# Properties

 Label 950.3 Modulus $950$ Conductor $475$ Order $180$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(950, base_ring=CyclotomicField(180))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([63,130]))

pari: [g,chi] = znchar(Mod(3,950))

## Basic properties

 Modulus: $$950$$ Conductor: $$475$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$180$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{475}(3,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 950.bi

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Values on generators

$$(77,401)$$ → $$(e\left(\frac{7}{20}\right),e\left(\frac{13}{18}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$21$$ $$23$$ $$27$$ $$29$$ $$1$$ $$1$$ $$e\left(\frac{151}{180}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{47}{180}\right)$$ $$e\left(\frac{139}{180}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{53}{180}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{44}{45}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{180})$ Fixed field: Number field defined by a degree 180 polynomial (not computed)

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 950 }(3,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{950}(3,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(3,r) e\left(\frac{r}{475}\right) = 18.3940742754+11.6900826153i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 950 }(3,·),\chi_{ 950 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{950}(3,\cdot),\chi_{950}(1,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(3,r) \chi_{950}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 950 }(3,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{950}(3,·)) = \sum_{r \in \Z/950\Z} \chi_{950}(3,r) e\left(\frac{1 r + 2 r^{-1}}{950}\right) = 0.0$$