Properties

Label 920.869
Modulus $920$
Conductor $920$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(920, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11,11,12]))
 
pari: [g,chi] = znchar(Mod(869,920))
 

Basic properties

Modulus: \(920\)
Conductor: \(920\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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.bf

\(\chi_{920}(29,\cdot)\) \(\chi_{920}(269,\cdot)\) \(\chi_{920}(349,\cdot)\) \(\chi_{920}(469,\cdot)\) \(\chi_{920}(509,\cdot)\) \(\chi_{920}(629,\cdot)\) \(\chi_{920}(669,\cdot)\) \(\chi_{920}(749,\cdot)\) \(\chi_{920}(869,\cdot)\) \(\chi_{920}(909,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 22.22.719807926798998083647106533713510400000000000.1

Values on generators

\((231,461,737,281)\) → \((1,-1,-1,e\left(\frac{6}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{7}{22}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 920 }(869,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{920}(869,\cdot)) = \sum_{r\in \Z/920\Z} \chi_{920}(869,r) e\left(\frac{r}{460}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 920 }(869,·),\chi_{ 920 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{920}(869,\cdot),\chi_{920}(1,\cdot)) = \sum_{r\in \Z/920\Z} \chi_{920}(869,r) \chi_{920}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 920 }(869,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{920}(869,·)) = \sum_{r \in \Z/920\Z} \chi_{920}(869,r) e\left(\frac{1 r + 2 r^{-1}}{920}\right) = -0.0 \)