Properties

Label 405.2
Modulus $405$
Conductor $405$
Order $108$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(405, base_ring=CyclotomicField(108))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,27]))
 
pari: [g,chi] = znchar(Mod(2,405))
 

Basic properties

Modulus: \(405\)
Conductor: \(405\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
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 405.x

\(\chi_{405}(2,\cdot)\) \(\chi_{405}(23,\cdot)\) \(\chi_{405}(32,\cdot)\) \(\chi_{405}(38,\cdot)\) \(\chi_{405}(47,\cdot)\) \(\chi_{405}(68,\cdot)\) \(\chi_{405}(77,\cdot)\) \(\chi_{405}(83,\cdot)\) \(\chi_{405}(92,\cdot)\) \(\chi_{405}(113,\cdot)\) \(\chi_{405}(122,\cdot)\) \(\chi_{405}(128,\cdot)\) \(\chi_{405}(137,\cdot)\) \(\chi_{405}(158,\cdot)\) \(\chi_{405}(167,\cdot)\) \(\chi_{405}(173,\cdot)\) \(\chi_{405}(182,\cdot)\) \(\chi_{405}(203,\cdot)\) \(\chi_{405}(212,\cdot)\) \(\chi_{405}(218,\cdot)\) \(\chi_{405}(227,\cdot)\) \(\chi_{405}(248,\cdot)\) \(\chi_{405}(257,\cdot)\) \(\chi_{405}(263,\cdot)\) \(\chi_{405}(272,\cdot)\) \(\chi_{405}(293,\cdot)\) \(\chi_{405}(302,\cdot)\) \(\chi_{405}(308,\cdot)\) \(\chi_{405}(317,\cdot)\) \(\chi_{405}(338,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((326,82)\) → \((e\left(\frac{1}{54}\right),i)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{29}{108}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{59}{108}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{13}{54}\right)\)\(e\left(\frac{97}{108}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{7}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 405 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 405 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 405 }(2,·),\chi_{ 405 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 405 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2