Properties

Label 345.301
Modulus $345$
Conductor $23$
Order $11$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,2]))
 
pari: [g,chi] = znchar(Mod(301,345))
 

Basic properties

Modulus: \(345\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 345.m

\(\chi_{345}(16,\cdot)\) \(\chi_{345}(31,\cdot)\) \(\chi_{345}(121,\cdot)\) \(\chi_{345}(151,\cdot)\) \(\chi_{345}(196,\cdot)\) \(\chi_{345}(211,\cdot)\) \(\chi_{345}(256,\cdot)\) \(\chi_{345}(271,\cdot)\) \(\chi_{345}(301,\cdot)\) \(\chi_{345}(331,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{23})^+\)

Values on generators

\((116,277,166)\) → \((1,1,e\left(\frac{1}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 345 }(301, a) \) \(1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{4}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 345 }(301,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 345 }(301,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 345 }(301,·),\chi_{ 345 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 345 }(301,·)) \;\) at \(\; a,b = \) e.g. 1,2