Properties

Conductor 181
Order 90
Real No
Primitive Yes
Parity Even
Orbit Label 181.q

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(181)
sage: chi = H[37]
pari: [g,chi] = znchar(Mod(37,181))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 181
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 90
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 = 181.q
Orbit index = 17

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 ]

\(\chi_{181}(4,\cdot)\) \(\chi_{181}(11,\cdot)\) \(\chi_{181}(12,\cdot)\) \(\chi_{181}(20,\cdot)\) \(\chi_{181}(33,\cdot)\) \(\chi_{181}(37,\cdot)\) \(\chi_{181}(52,\cdot)\) \(\chi_{181}(55,\cdot)\) \(\chi_{181}(60,\cdot)\) \(\chi_{181}(79,\cdot)\) \(\chi_{181}(94,\cdot)\) \(\chi_{181}(100,\cdot)\) \(\chi_{181}(106,\cdot)\) \(\chi_{181}(111,\cdot)\) \(\chi_{181}(136,\cdot)\) \(\chi_{181}(137,\cdot)\) \(\chi_{181}(143,\cdot)\) \(\chi_{181}(147,\cdot)\) \(\chi_{181}(165,\cdot)\) \(\chi_{181}(166,\cdot)\) \(\chi_{181}(167,\cdot)\) \(\chi_{181}(168,\cdot)\) \(\chi_{181}(172,\cdot)\) \(\chi_{181}(178,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{13}{90}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{13}{90}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{43}{45}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{45})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 181 }(37,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{181}(37,\cdot)) = \sum_{r\in \Z/181\Z} \chi_{181}(37,r) e\left(\frac{2r}{181}\right) = 3.6607410852+12.9460022674i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 181 }(37,·),\chi_{ 181 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{181}(37,\cdot),\chi_{181}(1,\cdot)) = \sum_{r\in \Z/181\Z} \chi_{181}(37,r) \chi_{181}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 181 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{181}(37,·)) = \sum_{r \in \Z/181\Z} \chi_{181}(37,r) e\left(\frac{1 r + 2 r^{-1}}{181}\right) = -14.9017774248+-7.2680824777i \)