Properties

Label 175.38
Modulus $175$
Conductor $175$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([57,10]))
 
pari: [g,chi] = znchar(Mod(38,175))
 

Basic properties

Modulus: \(175\)
Conductor: \(175\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 175.x

\(\chi_{175}(3,\cdot)\) \(\chi_{175}(12,\cdot)\) \(\chi_{175}(17,\cdot)\) \(\chi_{175}(33,\cdot)\) \(\chi_{175}(38,\cdot)\) \(\chi_{175}(47,\cdot)\) \(\chi_{175}(52,\cdot)\) \(\chi_{175}(73,\cdot)\) \(\chi_{175}(87,\cdot)\) \(\chi_{175}(103,\cdot)\) \(\chi_{175}(108,\cdot)\) \(\chi_{175}(117,\cdot)\) \(\chi_{175}(122,\cdot)\) \(\chi_{175}(138,\cdot)\) \(\chi_{175}(152,\cdot)\) \(\chi_{175}(173,\cdot)\)

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

\((127,101)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 175 }(38, a) \) \(1\)\(1\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{2}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 175 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 175 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 175 }(38,·),\chi_{ 175 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 175 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2