Properties

Label 700.87
Modulus $700$
Conductor $700$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,27,10]))
 
pari: [g,chi] = znchar(Mod(87,700))
 

Basic properties

Modulus: \(700\)
Conductor: \(700\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 700.bs

\(\chi_{700}(3,\cdot)\) \(\chi_{700}(47,\cdot)\) \(\chi_{700}(87,\cdot)\) \(\chi_{700}(103,\cdot)\) \(\chi_{700}(187,\cdot)\) \(\chi_{700}(227,\cdot)\) \(\chi_{700}(283,\cdot)\) \(\chi_{700}(327,\cdot)\) \(\chi_{700}(367,\cdot)\) \(\chi_{700}(383,\cdot)\) \(\chi_{700}(423,\cdot)\) \(\chi_{700}(467,\cdot)\) \(\chi_{700}(523,\cdot)\) \(\chi_{700}(563,\cdot)\) \(\chi_{700}(647,\cdot)\) \(\chi_{700}(663,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((351,477,101)\) → \((-1,e\left(\frac{9}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 700 }(87, a) \) \(-1\)\(1\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 700 }(87,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 700 }(87,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 700 }(87,·),\chi_{ 700 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 700 }(87,·)) \;\) at \(\; a,b = \) e.g. 1,2