $223$
Order $37$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(223, base_ring=CyclotomicField(74))
 
M = H._module
 
chi = DirichletCharacter(H, M([42]))
 
pari: [g,chi] = znchar(Mod(28,223))
 

Basic properties

Modulus: \(223\)
Conductor: \(223\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(37\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 223.e

\(\chi_{223}(2,\cdot)\) \(\chi_{223}(4,\cdot)\) \(\chi_{223}(7,\cdot)\) \(\chi_{223}(8,\cdot)\) \(\chi_{223}(14,\cdot)\) \(\chi_{223}(15,\cdot)\) \(\chi_{223}(16,\cdot)\) \(\chi_{223}(17,\cdot)\) \(\chi_{223}(28,\cdot)\) \(\chi_{223}(30,\cdot)\) \(\chi_{223}(32,\cdot)\) \(\chi_{223}(33,\cdot)\) \(\chi_{223}(34,\cdot)\) \(\chi_{223}(41,\cdot)\) \(\chi_{223}(49,\cdot)\) \(\chi_{223}(56,\cdot)\) \(\chi_{223}(60,\cdot)\) \(\chi_{223}(64,\cdot)\) \(\chi_{223}(66,\cdot)\) \(\chi_{223}(68,\cdot)\) \(\chi_{223}(82,\cdot)\) \(\chi_{223}(98,\cdot)\) \(\chi_{223}(105,\cdot)\) \(\chi_{223}(112,\cdot)\) \(\chi_{223}(115,\cdot)\) \(\chi_{223}(119,\cdot)\) \(\chi_{223}(120,\cdot)\) \(\chi_{223}(128,\cdot)\) \(\chi_{223}(132,\cdot)\) \(\chi_{223}(136,\cdot)\) ...

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: $\Q(\zeta_{37})$
Fixed field: Number field defined by a degree 37 polynomial

Values on generators

\(3\) → \(e\left(\frac{21}{37}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 223 }(28, a) \) \(1\)\(1\)\(e\left(\frac{6}{37}\right)\)\(e\left(\frac{21}{37}\right)\)\(e\left(\frac{12}{37}\right)\)\(e\left(\frac{19}{37}\right)\)\(e\left(\frac{27}{37}\right)\)\(e\left(\frac{7}{37}\right)\)\(e\left(\frac{18}{37}\right)\)\(e\left(\frac{5}{37}\right)\)\(e\left(\frac{25}{37}\right)\)\(e\left(\frac{27}{37}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 223 }(28,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 223 }(28,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 223 }(28,·),\chi_{ 223 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 223 }(28,·)) \;\) at \(\; a,b = \) e.g. 1,2