Properties

 Label 50.23 Modulus $50$ Conductor $25$ Order $20$ Real no Primitive no Minimal yes Parity odd

Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(20))

M = H._module

chi = DirichletCharacter(H, M([11]))

pari: [g,chi] = znchar(Mod(23,50))

Basic properties

 Modulus: $$50$$ Conductor: $$25$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$20$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{25}(23,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 50.f

sage: chi.galois_orbit()

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Related number fields

 Field of values: $$\Q(\zeta_{20})$$ Fixed field: $$\Q(\zeta_{25})$$

Values on generators

$$27$$ → $$e\left(\frac{11}{20}\right)$$

Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$\chi_{ 50 }(23, a)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$-i$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 50 }(23,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 50 }(23,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 50 }(23,·),\chi_{ 50 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 50 }(23,·)) \;$$ at $$\; a,b =$$ e.g. 1,2