Properties

Label 344.333
Modulus $344$
Conductor $344$
Order $14$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(344, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,7,3]))
 
pari: [g,chi] = znchar(Mod(333,344))
 

Basic properties

Modulus: \(344\)
Conductor: \(344\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 344.r

\(\chi_{344}(45,\cdot)\) \(\chi_{344}(125,\cdot)\) \(\chi_{344}(237,\cdot)\) \(\chi_{344}(285,\cdot)\) \(\chi_{344}(309,\cdot)\) \(\chi_{344}(333,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 14.0.3603461044766854684853927936.1

Values on generators

\((87,173,89)\) → \((1,-1,e\left(\frac{3}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 344 }(333, a) \) \(-1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(-1\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 344 }(333,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 344 }(333,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 344 }(333,·),\chi_{ 344 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 344 }(333,·)) \;\) at \(\; a,b = \) e.g. 1,2