Properties

Label 245.89
Modulus $245$
Conductor $245$
Order $42$
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(245, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,23]))
 
pari: [g,chi] = znchar(Mod(89,245))
 

Basic properties

Modulus: \(245\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 245.u

\(\chi_{245}(24,\cdot)\) \(\chi_{245}(54,\cdot)\) \(\chi_{245}(59,\cdot)\) \(\chi_{245}(89,\cdot)\) \(\chi_{245}(94,\cdot)\) \(\chi_{245}(124,\cdot)\) \(\chi_{245}(159,\cdot)\) \(\chi_{245}(164,\cdot)\) \(\chi_{245}(194,\cdot)\) \(\chi_{245}(199,\cdot)\) \(\chi_{245}(229,\cdot)\) \(\chi_{245}(234,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.56353276529596271503862578540802938668269419115433656434196014026165008544921875.1

Values on generators

\((197,101)\) → \((-1,e\left(\frac{23}{42}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 245 }(89, a) \) \(-1\)\(1\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{20}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 245 }(89,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 245 }(89,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 245 }(89,·),\chi_{ 245 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 245 }(89,·)) \;\) at \(\; a,b = \) e.g. 1,2