Properties

Label 414.55
Modulus $414$
Conductor $23$
Order $11$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(414, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10]))
 
pari: [g,chi] = znchar(Mod(55,414))
 

Basic properties

Modulus: \(414\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 414.i

\(\chi_{414}(55,\cdot)\) \(\chi_{414}(73,\cdot)\) \(\chi_{414}(127,\cdot)\) \(\chi_{414}(163,\cdot)\) \(\chi_{414}(271,\cdot)\) \(\chi_{414}(289,\cdot)\) \(\chi_{414}(307,\cdot)\) \(\chi_{414}(325,\cdot)\) \(\chi_{414}(361,\cdot)\) \(\chi_{414}(397,\cdot)\)

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: \(\Q(\zeta_{23})^+\)

Values on generators

\((47,235)\) → \((1,e\left(\frac{5}{11}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 414 }(55, a) \) \(1\)\(1\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{1}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 414 }(55,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 414 }(55,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 414 }(55,·),\chi_{ 414 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 414 }(55,·)) \;\) at \(\; a,b = \) e.g. 1,2