Properties

Modulus 94
Conductor 47
Order 23
Real no
Primitive no
Minimal yes
Parity even
Orbit label 94.c

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(94)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([17]))
 
pari: [g,chi] = znchar(Mod(81,94))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 94
Conductor = 47
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 23
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 94.c
Orbit index = 3

Galois orbit

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

\(\chi_{94}(3,\cdot)\) \(\chi_{94}(7,\cdot)\) \(\chi_{94}(9,\cdot)\) \(\chi_{94}(17,\cdot)\) \(\chi_{94}(21,\cdot)\) \(\chi_{94}(25,\cdot)\) \(\chi_{94}(27,\cdot)\) \(\chi_{94}(37,\cdot)\) \(\chi_{94}(49,\cdot)\) \(\chi_{94}(51,\cdot)\) \(\chi_{94}(53,\cdot)\) \(\chi_{94}(55,\cdot)\) \(\chi_{94}(59,\cdot)\) \(\chi_{94}(61,\cdot)\) \(\chi_{94}(63,\cdot)\) \(\chi_{94}(65,\cdot)\) \(\chi_{94}(71,\cdot)\) \(\chi_{94}(75,\cdot)\) \(\chi_{94}(79,\cdot)\) \(\chi_{94}(81,\cdot)\) \(\chi_{94}(83,\cdot)\) \(\chi_{94}(89,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{17}{23}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{17}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{10}{23}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{23})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 94 }(81,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{94}(81,\cdot)) = \sum_{r\in \Z/94\Z} \chi_{94}(81,r) e\left(\frac{r}{47}\right) = 5.2207183307+-4.4434333698i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 94 }(81,·),\chi_{ 94 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{94}(81,\cdot),\chi_{94}(1,\cdot)) = \sum_{r\in \Z/94\Z} \chi_{94}(81,r) \chi_{94}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 94 }(81,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{94}(81,·)) = \sum_{r \in \Z/94\Z} \chi_{94}(81,r) e\left(\frac{1 r + 2 r^{-1}}{94}\right) = 2.2001705373+3.1169315469i \)