# Properties

 Label 245.151 Modulus $245$ Conductor $49$ Order $21$ Real no Primitive no Minimal yes Parity even

# Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(245, base_ring=CyclotomicField(42))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,10]))

pari: [g,chi] = znchar(Mod(151,245))

## Basic properties

 Modulus: $$245$$ Conductor: $$49$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$21$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{49}(4,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 245.q

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\zeta_{21})$$ Fixed field: $$\Q(\zeta_{49})^+$$

## Values on generators

$$(197,101)$$ → $$(1,e\left(\frac{5}{21}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 245 }(151,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{245}(151,\cdot)) = \sum_{r\in \Z/245\Z} \chi_{245}(151,r) e\left(\frac{2r}{245}\right) = -6.9744383661+-0.5976700401i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 245 }(151,·),\chi_{ 245 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{245}(151,\cdot),\chi_{245}(1,\cdot)) = \sum_{r\in \Z/245\Z} \chi_{245}(151,r) \chi_{245}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 245 }(151,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{245}(151,·)) = \sum_{r \in \Z/245\Z} \chi_{245}(151,r) e\left(\frac{1 r + 2 r^{-1}}{245}\right) = 9.8813408808+6.7369841403i$$