Properties

Label 115.3
Modulus $115$
Conductor $115$
Order $44$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(115, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,32]))
 
pari: [g,chi] = znchar(Mod(3,115))
 

Basic properties

Modulus: \(115\)
Conductor: \(115\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 115.k

\(\chi_{115}(2,\cdot)\) \(\chi_{115}(3,\cdot)\) \(\chi_{115}(8,\cdot)\) \(\chi_{115}(12,\cdot)\) \(\chi_{115}(13,\cdot)\) \(\chi_{115}(18,\cdot)\) \(\chi_{115}(27,\cdot)\) \(\chi_{115}(32,\cdot)\) \(\chi_{115}(48,\cdot)\) \(\chi_{115}(52,\cdot)\) \(\chi_{115}(58,\cdot)\) \(\chi_{115}(62,\cdot)\) \(\chi_{115}(72,\cdot)\) \(\chi_{115}(73,\cdot)\) \(\chi_{115}(77,\cdot)\) \(\chi_{115}(78,\cdot)\) \(\chi_{115}(82,\cdot)\) \(\chi_{115}(87,\cdot)\) \(\chi_{115}(98,\cdot)\) \(\chi_{115}(108,\cdot)\)

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: 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1

Values on generators

\((47,51)\) → \((-i,e\left(\frac{8}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{19}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 115 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 115 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 115 }(3,·),\chi_{ 115 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 115 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2