Properties

Conductor 187
Order 80
Real No
Primitive Yes
Parity Odd
Orbit Label 187.s

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(187)
sage: chi = H[14]
pari: [g,chi] = znchar(Mod(14,187))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 187
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 80
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 = 187.s
Orbit index = 19

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 ]

\(\chi_{187}(3,\cdot)\) \(\chi_{187}(5,\cdot)\) \(\chi_{187}(14,\cdot)\) \(\chi_{187}(20,\cdot)\) \(\chi_{187}(27,\cdot)\) \(\chi_{187}(31,\cdot)\) \(\chi_{187}(37,\cdot)\) \(\chi_{187}(48,\cdot)\) \(\chi_{187}(58,\cdot)\) \(\chi_{187}(71,\cdot)\) \(\chi_{187}(75,\cdot)\) \(\chi_{187}(80,\cdot)\) \(\chi_{187}(82,\cdot)\) \(\chi_{187}(91,\cdot)\) \(\chi_{187}(92,\cdot)\) \(\chi_{187}(97,\cdot)\) \(\chi_{187}(108,\cdot)\) \(\chi_{187}(113,\cdot)\) \(\chi_{187}(114,\cdot)\) \(\chi_{187}(124,\cdot)\) \(\chi_{187}(125,\cdot)\) \(\chi_{187}(126,\cdot)\) \(\chi_{187}(130,\cdot)\) \(\chi_{187}(141,\cdot)\) \(\chi_{187}(146,\cdot)\) \(\chi_{187}(147,\cdot)\) \(\chi_{187}(148,\cdot)\) \(\chi_{187}(158,\cdot)\) \(\chi_{187}(159,\cdot)\) \(\chi_{187}(163,\cdot)\) ...

Values on generators

\((35,122)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{9}{16}\right))\)

Values

-11234567891012
\(-1\)\(1\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{77}{80}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{80}\right)\)\(e\left(\frac{51}{80}\right)\)\(e\left(\frac{63}{80}\right)\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{5}{16}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{80})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 187 }(14,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{187}(14,\cdot)) = \sum_{r\in \Z/187\Z} \chi_{187}(14,r) e\left(\frac{2r}{187}\right) = -7.0702777711+11.7051771554i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 187 }(14,·),\chi_{ 187 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{187}(14,\cdot),\chi_{187}(1,\cdot)) = \sum_{r\in \Z/187\Z} \chi_{187}(14,r) \chi_{187}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 187 }(14,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{187}(14,·)) = \sum_{r \in \Z/187\Z} \chi_{187}(14,r) e\left(\frac{1 r + 2 r^{-1}}{187}\right) = 8.1582313279+4.9993705876i \)