Properties

 Label 108.13 Modulus $108$ Conductor $27$ Order $9$ 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(108, base_ring=CyclotomicField(18))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(13,108))

Basic properties

 Modulus: $$108$$ Conductor: $$27$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$9$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{27}(13,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 108.i

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

$$(55,29)$$ → $$(1,e\left(\frac{4}{9}\right))$$

Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $$\Q(\zeta_{9})$$ Fixed field: $$\Q(\zeta_{27})^+$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 108 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{108}(13,\cdot)) = \sum_{r\in \Z/108\Z} \chi_{108}(13,r) e\left(\frac{r}{54}\right) = 9.5423816967+4.1161816718i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 108 }(13,·),\chi_{ 108 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{108}(13,\cdot),\chi_{108}(1,\cdot)) = \sum_{r\in \Z/108\Z} \chi_{108}(13,r) \chi_{108}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 108 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{108}(13,·)) = \sum_{r \in \Z/108\Z} \chi_{108}(13,r) e\left(\frac{1 r + 2 r^{-1}}{108}\right) = 0.0$$