Properties

Label 400.363
Modulus $400$
Conductor $400$
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(400, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,5,19]))
 
pari: [g,chi] = znchar(Mod(363,400))
 

Basic properties

Modulus: \(400\)
Conductor: \(400\)
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 400.bm

\(\chi_{400}(67,\cdot)\) \(\chi_{400}(123,\cdot)\) \(\chi_{400}(147,\cdot)\) \(\chi_{400}(203,\cdot)\) \(\chi_{400}(227,\cdot)\) \(\chi_{400}(283,\cdot)\) \(\chi_{400}(363,\cdot)\) \(\chi_{400}(387,\cdot)\)

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: 20.20.104857600000000000000000000000000000000000.1

Values on generators

\((351,101,177)\) → \((-1,i,e\left(\frac{19}{20}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(-i\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{7}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 400 }(363,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 400 }(363,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 400 }(363,·),\chi_{ 400 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 400 }(363,·)) \;\) at \(\; a,b = \) e.g. 1,2