Properties

Label 275.252
Modulus $275$
Conductor $275$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(275, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,10]))
 
pari: [g,chi] = znchar(Mod(252,275))
 

Basic properties

Modulus: \(275\)
Conductor: \(275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 275.bo

\(\chi_{275}(87,\cdot)\) \(\chi_{275}(98,\cdot)\) \(\chi_{275}(142,\cdot)\) \(\chi_{275}(153,\cdot)\) \(\chi_{275}(197,\cdot)\) \(\chi_{275}(208,\cdot)\) \(\chi_{275}(252,\cdot)\) \(\chi_{275}(263,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.75487840807181783020496368408203125.1

Values on generators

\((177,101)\) → \((e\left(\frac{1}{20}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(-i\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 275 }(252,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 275 }(252,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 275 }(252,·),\chi_{ 275 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 275 }(252,·)) \;\) at \(\; a,b = \) e.g. 1,2