Properties

Label 215.9
Modulus $215$
Conductor $215$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(215, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,2]))
 
pari: [g,chi] = znchar(Mod(9,215))
 

Basic properties

Modulus: \(215\)
Conductor: \(215\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 215.u

\(\chi_{215}(9,\cdot)\) \(\chi_{215}(14,\cdot)\) \(\chi_{215}(24,\cdot)\) \(\chi_{215}(74,\cdot)\) \(\chi_{215}(99,\cdot)\) \(\chi_{215}(109,\cdot)\) \(\chi_{215}(124,\cdot)\) \(\chi_{215}(139,\cdot)\) \(\chi_{215}(144,\cdot)\) \(\chi_{215}(154,\cdot)\) \(\chi_{215}(169,\cdot)\) \(\chi_{215}(189,\cdot)\)

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: 42.42.104017955712751803355033526522081856753017553948018605377854818344593048095703125.1

Values on generators

\((87,46)\) → \((-1,e\left(\frac{1}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 215 }(9, a) \) \(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{1}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 215 }(9,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 215 }(9,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 215 }(9,·),\chi_{ 215 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 215 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2