Properties

Conductor 547
Order 91
Real No
Primitive Yes
Parity Even
Orbit Label 547.m

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(547)
sage: chi = H[10]
pari: [g,chi] = znchar(Mod(10,547))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 547
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 91
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 = 547.m
Orbit index = 13

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 ]

\(\chi_{547}(10,\cdot)\) \(\chi_{547}(24,\cdot)\) \(\chi_{547}(29,\cdot)\) \(\chi_{547}(35,\cdot)\) \(\chi_{547}(44,\cdot)\) \(\chi_{547}(52,\cdot)\) \(\chi_{547}(64,\cdot)\) \(\chi_{547}(84,\cdot)\) \(\chi_{547}(85,\cdot)\) \(\chi_{547}(90,\cdot)\) \(\chi_{547}(93,\cdot)\) \(\chi_{547}(100,\cdot)\) \(\chi_{547}(114,\cdot)\) \(\chi_{547}(131,\cdot)\) \(\chi_{547}(149,\cdot)\) \(\chi_{547}(154,\cdot)\) \(\chi_{547}(161,\cdot)\) \(\chi_{547}(165,\cdot)\) \(\chi_{547}(167,\cdot)\) \(\chi_{547}(179,\cdot)\) \(\chi_{547}(185,\cdot)\) \(\chi_{547}(195,\cdot)\) \(\chi_{547}(204,\cdot)\) \(\chi_{547}(205,\cdot)\) \(\chi_{547}(209,\cdot)\) \(\chi_{547}(212,\cdot)\) \(\chi_{547}(215,\cdot)\) \(\chi_{547}(216,\cdot)\) \(\chi_{547}(218,\cdot)\) \(\chi_{547}(224,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{30}{91}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{30}{91}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{60}{91}\right)\)\(e\left(\frac{1}{91}\right)\)\(e\left(\frac{69}{91}\right)\)\(e\left(\frac{87}{91}\right)\)\(e\left(\frac{90}{91}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{31}{91}\right)\)\(e\left(\frac{1}{13}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{91})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 547 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{547}(10,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(10,r) e\left(\frac{2r}{547}\right) = 6.5018508272+-22.4661063787i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 547 }(10,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{547}(10,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(10,r) \chi_{547}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 547 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{547}(10,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(10,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = 0.1355462089+0.2286553044i \)