Properties

Label 414.5
Modulus $414$
Conductor $207$
Order $66$
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(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,3]))
 
pari: [g,chi] = znchar(Mod(5,414))
 

Basic properties

Modulus: \(414\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{207}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 414.p

\(\chi_{414}(5,\cdot)\) \(\chi_{414}(11,\cdot)\) \(\chi_{414}(65,\cdot)\) \(\chi_{414}(83,\cdot)\) \(\chi_{414}(113,\cdot)\) \(\chi_{414}(149,\cdot)\) \(\chi_{414}(155,\cdot)\) \(\chi_{414}(191,\cdot)\) \(\chi_{414}(203,\cdot)\) \(\chi_{414}(221,\cdot)\) \(\chi_{414}(227,\cdot)\) \(\chi_{414}(245,\cdot)\) \(\chi_{414}(263,\cdot)\) \(\chi_{414}(281,\cdot)\) \(\chi_{414}(293,\cdot)\) \(\chi_{414}(329,\cdot)\) \(\chi_{414}(365,\cdot)\) \(\chi_{414}(383,\cdot)\) \(\chi_{414}(389,\cdot)\) \(\chi_{414}(401,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((47,235)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{22}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 414 }(5, a) \) \(1\)\(1\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 414 }(5,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 414 }(5,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 414 }(5,·),\chi_{ 414 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 414 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2