Properties

Label 301.239
Modulus $301$
Conductor $43$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(301, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,40]))
 
pari: [g,chi] = znchar(Mod(239,301))
 

Basic properties

Modulus: \(301\)
Conductor: \(43\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{43}(24,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 301.y

\(\chi_{301}(15,\cdot)\) \(\chi_{301}(57,\cdot)\) \(\chi_{301}(99,\cdot)\) \(\chi_{301}(169,\cdot)\) \(\chi_{301}(197,\cdot)\) \(\chi_{301}(225,\cdot)\) \(\chi_{301}(232,\cdot)\) \(\chi_{301}(239,\cdot)\) \(\chi_{301}(246,\cdot)\) \(\chi_{301}(253,\cdot)\) \(\chi_{301}(267,\cdot)\) \(\chi_{301}(281,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\((87,218)\) → \((1,e\left(\frac{20}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 301 }(239, a) \) \(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{8}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 301 }(239,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 301 }(239,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 301 }(239,·),\chi_{ 301 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 301 }(239,·)) \;\) at \(\; a,b = \) e.g. 1,2