Properties

Conductor 175
Order 20
Real No
Primitive Yes
Parity Even
Orbit Label 175.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(175)
sage: chi = H[83]
pari: [g,chi] = znchar(Mod(83,175))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 175
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 20
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 = 175.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 ]

\(\chi_{175}(13,\cdot)\) \(\chi_{175}(27,\cdot)\) \(\chi_{175}(48,\cdot)\) \(\chi_{175}(62,\cdot)\) \(\chi_{175}(83,\cdot)\) \(\chi_{175}(97,\cdot)\) \(\chi_{175}(153,\cdot)\) \(\chi_{175}(167,\cdot)\)

Values on generators

\((127,101)\) → \((e\left(\frac{3}{20}\right),-1)\)

Values

-1123468911121316
\(1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{5}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{20})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 175 }(83,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{175}(83,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(83,r) e\left(\frac{2r}{175}\right) = -4.8698300895+12.2997867827i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 175 }(83,·),\chi_{ 175 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{175}(83,\cdot),\chi_{175}(1,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(83,r) \chi_{175}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 175 }(83,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{175}(83,·)) = \sum_{r \in \Z/175\Z} \chi_{175}(83,r) e\left(\frac{1 r + 2 r^{-1}}{175}\right) = -0.0 \)