Properties

Conductor 197
Order 49
Real No
Primitive Yes
Parity Even
Orbit Label 197.g

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(197)
sage: chi = H[34]
pari: [g,chi] = znchar(Mod(34,197))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 197
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 49
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 = 197.g
Orbit index = 7

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 ]

\(\chi_{197}(16,\cdot)\) \(\chi_{197}(23,\cdot)\) \(\chi_{197}(24,\cdot)\) \(\chi_{197}(28,\cdot)\) \(\chi_{197}(29,\cdot)\) \(\chi_{197}(34,\cdot)\) \(\chi_{197}(37,\cdot)\) \(\chi_{197}(40,\cdot)\) \(\chi_{197}(42,\cdot)\) \(\chi_{197}(49,\cdot)\) \(\chi_{197}(51,\cdot)\) \(\chi_{197}(53,\cdot)\) \(\chi_{197}(54,\cdot)\) \(\chi_{197}(59,\cdot)\) \(\chi_{197}(60,\cdot)\) \(\chi_{197}(61,\cdot)\) \(\chi_{197}(63,\cdot)\) \(\chi_{197}(70,\cdot)\) \(\chi_{197}(76,\cdot)\) \(\chi_{197}(81,\cdot)\) \(\chi_{197}(85,\cdot)\) \(\chi_{197}(88,\cdot)\) \(\chi_{197}(90,\cdot)\) \(\chi_{197}(100,\cdot)\) \(\chi_{197}(101,\cdot)\) \(\chi_{197}(105,\cdot)\) \(\chi_{197}(132,\cdot)\) \(\chi_{197}(133,\cdot)\) \(\chi_{197}(135,\cdot)\) \(\chi_{197}(142,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{40}{49}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{40}{49}\right)\)\(e\left(\frac{37}{49}\right)\)\(e\left(\frac{31}{49}\right)\)\(e\left(\frac{32}{49}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{9}{49}\right)\)\(e\left(\frac{22}{49}\right)\)\(e\left(\frac{25}{49}\right)\)\(e\left(\frac{23}{49}\right)\)\(e\left(\frac{33}{49}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{49})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 197 }(34,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{197}(34,\cdot)) = \sum_{r\in \Z/197\Z} \chi_{197}(34,r) e\left(\frac{2r}{197}\right) = -12.1758025452+-6.9821080184i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 197 }(34,·),\chi_{ 197 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{197}(34,\cdot),\chi_{197}(1,\cdot)) = \sum_{r\in \Z/197\Z} \chi_{197}(34,r) \chi_{197}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 197 }(34,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{197}(34,·)) = \sum_{r \in \Z/197\Z} \chi_{197}(34,r) e\left(\frac{1 r + 2 r^{-1}}{197}\right) = 18.89761466+-12.3009839735i \)