Properties

Label 407.327
Modulus $407$
Conductor $407$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,5]))
 
pari: [g,chi] = znchar(Mod(327,407))
 

Basic properties

Modulus: \(407\)
Conductor: \(407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 407.w

\(\chi_{407}(6,\cdot)\) \(\chi_{407}(68,\cdot)\) \(\chi_{407}(105,\cdot)\) \(\chi_{407}(117,\cdot)\) \(\chi_{407}(216,\cdot)\) \(\chi_{407}(228,\cdot)\) \(\chi_{407}(327,\cdot)\) \(\chi_{407}(376,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.1853933678304667725967568739025947707448133.1

Values on generators

\((112,298)\) → \((e\left(\frac{3}{10}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 407 }(327, a) \) \(1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(-1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 407 }(327,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 407 }(327,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 407 }(327,·),\chi_{ 407 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 407 }(327,·)) \;\) at \(\; a,b = \) e.g. 1,2