Properties

Label 605.344
Modulus $605$
Conductor $605$
Order $110$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(605, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,28]))
 
pari: [g,chi] = znchar(Mod(344,605))
 

Basic properties

Modulus: \(605\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
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 605.u

\(\chi_{605}(4,\cdot)\) \(\chi_{605}(14,\cdot)\) \(\chi_{605}(49,\cdot)\) \(\chi_{605}(59,\cdot)\) \(\chi_{605}(64,\cdot)\) \(\chi_{605}(69,\cdot)\) \(\chi_{605}(104,\cdot)\) \(\chi_{605}(114,\cdot)\) \(\chi_{605}(119,\cdot)\) \(\chi_{605}(159,\cdot)\) \(\chi_{605}(169,\cdot)\) \(\chi_{605}(174,\cdot)\) \(\chi_{605}(179,\cdot)\) \(\chi_{605}(214,\cdot)\) \(\chi_{605}(224,\cdot)\) \(\chi_{605}(229,\cdot)\) \(\chi_{605}(234,\cdot)\) \(\chi_{605}(279,\cdot)\) \(\chi_{605}(284,\cdot)\) \(\chi_{605}(289,\cdot)\) \(\chi_{605}(324,\cdot)\) \(\chi_{605}(334,\cdot)\) \(\chi_{605}(339,\cdot)\) \(\chi_{605}(344,\cdot)\) \(\chi_{605}(379,\cdot)\) \(\chi_{605}(389,\cdot)\) \(\chi_{605}(394,\cdot)\) \(\chi_{605}(399,\cdot)\) \(\chi_{605}(434,\cdot)\) \(\chi_{605}(449,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((122,486)\) → \((-1,e\left(\frac{14}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{28}{55}\right)\)\(e\left(\frac{36}{55}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{29}{110}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{23}{110}\right)\)\(e\left(\frac{2}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 605 }(344,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 605 }(344,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 605 }(344,·),\chi_{ 605 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 605 }(344,·)) \;\) at \(\; a,b = \) e.g. 1,2