Properties

Label 552.5
Modulus $552$
Conductor $552$
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(552, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11,11,1]))
 
pari: [g,chi] = znchar(Mod(5,552))
 

Basic properties

Modulus: \(552\)
Conductor: \(552\)
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 552.bf

\(\chi_{552}(5,\cdot)\) \(\chi_{552}(53,\cdot)\) \(\chi_{552}(125,\cdot)\) \(\chi_{552}(149,\cdot)\) \(\chi_{552}(221,\cdot)\) \(\chi_{552}(245,\cdot)\) \(\chi_{552}(293,\cdot)\) \(\chi_{552}(341,\cdot)\) \(\chi_{552}(365,\cdot)\) \(\chi_{552}(389,\cdot)\)

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: 22.22.60063165247472201954266758470414053725437952.1

Values on generators

\((415,277,185,97)\) → \((1,-1,-1,e\left(\frac{1}{22}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{10}{11}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 552 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{552}(5,\cdot)) = \sum_{r\in \Z/552\Z} \chi_{552}(5,r) e\left(\frac{r}{276}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 552 }(5,·),\chi_{ 552 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{552}(5,\cdot),\chi_{552}(1,\cdot)) = \sum_{r\in \Z/552\Z} \chi_{552}(5,r) \chi_{552}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 552 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{552}(5,·)) = \sum_{r \in \Z/552\Z} \chi_{552}(5,r) e\left(\frac{1 r + 2 r^{-1}}{552}\right) = 0.0 \)