Properties

Label 175.23
Modulus $175$
Conductor $175$
Order $60$
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(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,20]))
 
pari: [g,chi] = znchar(Mod(23,175))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 175.w

\(\chi_{175}(2,\cdot)\) \(\chi_{175}(23,\cdot)\) \(\chi_{175}(37,\cdot)\) \(\chi_{175}(53,\cdot)\) \(\chi_{175}(58,\cdot)\) \(\chi_{175}(67,\cdot)\) \(\chi_{175}(72,\cdot)\) \(\chi_{175}(88,\cdot)\) \(\chi_{175}(102,\cdot)\) \(\chi_{175}(123,\cdot)\) \(\chi_{175}(128,\cdot)\) \(\chi_{175}(137,\cdot)\) \(\chi_{175}(142,\cdot)\) \(\chi_{175}(158,\cdot)\) \(\chi_{175}(163,\cdot)\) \(\chi_{175}(172,\cdot)\)

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

Values on generators

\((127,101)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 175 }(23,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 175 }(23,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 175 }(23,·),\chi_{ 175 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 175 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2