Properties

Label 837.418
Modulus $837$
Conductor $837$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(837, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([40,63]))
 
pari: [g,chi] = znchar(Mod(418,837))
 

Basic properties

Modulus: \(837\)
Conductor: \(837\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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 837.ck

\(\chi_{837}(58,\cdot)\) \(\chi_{837}(85,\cdot)\) \(\chi_{837}(139,\cdot)\) \(\chi_{837}(151,\cdot)\) \(\chi_{837}(178,\cdot)\) \(\chi_{837}(184,\cdot)\) \(\chi_{837}(232,\cdot)\) \(\chi_{837}(277,\cdot)\) \(\chi_{837}(337,\cdot)\) \(\chi_{837}(364,\cdot)\) \(\chi_{837}(418,\cdot)\) \(\chi_{837}(430,\cdot)\) \(\chi_{837}(457,\cdot)\) \(\chi_{837}(463,\cdot)\) \(\chi_{837}(511,\cdot)\) \(\chi_{837}(556,\cdot)\) \(\chi_{837}(616,\cdot)\) \(\chi_{837}(643,\cdot)\) \(\chi_{837}(697,\cdot)\) \(\chi_{837}(709,\cdot)\) \(\chi_{837}(736,\cdot)\) \(\chi_{837}(742,\cdot)\) \(\chi_{837}(790,\cdot)\) \(\chi_{837}(835,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((218,406)\) → \((e\left(\frac{4}{9}\right),e\left(\frac{7}{10}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 837 }(418, a) \) \(-1\)\(1\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{23}{90}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{44}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 837 }(418,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 837 }(418,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 837 }(418,·),\chi_{ 837 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 837 }(418,·)) \;\) at \(\; a,b = \) e.g. 1,2