Properties

Conductor 175
Order 60
Real No
Primitive Yes
Parity Even
Orbit Label 175.x

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[3]
pari: [g,chi] = znchar(Mod(3,175))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 175
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 60
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.x
Orbit index = 24

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 ]

\(\chi_{175}(3,\cdot)\) \(\chi_{175}(12,\cdot)\) \(\chi_{175}(17,\cdot)\) \(\chi_{175}(33,\cdot)\) \(\chi_{175}(38,\cdot)\) \(\chi_{175}(47,\cdot)\) \(\chi_{175}(52,\cdot)\) \(\chi_{175}(73,\cdot)\) \(\chi_{175}(87,\cdot)\) \(\chi_{175}(103,\cdot)\) \(\chi_{175}(108,\cdot)\) \(\chi_{175}(117,\cdot)\) \(\chi_{175}(122,\cdot)\) \(\chi_{175}(138,\cdot)\) \(\chi_{175}(152,\cdot)\) \(\chi_{175}(173,\cdot)\)

Values on generators

\((127,101)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{1}{6}\right))\)

Values

-1123468911121316
\(1\)\(1\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{11}{15}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{60})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 175 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{175}(3,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(3,r) e\left(\frac{2r}{175}\right) = 3.2759320344+12.8167183517i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 175 }(3,·),\chi_{ 175 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{175}(3,\cdot),\chi_{175}(1,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(3,r) \chi_{175}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 175 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{175}(3,·)) = \sum_{r \in \Z/175\Z} \chi_{175}(3,r) e\left(\frac{1 r + 2 r^{-1}}{175}\right) = 0.0 \)