Properties

Label 209.112
Modulus $209$
Conductor $209$
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(209, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,50]))
 
pari: [g,chi] = znchar(Mod(112,209))
 

Basic properties

Modulus: \(209\)
Conductor: \(209\)
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 209.v

\(\chi_{209}(6,\cdot)\) \(\chi_{209}(17,\cdot)\) \(\chi_{209}(24,\cdot)\) \(\chi_{209}(28,\cdot)\) \(\chi_{209}(35,\cdot)\) \(\chi_{209}(61,\cdot)\) \(\chi_{209}(62,\cdot)\) \(\chi_{209}(63,\cdot)\) \(\chi_{209}(73,\cdot)\) \(\chi_{209}(74,\cdot)\) \(\chi_{209}(85,\cdot)\) \(\chi_{209}(101,\cdot)\) \(\chi_{209}(112,\cdot)\) \(\chi_{209}(118,\cdot)\) \(\chi_{209}(123,\cdot)\) \(\chi_{209}(138,\cdot)\) \(\chi_{209}(139,\cdot)\) \(\chi_{209}(149,\cdot)\) \(\chi_{209}(150,\cdot)\) \(\chi_{209}(156,\cdot)\) \(\chi_{209}(161,\cdot)\) \(\chi_{209}(194,\cdot)\) \(\chi_{209}(195,\cdot)\) \(\chi_{209}(206,\cdot)\)

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

\((134,78)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{5}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 209 }(112, a) \) \(-1\)\(1\)\(e\left(\frac{59}{90}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 209 }(112,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 209 }(112,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 209 }(112,·),\chi_{ 209 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 209 }(112,·)) \;\) at \(\; a,b = \) e.g. 1,2