Properties

Conductor 47
Order 46
Real No
Primitive No
Parity Odd
Orbit Label 94.d

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(94)
sage: chi = H[57]
pari: [g,chi] = znchar(Mod(57,94))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 47
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 46
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 94.d
Orbit index = 4

Galois orbit

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

\(\chi_{94}(5,\cdot)\) \(\chi_{94}(11,\cdot)\) \(\chi_{94}(13,\cdot)\) \(\chi_{94}(15,\cdot)\) \(\chi_{94}(19,\cdot)\) \(\chi_{94}(23,\cdot)\) \(\chi_{94}(29,\cdot)\) \(\chi_{94}(31,\cdot)\) \(\chi_{94}(33,\cdot)\) \(\chi_{94}(35,\cdot)\) \(\chi_{94}(39,\cdot)\) \(\chi_{94}(41,\cdot)\) \(\chi_{94}(43,\cdot)\) \(\chi_{94}(45,\cdot)\) \(\chi_{94}(57,\cdot)\) \(\chi_{94}(67,\cdot)\) \(\chi_{94}(69,\cdot)\) \(\chi_{94}(73,\cdot)\) \(\chi_{94}(77,\cdot)\) \(\chi_{94}(85,\cdot)\) \(\chi_{94}(87,\cdot)\) \(\chi_{94}(91,\cdot)\)

Inducing primitive character

\(\chi_{47}(10,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{19}{46}\right)\)

Values

-113579111315171921
\(-1\)\(1\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{19}{46}\right)\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{41}{46}\right)\)\(e\left(\frac{25}{46}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{27}{46}\right)\)\(e\left(\frac{11}{23}\right)\)
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 94 }(57,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{94}(57,\cdot)) = \sum_{r\in \Z/94\Z} \chi_{94}(57,r) e\left(\frac{r}{47}\right) = 4.7953005785+-4.899499195i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 94 }(57,·),\chi_{ 94 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{94}(57,\cdot),\chi_{94}(1,\cdot)) = \sum_{r\in \Z/94\Z} \chi_{94}(57,r) \chi_{94}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 94 }(57,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{94}(57,·)) = \sum_{r \in \Z/94\Z} \chi_{94}(57,r) e\left(\frac{1 r + 2 r^{-1}}{94}\right) = -3.1661850857+0.6579408268i \)