Properties

 Label 44.43 Modulus $44$ Conductor $44$ Order $2$ Real yes Primitive yes Minimal yes Parity even

Related objects

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

sage: H = DirichletGroup(44)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,1]))

pari: [g,chi] = znchar(Mod(43,44))

Kronecker symbol representation

sage: kronecker_character(44)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{44}{\bullet}\right)$$

Basic properties

 Modulus: $$44$$ Conductor: $$44$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 44.c

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

$$(23,13)$$ → $$(-1,-1)$$

Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$23$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$
 value at e.g. 2

Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{11})$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 44 }(43,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{44}(43,\cdot)) = \sum_{r\in \Z/44\Z} \chi_{44}(43,r) e\left(\frac{r}{22}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 44 }(43,·),\chi_{ 44 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{44}(43,\cdot),\chi_{44}(1,\cdot)) = \sum_{r\in \Z/44\Z} \chi_{44}(43,r) \chi_{44}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 44 }(43,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{44}(43,·)) = \sum_{r \in \Z/44\Z} \chi_{44}(43,r) e\left(\frac{1 r + 2 r^{-1}}{44}\right) = 0.0$$