Properties

Label 196.39
Modulus $196$
Conductor $196$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,34]))
 
pari: [g,chi] = znchar(Mod(39,196))
 

Basic properties

Modulus: \(196\)
Conductor: \(196\)
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 196.o

\(\chi_{196}(11,\cdot)\) \(\chi_{196}(23,\cdot)\) \(\chi_{196}(39,\cdot)\) \(\chi_{196}(51,\cdot)\) \(\chi_{196}(95,\cdot)\) \(\chi_{196}(107,\cdot)\) \(\chi_{196}(123,\cdot)\) \(\chi_{196}(135,\cdot)\) \(\chi_{196}(151,\cdot)\) \(\chi_{196}(163,\cdot)\) \(\chi_{196}(179,\cdot)\) \(\chi_{196}(191,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.74252462132603256348231837398371002884673933378885582779211491265789772693504.1

Values on generators

\((99,101)\) → \((-1,e\left(\frac{17}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 196 }(39, a) \) \(-1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{20}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 196 }(39,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 196 }(39,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 196 }(39,·),\chi_{ 196 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 196 }(39,·)) \;\) at \(\; a,b = \) e.g. 1,2