Properties

Label 700.89
Modulus $700$
Conductor $175$
Order $30$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,25]))
 
pari: [g,chi] = znchar(Mod(89,700))
 

Basic properties

Modulus: \(700\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{175}(89,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 700.br

\(\chi_{700}(89,\cdot)\) \(\chi_{700}(129,\cdot)\) \(\chi_{700}(229,\cdot)\) \(\chi_{700}(269,\cdot)\) \(\chi_{700}(369,\cdot)\) \(\chi_{700}(409,\cdot)\) \(\chi_{700}(509,\cdot)\) \(\chi_{700}(689,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.595554103661284317897450790724178659729659557342529296875.1

Values on generators

\((351,477,101)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 700 }(89, a) \) \(-1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 700 }(89,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 700 }(89,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 700 }(89,·),\chi_{ 700 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 700 }(89,·)) \;\) at \(\; a,b = \) e.g. 1,2