Properties

Conductor 203
Order 84
Real No
Primitive Yes
Parity Odd
Orbit Label 203.w

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[2]
pari: [g,chi] = znchar(Mod(2,203))

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 = Odd
Orbit label = 203.w
Orbit index = 23

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 ]

\(\chi_{203}(2,\cdot)\) \(\chi_{203}(11,\cdot)\) \(\chi_{203}(18,\cdot)\) \(\chi_{203}(32,\cdot)\) \(\chi_{203}(37,\cdot)\) \(\chi_{203}(39,\cdot)\) \(\chi_{203}(44,\cdot)\) \(\chi_{203}(60,\cdot)\) \(\chi_{203}(72,\cdot)\) \(\chi_{203}(79,\cdot)\) \(\chi_{203}(95,\cdot)\) \(\chi_{203}(102,\cdot)\) \(\chi_{203}(114,\cdot)\) \(\chi_{203}(130,\cdot)\) \(\chi_{203}(135,\cdot)\) \(\chi_{203}(137,\cdot)\) \(\chi_{203}(142,\cdot)\) \(\chi_{203}(156,\cdot)\) \(\chi_{203}(163,\cdot)\) \(\chi_{203}(172,\cdot)\) \(\chi_{203}(177,\cdot)\) \(\chi_{203}(184,\cdot)\) \(\chi_{203}(193,\cdot)\) \(\chi_{203}(200,\cdot)\)

Values on generators

\((59,176)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{28}\right))\)

Values

-112345689101112
\(-1\)\(1\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{11}{12}\right)\)
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)
\( \tau_{ a }( \chi_{ 203 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{203}(2,\cdot)) = \sum_{r\in \Z/203\Z} \chi_{203}(2,r) e\left(\frac{2r}{203}\right) = 13.9819936039+2.7393164953i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 203 }(2,·),\chi_{ 203 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{203}(2,\cdot),\chi_{203}(1,\cdot)) = \sum_{r\in \Z/203\Z} \chi_{203}(2,r) \chi_{203}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 203 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{203}(2,·)) = \sum_{r \in \Z/203\Z} \chi_{203}(2,r) e\left(\frac{1 r + 2 r^{-1}}{203}\right) = 6.7876004804+5.0094771303i \)