Properties

Conductor 525
Order 60
Real No
Primitive Yes
Parity Even
Orbit Label 525.bs

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(525)
sage: chi = H[53]
pari: [g,chi] = znchar(Mod(53,525))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 525
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 = 525.bs
Orbit index = 45

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 ]

\(\chi_{525}(2,\cdot)\) \(\chi_{525}(23,\cdot)\) \(\chi_{525}(53,\cdot)\) \(\chi_{525}(128,\cdot)\) \(\chi_{525}(137,\cdot)\) \(\chi_{525}(158,\cdot)\) \(\chi_{525}(212,\cdot)\) \(\chi_{525}(233,\cdot)\) \(\chi_{525}(242,\cdot)\) \(\chi_{525}(263,\cdot)\) \(\chi_{525}(317,\cdot)\) \(\chi_{525}(338,\cdot)\) \(\chi_{525}(347,\cdot)\) \(\chi_{525}(422,\cdot)\) \(\chi_{525}(452,\cdot)\) \(\chi_{525}(473,\cdot)\)

Values on generators

\((176,127,451)\) → \((-1,e\left(\frac{7}{20}\right),e\left(\frac{2}{3}\right))\)

Values

-1124811131617192223
\(1\)\(1\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{41}{60}\right)\)
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)
\( \tau_{ a }( \chi_{ 525 }(53,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{525}(53,\cdot)) = \sum_{r\in \Z/525\Z} \chi_{525}(53,r) e\left(\frac{2r}{525}\right) = -5.2442956973+-22.3046489019i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 525 }(53,·),\chi_{ 525 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{525}(53,\cdot),\chi_{525}(1,\cdot)) = \sum_{r\in \Z/525\Z} \chi_{525}(53,r) \chi_{525}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 525 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{525}(53,·)) = \sum_{r \in \Z/525\Z} \chi_{525}(53,r) e\left(\frac{1 r + 2 r^{-1}}{525}\right) = 0.0 \)