Properties

Label 500.41
Modulus $500$
Conductor $125$
Order $25$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(500, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,22]))
 
pari: [g,chi] = znchar(Mod(41,500))
 

Basic properties

Modulus: \(500\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(25\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{125}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 500.m

\(\chi_{500}(21,\cdot)\) \(\chi_{500}(41,\cdot)\) \(\chi_{500}(61,\cdot)\) \(\chi_{500}(81,\cdot)\) \(\chi_{500}(121,\cdot)\) \(\chi_{500}(141,\cdot)\) \(\chi_{500}(161,\cdot)\) \(\chi_{500}(181,\cdot)\) \(\chi_{500}(221,\cdot)\) \(\chi_{500}(241,\cdot)\) \(\chi_{500}(261,\cdot)\) \(\chi_{500}(281,\cdot)\) \(\chi_{500}(321,\cdot)\) \(\chi_{500}(341,\cdot)\) \(\chi_{500}(361,\cdot)\) \(\chi_{500}(381,\cdot)\) \(\chi_{500}(421,\cdot)\) \(\chi_{500}(441,\cdot)\) \(\chi_{500}(461,\cdot)\) \(\chi_{500}(481,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: 25.25.338813178901720135627329000271856784820556640625.1

Values on generators

\((251,377)\) → \((1,e\left(\frac{11}{25}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 500 }(41, a) \) \(1\)\(1\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{11}{25}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{6}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 500 }(41,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 500 }(41,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 500 }(41,·),\chi_{ 500 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 500 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2