Properties

Conductor 203
Order 84
Real No
Primitive Yes
Parity Even
Orbit Label 203.x

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(203)
sage: chi = H[3]
pari: [g,chi] = znchar(Mod(3,203))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 203
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 84
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 203.x
Orbit index = 24

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 ]

\(\chi_{203}(3,\cdot)\) \(\chi_{203}(10,\cdot)\) \(\chi_{203}(19,\cdot)\) \(\chi_{203}(26,\cdot)\) \(\chi_{203}(31,\cdot)\) \(\chi_{203}(40,\cdot)\) \(\chi_{203}(47,\cdot)\) \(\chi_{203}(61,\cdot)\) \(\chi_{203}(66,\cdot)\) \(\chi_{203}(68,\cdot)\) \(\chi_{203}(73,\cdot)\) \(\chi_{203}(89,\cdot)\) \(\chi_{203}(101,\cdot)\) \(\chi_{203}(108,\cdot)\) \(\chi_{203}(124,\cdot)\) \(\chi_{203}(131,\cdot)\) \(\chi_{203}(143,\cdot)\) \(\chi_{203}(159,\cdot)\) \(\chi_{203}(164,\cdot)\) \(\chi_{203}(166,\cdot)\) \(\chi_{203}(171,\cdot)\) \(\chi_{203}(185,\cdot)\) \(\chi_{203}(192,\cdot)\) \(\chi_{203}(201,\cdot)\)

Values on generators

\((59,176)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{28}\right))\)

Values

-112345689101112
\(1\)\(1\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{1}{12}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{84})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 203 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{203}(3,\cdot)) = \sum_{r\in \Z/203\Z} \chi_{203}(3,r) e\left(\frac{2r}{203}\right) = -13.2745106355+5.1756513974i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 203 }(3,·),\chi_{ 203 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{203}(3,\cdot),\chi_{203}(1,\cdot)) = \sum_{r\in \Z/203\Z} \chi_{203}(3,r) \chi_{203}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 203 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{203}(3,·)) = \sum_{r \in \Z/203\Z} \chi_{203}(3,r) e\left(\frac{1 r + 2 r^{-1}}{203}\right) = 0.2270623836+-6.0683699548i \)