Properties

Label 460.13
Modulus $460$
Conductor $115$
Order $44$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(460, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,33,28]))
 
pari: [g,chi] = znchar(Mod(13,460))
 

Basic properties

Modulus: \(460\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 460.u

\(\chi_{460}(13,\cdot)\) \(\chi_{460}(73,\cdot)\) \(\chi_{460}(77,\cdot)\) \(\chi_{460}(117,\cdot)\) \(\chi_{460}(133,\cdot)\) \(\chi_{460}(173,\cdot)\) \(\chi_{460}(177,\cdot)\) \(\chi_{460}(193,\cdot)\) \(\chi_{460}(197,\cdot)\) \(\chi_{460}(213,\cdot)\) \(\chi_{460}(233,\cdot)\) \(\chi_{460}(257,\cdot)\) \(\chi_{460}(317,\cdot)\) \(\chi_{460}(353,\cdot)\) \(\chi_{460}(357,\cdot)\) \(\chi_{460}(377,\cdot)\) \(\chi_{460}(393,\cdot)\) \(\chi_{460}(397,\cdot)\) \(\chi_{460}(417,\cdot)\) \(\chi_{460}(453,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1

Values on generators

\((231,277,281)\) → \((1,-i,e\left(\frac{7}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(-1\)\(1\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 460 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 460 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 460 }(13,·),\chi_{ 460 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 460 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2