Properties

Label 500.37
Modulus $500$
Conductor $125$
Order $100$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(500, base_ring=CyclotomicField(100))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,29]))
 
pari: [g,chi] = znchar(Mod(37,500))
 

Basic properties

Modulus: \(500\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{125}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 500.q

\(\chi_{500}(13,\cdot)\) \(\chi_{500}(17,\cdot)\) \(\chi_{500}(33,\cdot)\) \(\chi_{500}(37,\cdot)\) \(\chi_{500}(53,\cdot)\) \(\chi_{500}(73,\cdot)\) \(\chi_{500}(77,\cdot)\) \(\chi_{500}(97,\cdot)\) \(\chi_{500}(113,\cdot)\) \(\chi_{500}(117,\cdot)\) \(\chi_{500}(133,\cdot)\) \(\chi_{500}(137,\cdot)\) \(\chi_{500}(153,\cdot)\) \(\chi_{500}(173,\cdot)\) \(\chi_{500}(177,\cdot)\) \(\chi_{500}(197,\cdot)\) \(\chi_{500}(213,\cdot)\) \(\chi_{500}(217,\cdot)\) \(\chi_{500}(233,\cdot)\) \(\chi_{500}(237,\cdot)\) \(\chi_{500}(253,\cdot)\) \(\chi_{500}(273,\cdot)\) \(\chi_{500}(277,\cdot)\) \(\chi_{500}(297,\cdot)\) \(\chi_{500}(313,\cdot)\) \(\chi_{500}(317,\cdot)\) \(\chi_{500}(333,\cdot)\) \(\chi_{500}(337,\cdot)\) \(\chi_{500}(353,\cdot)\) \(\chi_{500}(373,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((251,377)\) → \((1,e\left(\frac{29}{100}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 500 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{3}{100}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{31}{100}\right)\)\(e\left(\frac{17}{100}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{99}{100}\right)\)\(e\left(\frac{9}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 500 }(37,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 500 }(37,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 500 }(37,·),\chi_{ 500 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 500 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2