Properties

Conductor 317
Order 79
Real No
Primitive Yes
Parity Even
Orbit Label 317.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(317)
sage: chi = H[11]
pari: [g,chi] = znchar(Mod(11,317))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 317
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 79
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 = 317.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 ]

\(\chi_{317}(10,\cdot)\) \(\chi_{317}(11,\cdot)\) \(\chi_{317}(15,\cdot)\) \(\chi_{317}(16,\cdot)\) \(\chi_{317}(23,\cdot)\) \(\chi_{317}(24,\cdot)\) \(\chi_{317}(28,\cdot)\) \(\chi_{317}(31,\cdot)\) \(\chi_{317}(34,\cdot)\) \(\chi_{317}(36,\cdot)\) \(\chi_{317}(38,\cdot)\) \(\chi_{317}(42,\cdot)\) \(\chi_{317}(43,\cdot)\) \(\chi_{317}(49,\cdot)\) \(\chi_{317}(51,\cdot)\) \(\chi_{317}(54,\cdot)\) \(\chi_{317}(57,\cdot)\) \(\chi_{317}(59,\cdot)\) \(\chi_{317}(63,\cdot)\) \(\chi_{317}(65,\cdot)\) \(\chi_{317}(67,\cdot)\) \(\chi_{317}(79,\cdot)\) \(\chi_{317}(81,\cdot)\) \(\chi_{317}(89,\cdot)\) \(\chi_{317}(100,\cdot)\) \(\chi_{317}(101,\cdot)\) \(\chi_{317}(103,\cdot)\) \(\chi_{317}(104,\cdot)\) \(\chi_{317}(110,\cdot)\) \(\chi_{317}(113,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{72}{79}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{72}{79}\right)\)\(e\left(\frac{74}{79}\right)\)\(e\left(\frac{65}{79}\right)\)\(e\left(\frac{37}{79}\right)\)\(e\left(\frac{67}{79}\right)\)\(e\left(\frac{45}{79}\right)\)\(e\left(\frac{58}{79}\right)\)\(e\left(\frac{69}{79}\right)\)\(e\left(\frac{30}{79}\right)\)\(e\left(\frac{38}{79}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{79})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 317 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{317}(11,\cdot)) = \sum_{r\in \Z/317\Z} \chi_{317}(11,r) e\left(\frac{2r}{317}\right) = -2.2586384354+-17.66064983i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 317 }(11,·),\chi_{ 317 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{317}(11,\cdot),\chi_{317}(1,\cdot)) = \sum_{r\in \Z/317\Z} \chi_{317}(11,r) \chi_{317}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 317 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{317}(11,·)) = \sum_{r \in \Z/317\Z} \chi_{317}(11,r) e\left(\frac{1 r + 2 r^{-1}}{317}\right) = -8.842249543+2.5270199137i \)