Properties

 Label 920.677 Modulus $920$ Conductor $920$ Order $44$ Real no Primitive yes Minimal yes Parity even

Related objects

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

sage: H = DirichletGroup(920, base_ring=CyclotomicField(44))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,22,11,6]))

pari: [g,chi] = znchar(Mod(677,920))

Basic properties

 Modulus: $$920$$ Conductor: $$920$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$44$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no 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 920.bo

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

$$(231,461,737,281)$$ → $$(1,-1,i,e\left(\frac{3}{22}\right))$$

Values

 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$27$$ $$29$$ $$1$$ $$1$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{5}{11}\right)$$
 value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 920 }(677,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{920}(677,\cdot)) = \sum_{r\in \Z/920\Z} \chi_{920}(677,r) e\left(\frac{r}{460}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 920 }(677,·),\chi_{ 920 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{920}(677,\cdot),\chi_{920}(1,\cdot)) = \sum_{r\in \Z/920\Z} \chi_{920}(677,r) \chi_{920}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 920 }(677,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{920}(677,·)) = \sum_{r \in \Z/920\Z} \chi_{920}(677,r) e\left(\frac{1 r + 2 r^{-1}}{920}\right) = -1.1377157868+15.9073452553i$$