Properties

Label 353.9
Modulus $353$
Conductor $353$
Order $176$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(353, base_ring=CyclotomicField(176))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(9,353))
 

Basic properties

Modulus: \(353\)
Conductor: \(353\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(176\)
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 353.k

\(\chi_{353}(9,\cdot)\) \(\chi_{353}(15,\cdot)\) \(\chi_{353}(18,\cdot)\) \(\chi_{353}(19,\cdot)\) \(\chi_{353}(23,\cdot)\) \(\chi_{353}(25,\cdot)\) \(\chi_{353}(30,\cdot)\) \(\chi_{353}(38,\cdot)\) \(\chi_{353}(39,\cdot)\) \(\chi_{353}(41,\cdot)\) \(\chi_{353}(43,\cdot)\) \(\chi_{353}(46,\cdot)\) \(\chi_{353}(47,\cdot)\) \(\chi_{353}(50,\cdot)\) \(\chi_{353}(65,\cdot)\) \(\chi_{353}(72,\cdot)\) \(\chi_{353}(76,\cdot)\) \(\chi_{353}(78,\cdot)\) \(\chi_{353}(82,\cdot)\) \(\chi_{353}(86,\cdot)\) \(\chi_{353}(92,\cdot)\) \(\chi_{353}(93,\cdot)\) \(\chi_{353}(94,\cdot)\) \(\chi_{353}(98,\cdot)\) \(\chi_{353}(99,\cdot)\) \(\chi_{353}(113,\cdot)\) \(\chi_{353}(120,\cdot)\) \(\chi_{353}(127,\cdot)\) \(\chi_{353}(130,\cdot)\) \(\chi_{353}(144,\cdot)\) ...

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: $\Q(\zeta_{176})$
Fixed field: Number field defined by a degree 176 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1}{176}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 353 }(9, a) \) \(1\)\(1\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{1}{176}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{133}{176}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{1}{88}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{15}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 353 }(9,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 353 }(9,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 353 }(9,·),\chi_{ 353 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 353 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2