Properties

 Label 403.144 Modulus $403$ Conductor $31$ Order $15$ 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(403, base_ring=CyclotomicField(30))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,8]))

pari: [g,chi] = znchar(Mod(144,403))

Basic properties

 Modulus: $$403$$ Conductor: $$31$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$15$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{31}(20,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 403.bi

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: $$\Q(\zeta_{15})$$ Fixed field: $$\Q(\zeta_{31})^+$$

Values on generators

$$(249,313)$$ → $$(1,e\left(\frac{4}{15}\right))$$

Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$
 value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 403 }(144,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{403}(144,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(144,r) e\left(\frac{2r}{403}\right) = -4.2138648197+3.6391404592i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 403 }(144,·),\chi_{ 403 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{403}(144,\cdot),\chi_{403}(1,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(144,r) \chi_{403}(1,1-r) = -11$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 403 }(144,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{403}(144,·)) = \sum_{r \in \Z/403\Z} \chi_{403}(144,r) e\left(\frac{1 r + 2 r^{-1}}{403}\right) = -0.1513405919+-0.4657784482i$$