Properties

Label 469.263
Modulus $469$
Conductor $469$
Order $33$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(469, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([44,48]))
 
pari: [g,chi] = znchar(Mod(263,469))
 

Basic properties

Modulus: \(469\)
Conductor: \(469\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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 469.ba

\(\chi_{469}(9,\cdot)\) \(\chi_{469}(25,\cdot)\) \(\chi_{469}(81,\cdot)\) \(\chi_{469}(107,\cdot)\) \(\chi_{469}(149,\cdot)\) \(\chi_{469}(156,\cdot)\) \(\chi_{469}(158,\cdot)\) \(\chi_{469}(193,\cdot)\) \(\chi_{469}(198,\cdot)\) \(\chi_{469}(226,\cdot)\) \(\chi_{469}(263,\cdot)\) \(\chi_{469}(277,\cdot)\) \(\chi_{469}(282,\cdot)\) \(\chi_{469}(359,\cdot)\) \(\chi_{469}(375,\cdot)\) \(\chi_{469}(394,\cdot)\) \(\chi_{469}(417,\cdot)\) \(\chi_{469}(424,\cdot)\) \(\chi_{469}(464,\cdot)\) \(\chi_{469}(466,\cdot)\)

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: 33.33.23681050358190252966666038984115482490423545779778728084912954382239302601.1

Values on generators

\((269,337)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{8}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{5}{33}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 469 }(263,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{469}(263,\cdot)) = \sum_{r\in \Z/469\Z} \chi_{469}(263,r) e\left(\frac{2r}{469}\right) = 10.7423043441+18.804331878i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 469 }(263,·),\chi_{ 469 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{469}(263,\cdot),\chi_{469}(1,\cdot)) = \sum_{r\in \Z/469\Z} \chi_{469}(263,r) \chi_{469}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 469 }(263,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{469}(263,·)) = \sum_{r \in \Z/469\Z} \chi_{469}(263,r) e\left(\frac{1 r + 2 r^{-1}}{469}\right) = 3.7790706359+0.7283561651i \)