Properties

Label 605.142
Modulus $605$
Conductor $605$
Order $44$
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(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,38]))
 
pari: [g,chi] = znchar(Mod(142,605))
 

Basic properties

Modulus: \(605\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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.r

\(\chi_{605}(32,\cdot)\) \(\chi_{605}(43,\cdot)\) \(\chi_{605}(87,\cdot)\) \(\chi_{605}(98,\cdot)\) \(\chi_{605}(142,\cdot)\) \(\chi_{605}(153,\cdot)\) \(\chi_{605}(197,\cdot)\) \(\chi_{605}(208,\cdot)\) \(\chi_{605}(252,\cdot)\) \(\chi_{605}(263,\cdot)\) \(\chi_{605}(307,\cdot)\) \(\chi_{605}(318,\cdot)\) \(\chi_{605}(373,\cdot)\) \(\chi_{605}(417,\cdot)\) \(\chi_{605}(428,\cdot)\) \(\chi_{605}(472,\cdot)\) \(\chi_{605}(527,\cdot)\) \(\chi_{605}(538,\cdot)\) \(\chi_{605}(582,\cdot)\) \(\chi_{605}(593,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1

Values on generators

\((122,486)\) → \((i,e\left(\frac{19}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{5}{44}\right)\)\(-i\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{15}{44}\right)\)\(-1\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 605 }(142,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 605 }(142,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 605 }(142,·),\chi_{ 605 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 605 }(142,·)) \;\) at \(\; a,b = \) e.g. 1,2