Properties

Label 175.31
Modulus $175$
Conductor $175$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(175, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12,5]))
 
pari: [g,chi] = znchar(Mod(31,175))
 

Basic properties

Modulus: \(175\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 175.v

\(\chi_{175}(31,\cdot)\) \(\chi_{175}(61,\cdot)\) \(\chi_{175}(66,\cdot)\) \(\chi_{175}(96,\cdot)\) \(\chi_{175}(131,\cdot)\) \(\chi_{175}(136,\cdot)\) \(\chi_{175}(166,\cdot)\) \(\chi_{175}(171,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.4764432829290274543179606325793429277837276458740234375.1

Values on generators

\((127,101)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{14}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 175 }(31,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 175 }(31,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 175 }(31,·),\chi_{ 175 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 175 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2